Lesson 9: Construction B3

We're still looking for minimal strongly asteroidal graphs, and it's time to move on to the next category B construction. Recall (from last time) that this construction will involve adding a vertex which is adjacent to all three interior vertices; we will call this vertex v₁.

Construction B3: v₁ is adjacent to both a₂ and a₃, thus creating the path a₂-v₁-a₃, which is a₁-light. Combining this construction with previous constructions is tricky. Adding constructions A1 or A2 to both a₂ and a₃ creates a copy of the bad aster. However, if we modify these constructions by making the added vertices adjacent to v₁ as well, this creates a minimal graph (if a₂ uses the modified construction, but a₃ does not, then b₂ could be removed to make a smaller strongly asteroidal graph). Similarly, adding construction B1 to both X₂ and X₃ creates a sun, unless we modify the construction by making the added vertices adjacent to v₁. This modification does not work for construction A3, however, as it creates a copy of the parasol.

If we combine constructions of type A with construction B1, we must still use the modified construction of B1; otherwise, we will create a graph in which a₁ can be removed to make a smaller strongly asteroidal graph. We may, however, use modified or unmodified type A constructions. Graphs 19-21 use only type A constructions with construction B3, and graphs 22-26 use construction B1 at least once.

Construction B3 can also be combined with construction B2. If vertices u₂, V₂, and W₂ are added to create the path a₁-b₁-u₂-v₂-w₂-b₃-a₃, then there is a parasol, unless v₂ and/or w₂ is adjacent to v₁. If only w₂ is adjacent to v₁, a chordless cycle is created, and if only v₂ is adjacent to v₁, a₁ and b₁ can be removed to make a smaller strongly asteroidal graph, regardless of the construction added to a₃ (which must be a type A construction, because B2 does not combine with B1, and applying B2 twice will produce a copy of graph 18). If both are adjacent, then removing a₁ and b₁ leaves a graph equivalent to combining B3 with B1 (with u₂ and v₁ standing in for a₁ and b₁). So, u₂, v₂, and w₂ must all be adjacent to v₁. Then, either a modified or unmodified type A construction can be added to a₃, producing graphs 27-30.

If construction B3 is applied twice, with vertex v₂ adjacent to a₁ and a₃ (as well as each vertex b_i), then v₁ and v₂ must be adjacent to avoid chordless cycles. Adding construction A1 or A2 to a₃ will produce a copy of graph 9, so we must add a type B construction to X₃ to obtain a minimal graph. Applying B3 a third time will result in a 3-sun, but both B1 and B2 will work, as long as all of the added vertices are adjacent to both v₁ and v₂. Construction B1 produces graph 31, and construction B2 produces graph 32.

Next time, we face the horror that is construction B4. For those who couldn't give a rip about all of this graph theory, there will be trivia posted tonight or tomorrow.

Lesson 8: B2, or not-B2

Welcome to the latest installment in the continuing struggle to find minimal strongly asteroidal graphs. We are creating these graphs by adding vertices to the minimal asteroidal graphs, as discussed in lesson 5. Specifically, we're adding these vertices to the graph S₃; see lesson 7 for the terminology I use in referring to the various vertices in that graph. Last time, we looked at category A constructions, which add a neighbor to the vertex a_i, in order to prevent it from being a middle vertex. In today's lesson, we'll start to look at the category B constructions, which involve adding vertices to the path X_i to make it a_i-light. The number of vertices we can add is limited somewhat; I'll spare the details, but an a_i-light path in a minimal sAT will have a maximum of three a_i-heavy vertices. This is a small result, but it is important, because it a) means that there are only a small number of constructions in this category, and b) it places an upper bound on the length of the a_i-light paths, which in turn places an upper bound on the order of the minimal graphs we can get by adding vertices to S₃, which means the number of sporadic cases is finite. Before we begin, recall that, when a construction of category B (or C) is applied, the added vertices can be adjacent to the added vertices from other constructions from category B or C. This adds a level of complexity not found in the category A constructions.

Construction B1: a single vertex u₁ is added, adjacent to b₂ and b₃. This creates an a₁-light path, and thus prevents a₁ from being a middle vertex. If this construction is applied twice, say by adding a vertex u₂ to prevent a₂ from being the middle vertex, then the new vertices cannot be adjacent: this would create a chordless cycle u₁−u₂−b₁−b₂−u₁, and adding a chord between b_i and u_i (where i is 1 or 2) would make u_i a_i-heavy, defeating the purpose of the construction. Then B1 cannot be applied to all three paths, as this will create a 3-sun. It may be applied up to twice, though, in combination with constructions A1 or A2, to produce graphs 10-14. Applying constructions A3 and B1 together produces a copy of the parasol, so this is not minimal.

Starting with construction B2, the remaining constructions all involve adding one or more vertices which are adjacent to all three interior vertices, and to other vertices (possibly one or more of the asteroidal vertices) to create a_i-light paths. The other constructions of type B involve adding only one such vertex; we will call this vertex v₁.

Construction B2: In addition to v₁, vertices u₁and w₁ are added, such that u₁ is adjacent only to b₂ and v₁, and w₁ is adjacent only to b₃ and v₁. Then a₂−b₂−u₁−v₁−w₁−b₃−a₃ is a₁-light. This construction can be applied once, in combination with constructions A1 and A2, to produce graphs 15-17. Combining B2 with A3, by adding a neighbor to a₂, does not produce a minimal graph: this combination will have a copy of the parasol, unless the vertex added to a₂ is adjacent to v₁. But, in this case, a₁, a₂, and w₁ are in a strongly asteroidal triple, so a₃ can be removed to make a smaller strongly asteroidal graph (it is, in fact, graph 35, which we will see later). Combining B2 and B1 (by adding u₂ to the path X₂) also fails to make a minimal graph; adding any of the previous constructions to a₃ or X₃ will produce a graph in which a₂ (at least) can be removed to make a smaller strongly asteroidal graph. Applying B2 a second time creates an irregular construction. If vertices u₂, v₂, and w₂ are added to create the path a₁−b₁−u₂−v₂−w₂−b₃−a₃, then there is a 3-sun unless v₁ and v₂ are adjacent. Adding construction B1 to X₃ will produce a sun. Adding a construction of type A to a₃ creates a copy of either graph 13 or 14 (in which u₁, u₂, and a₃ are a strongly asteroidal triple), unless u₂ is adjacent to v₁ and/or u₁ is adjacent to v₂; we may assume the former. In this case, though, v₂ and w₂ (and the construction added to a₃) can be removed to make a smaller strongly asteroidal graph. Specifically, this is graph 18, in which, for 1 ≤ i ≤ 3, there is a vertex u_i adjacent only to b_i and v₁. Each u_i is a_j-light for j ≠ i, and so for each i, there is an a_i-light path between the other two asteroidal vertices. Applying B2 a third time creates a graph which contains either a sun, or a copy of graph 18, so this does not give us an additional minimal graph.

That wasn't so bad, was it? Things get tougher, though, with construction B3, and I may need to spend the weekend re-checking my notes before I'm ready to show you B4.

Lesson 7: A is for Adjacent

In the previous lesson, I broke down the categories of construction which we can use to change an asteroidal triple into a strongly asteroidal triple. Today, we'll examine the category A constructions. Right now, we're only looking at S₃; let's label the vertices of the asteroidal triple as a₁, a₂, and a₃, and let b₁, b₂, and b₃ be their respective neighbors. Then the path X₁, for example, would be a₂-b₂-b₃-a₃. Each path X_i is a_i-heavy; our goal is to add vertices in such a way as to make either make this path a_i-light, or to create a new path that is a_i-light.

The category A constructions involve adding a single vertex adjacent to the vertex a_i, and possibly to additional vertices, specifically to prevent a_i from being a middle vertex. There are three constructions in this category, each of which involves adding only one vertex, which we will call v.

Before we go further: when describing each construction, in this lesson and in future lessons, I will assume that vertices are being added for the explicit purpose of preventing a₁ from being a middle vertex unless otherwise noted. Some constructions may affect other vertices or paths, and some constructions have special considerations when they are being used more than once; I will discuss these considerations when they come up.

Construction A1: v is a pendant vertex; in other words, the v is adjacent only to a₁. This means that a₁ is no longer simplicial, and thus cannot be a middle vertex. This construction can be applied to all three vertices to produce a minimal sAT; this is graph 3, shown below.

 

Construction A2: v is adjacent to both a₁ and b₁. While a₁ is still simplicial, v is not adjacent to any vertex on the path X₁, which makes this path a₁-light. This construction can be applied to all three vertices, to produce graph 4; or, it can be applied in combination with construction A1 to produce graphs 5 and 6, shown below.

Construction A3: v is adjacent to both a₁ and b₁, and to one of the other interior vertices; in the example at left, v is adjacent to b₃. Like construction A2, this makes the path X₁ a₁-light. However, the path a₁-v-b₃-a₃ is a₂-light, so this construction also prevents a₂ from being a middle vertex! Thus, only one more construction is necessary to create a minimal sAT. If we attempt to combine this with construction A1, then we create a graph which contains the bad aster. However, combining it with construction A2 creates graph 7, below. Construction A3 can be applied twice, but the second added vertex must be adjacent to the same interior vertices (in the example above, b₁ and b₃) as v  to create a minimal sAT. The second added vertex may be adjacent to v or not. If it is not, we create graph 8; if it is adjacent to v, we create graph 9.

Other adjacencies for v do not create new constructions in this category. If v is adjacent to both b₂ and b₃, then we create no new light paths, so adding v would not prevent any of the asteroidal vertices from being the middle vertex. If v is adjacent to any of the other asteroidal vertices, then this would create a new light path, but the other adjacencies required (to keep our graph strongly chordal) will actually fall under category B or C. In this vein, construction A3 might be considered a category B construction; I have placed this construction in category A because it fits the definition of adding exactly one vertex, which is adjacent to exactly one asteroidal vertex, and which prevents that asteroidal vertex from being a middle vertex.

Next time, we'll get started on the category B constructions. Because each new construction can be combined (in most cases) with previous constructions, I'll probably break this category up into two or more parts. This also means fewer breaks if I want to get through this, so maybe it will have the added benefit of speeding up the lessons.

He only dreamed of places now and of the lions on the beach.

As I continue to catch up on our trivia misses, this week's theme is Famous Characters in Literature (and other media). The Lone Ranger's nephew's horse is still not a question.

1. Which family did Hazel work for?

2. Who is the superhero alter ego of Steve Rogers?

3. What is the name of Andy Capp's wife?

4. What is the name of Oscar Madison's wife?

5. What well-known novel features a character named Aarfy Aardvark?

6. Who is Tom Sawyer's girlfriend?

7. In The Old Man and the Sea, what is the old man's name?

Lesson 6: There is NO lesson 6

We now know a few of the minimal forbidden subgraphs for the co-TT class of graphs:

1. Chordless cycles (co-TT graphs are chordal)
2. k-suns (co-TT graphs are, in fact, strongly chordal)
3. The bad aster
4. The parasol

Last time, we learned that we can find the minimal strongly asteroidal graphs by modifying the minimal asteroidal graphs to create light paths. Let a₁, a₂, and a₃ be our asteroidal triple, and X₁, X₂, and X₃ the paths connecting them, as described previously.

The only graphs we need concern ourselves with are graphs from the III_n and IV_n families, where n > 1 (graph IV₁, recall, is the 3-sun). We will begin with graph III₂, which is S₃. This graph is unique, because of its rotational symmetry; any of the three asteroidal vertices could be a middle vertex (the rest of the graphs which we have to consider have only one middle vertex, and only one heavy path). This means we must create light paths for each of these vertices. There are several constructions that we can use to prevent an asteroidal vertex a_i from being a middle vertex. I have grouped these constructions into three categories:

• Category A constructions involve adding a vertex adjacent to a_i. This vertex may or may not be adjacent to any neighbors of a_i. In fact, adding a pendant vertex to a_i is the easiest way to prevent a_i from being a middle vertex, because it is no longer simplicial.
• Category B constructions modify the a_i-heavy path X_i by adding a_i-light vertices. When category B constructions are added to more than one of the asteroidal vertices, we also gain the option of adding edges between the newly added vertices. This has to be done carefully; many graphs produced by these constructions are not strongly chordal.
• Category C constructions combine the other two constructions, adding a new neighbor to a_i which is adjacent to the a_i-heavy vertices on X_i, and then adding vertices to X_i which are not (all) adjacent to this new neighbor. These are the most difficult constructions to find, but in at least one case, applying this construction to one asteroidal vertex prevents the entire triple from having a middle vertex.

We'll start with the category A constructions next time. They are fairly simple, so we should be able to get through all of them in one go. This will not be the case for the other two categories.

Lesson 5: The Journey Begins

We're ready to start our search for minimal strongly asteroidal triples. There is a fairly important result which will make the search easier. In previous lessons, I've just been stating results and either linking to the relevant citation, or referring to my dissertation, but this result actually doesn't exist in print anywhere, so I need to prove it. As a result, this section will be much more technical than the first four; if you are not a trained mathematician, you may want to take proper safety precautions, or you may just want to skip to the end, where I sum everything up.

First, we need to introduce one more concept, that of the irreducible path. A path in a graph G is a sequence of distinct vertices x₁, x₂, ..., x_k, such that the vertex x_i is adjacent to the vertex x_i+1. If x_i and x_j are adjacent in G, and i and j differ by more than one, the path is reducible. For example, if a path has vertices x₁, x₂, ..., x₆, and x₂ is adjacent to x₅, then we can skip over vertices x₃ and x₄, making a shorter path between x₁ and x₆. A path which is not reducible is irreducible. Lekkerkerker and Boland showed that the paths in a minimal AT must be irreducible. Once more, let us consider the minimal asteroidal graphs, below.

Suppose that G is a chordal, strongly asteroidal graph, and that no proper subgraph of G is strongly asteroidal (in other words, G is minimal). As was mentioned in lesson 2, the bad aster and parasol (graphs I and II) are strongly asteroidal. These are also mimimally asteroidal, so these are our first two minimal strongly asteroidal graphs. In lesson 3, I stated that k-suns are also minimal strongly asteroidal graphs. Recall that a strongly chordal graph is one which is chordal and sun-free, so, as we search for the other minimal sATs, we may assume that G is strongly chordal, not just chordal.

Let a₁, a₂, and a₃ be a strongly asteroidal triple in G. Note that, even if G is minimal, it may contain more than one sAT. We may have to choose a specific triple in a few sentences, but, for now, let's just pick the first one we find.

Let W₁, W₂, and W₃ be paths such that W_i is a_i-light, and contains no neighbor of a_i. Each of these paths contains an irreducible path, if it is not itself irreducible; let W′₁, W′₂, and W′₃ be these irreducible paths.

Because G is minimal, it follows that
W₁ ∪W₂ ∪W₃ ∪ N(a₁) ∪ N(a₂) ∪ N(a₃) = G

(where N(v) refers to the open neighborhood of v - the set of all vertices adjacent to v.)
If we consider W′₁ ∪ W′₂ ∪ W′₃, each a_i must be simplicial: if a_i has two neighbors b_i and c_i (it will not have more, as each W′_i is irreducible), then the three paths contain a cycle which contains b_i−a_i−c_i as consecutive vertices, where b_i ≠ c_i and a_i has no other neighbors. Then (because G is chordal) b_i is adjacent to c_i. Further, because a_i is necessarily a_j-light for j ≠ i, there is no need to place a vertex of W_j between a_i and either b_i or c_i. Therefore, we may assume that each a_i is simplicial in W₁ ∪ W₂ ∪ W₃. Finally (and here's where we may have to pick a different triple), we may assume that each a_i has at least one neighbor adjacent to vertices on W_i: if not, then let b be any neighbor of a_i. Then, because a_i is not adjacent to any member of W_i, b is also in a sAT with the vertices a_j , j ≠ i. If b has a neighbor adjacent to a member of W_i, then replace a_i with b; if it does not, then G is not minimal.

Now, let G′ be a minimal subgraph such that a₁, a₂, and a₃ are asteroidal. In other words, remove as many vertices from G as we can while keeping those three vertices as an AT (not a sAT). Then G' is a minimal asteroidal triple - in other words, G' is one of the graphs above!

To see this, note that G′ is a subgraph of W′₁ ∪ W′₂ ∪ W′₃, and that each of the vertices a_i is necessarily simplicial in G′, and are the only simplicial vertices in G′. Let X₁, X₂, and X₃ be irreducible paths of G′ such that X_i contains no neighbor of a_i (while connecting the other two vertices in the triple). If G′ contains another minimal asteroidal triple, then at least one of a₁, a₂, or a₃ is not in this triple. Because this vertex is simplicial, it would not be part of an irreducible path; thus, it can be removed to make a smaller asteroidal graph. Without loss of generality, suppose that a₁ can be removed. Then G′−a₁ has an asteroidal triple of simplicial points (this is a result of Lekkerkerker and Boland, in the paper linked above). The only simplicial points are a₂, a₃, and (possibly) the neighbors of a₁, so a₁ has some neighbor b which is in an asteroidal triple with a₂ and a₃. Then b is the next vertex on one of the irreducible paths X₂ or X₃; without loss of generality, suppose b is on X₂. Consider W₂ and W₃; if b is not the second vertex (after a₁) of these paths, then it is adjacent to that vertex. Because b is not adjacent to every neighbor of a₂ or a₃, we can replace a₁ with b as the first vertex of W₂ and W₃, and these paths will still be a₂-light and a₃-light, respectively. If any a₂−a₃ path is b-light, then b, a₂, and a₃ are strongly asteroidal in G−a₁, which contradicts the minimality of G. So, X₁ must be b-heavy. Specifically, X₁ must have two consecutive vertices u and v (with v closer to a₂) adjacent to the next vertex on X₂; call this vertex c. None of these vertices can be a₃, so we may assume that u is not a neighbor of a₃. Neither u nor c is adjacent to a₁, so the portion of X₁ from a₂ to a, plus the portion of X₂ from c to a₃, contains no neighbor of a₁. Thus, we can eliminate v, and any vertices of X₁ between v and a₃, and obtain a proper subgraph of G′ in which a₁, a₂, and a₃ are still asteroidal, a contradiction. Phew!

This result implies that we can obtain the minimal strongly asteroidal graphs by simply appending vertices to the minimal asteroidal graphs in such a way as to make the asteroidal triples strongly asteroidal. We can break down our search for the strongly asteroidal graphs by looking at the minimal asteroidal graphs they contain. We'll get started on the first few cases next time.

Lesson 4: Stuck In the Middle

In lesson 2, I pointed out that, while two of Lekkerkerker and Boland's minimal asteroidal graphs also contain sATs, the two infinite families (except for the 3-sun) do not. This is because the lower path contains consecutive vertices adjacent to the neighbor(s) of the "top" vertex (and this is the only path between the "left" and "right" vertices not passing through these neighbors).

In their original paper, Monma, Reed, and Trotter showed that a graph is a co-TT graph if and only if each vertex x can be assigned two real numbers, a_x and b_x, such that vertices x and y are adjacent if and only if a_x ≤ b_y and a_y ≤ b_x. If, for vertex x, a_x ≤ b_x, then x is known as a bounded vertex; if a_x ≥ b_x, then x is unbounded. A graph which contains only bounded vertices is equivalent to an interval graph (Jamison, "Cross Comparison Graphs," no citation currently exists), so the unbounded vertices are the key to the difference between interval graphs and co-TT graphs.

At left is a graph with the a and b values assigned to each vertex. The "top" vertex is the only unbounded vertex in this assignment, though any or all of the pendant vertices could be made unbounded.

If a graph does not contain an AT, it is an interval graph, so we are most interested in what happens with asteroidal triples. For a vertex v in an AT to be given an unbounded assignment, it must satisfy the requirement that every path between the other two vertices in the AT that does not contain a neighbor of v must be v-heavy. A vertex is v-heavy if it is adjacent to every neighbor of v; a path is v-heavy if it contains two consecutive heavy vertices. A vertex or path which is not v-heavy is v-light.

A vertex in an AT which satisfies these two properties is a middle vertex. In families III and IV, the "top" vertex is a middle vertex; to obtain sATs, we must add vertices to these graphs to create a light path between the other two asteroidal vertices. How do we do that? Stay tuned.

Like Lazarus (which, by the way, was a question last week)

We've found a new place for trivia, so the weekly "trivia misses" post is back!

The new format is old-style pub quiz: rounds of ten questions each, so we've got a bunch of misses saved up. I'll dole them out slowly, just in case we miss a week. There's also a round or two of "celebrity photos," in which one must identify the celebrity face, which the quizmaster has cut from a magazine and pasted onto a cartoon body. These, sadly, do not translate well to the Internets, or we'd have a lot more questions. Here, anyway, are some of our recent misses. This week's theme: animal names. Everyone's favorite animal name question was not asked.

1. On Bonanza, what was the name of Ben Cartwright's horse?

2. What is the name of the Yale Bulldog? (as in, the dog himself has a name, what is it)

3. What was the name of the goldfish in the Disney version of Pinocchio?

4. Who is Porky Pig's girlfriend?

5. What was the name of the cross-eyed lion on Daktari?

Lesson 3: Here Comes The Sun

I've mentioned chordal graphs a few times already. Today, we'll look at strongly chordal graphs. If a graph has a cycle with k vertices in it (where k > 3), and we label the vertices 1, 2, ..., k, consecutively (so 1 is adjacent to 2 and to k), then if vertices i and j are adjacent, and i and j differ by more than one (and aren't 1 and k), then the edge between i and j is a chord. If the difference between i and j is odd (even), then this edge is an odd (even) chord. A strongly chordal graph is a graph in which every cycle of at least four vertices has a chord (which makes it chordal), and every cycle of at least six vertices has an odd chord.

Strongly chordal graphs are equivalently defined as graphs which have a strong elimination ordering (the definition begins at the bottom of page 76 in the linked text). Farber defined the strong elimination ordering in a 1983 paper, and also showed that strongly chordal graphs are precisely the graphs which are chordal and sun-free. A k-sun is an even cycle, with at 2k vertices (k > 2), in which the vertices can be numbered such that the even vertices have no additional edges (beyond the two you'd expect), and the odd vertices form a clique. If the odd vertices form a chordal graph, but not a complete graph, then the cycle is an incomplete sun. Some graph theorists prefer to call the latter graph a sun, and the former a complete sun. Every incomplete sun contains a k-sun, so the difference is semantic. Below are S₃ (again) and S₄.

As was mentioned yesterday, S₃ contains a sAT. So does S₄, and, in fact, every sun. This is not a surprise; Monma, Reed, and Trotter showed in their original paper that co-TT graphs are strongly chordal (by finding a strong elimination ordering). In my dissertation, I showed that the k-suns are minimal forbidden subgraphs; removing any vertex from a k-sun results in a co-TT graph. If we label the vertices of a k-sun as described above, any three of the even vertices form a sAT; simply moving along the cycle from one vertex to the next gives a light path, as opposed to a heavy path, which I mentioned yesterday, and will talk about tomorrow.

Lesson 2: More Triples Than Johnny Damon

Yesterday, I introduced (again) the strongly asteroidal triple, and began laying the groundwork for the search for minimal strongly asteroidal graphs. First, though, I should mention where the term "strongly asteroidal" comes from. The sAT is related to two weaker conditions:

Given three distinct vertices a, b, and c in a graph, if there are paths between each pair of vertices which contain no neighbor of the third, then a, b, and c are an asteroidal triple (AT). If the paths between each pair of vertices do not contain two consecutive neighbors of the third, then a, b, and c are a astral triple. Every strongly asteroidal triple is also asteroidal, and every asteroidal triple is also astral. The astral and asteroidal triples are known in graph theory for being forbidden subgraphs of other graph classes. Interval graphs were shown by Lekkerkerker and Boland to be precisely the graphs which are chordal and AT-free (in other words, containing no asteroidal triple). Jackowski showed that unit interval graphs (interval graphs in which each interval is the same length) are precisely the graphs which are astral triple-free.

The three graphs to the left are the minimal chordal graphs containing an astral triple (each cycle of length at least 4 also contains astral triples). The graph on the far left is known as the claw, and is also denoted K_1,3. The graph on the right, the 3-sun, is denoted S₃; the middle graph is its complement, denoted S₃.

These four graphs are the minimal chordal graphs containing an asteroidal triple. The first graph, which one of my advisors refers to as the "bad aster," is sometimes denoted T₂. The second graph has no name that I know of; I occasionally refer to it as the "parasol" (the "umbrella" is something else), or simply as Graph II. The remaining two graphs represent infinite families; the dashed line represents a path with as many vertices as desired (even 0, in which case the vertices on either end are the same vertex), all of which are adjacent to the neighbor(s) of the "top" vertex. The smallest member of family III, graph III₂, is S₃. In family IV, the entire path on the "bottom" may just have one vertex, so the smallest member of this family, graph IV₁, is the 3-sun.

A quick check of these graphs reveals that the asteroidal triples in the bad aster and the parasol are also strongly asteroidal, as is the 3-sun. However, for n ≥ 2, graphs III_n and IV_n do not contain a sAT, as every path between the "right" and "left" asteroidal vertices either passes through a neighbor of the "top" vertex, or contains consecutive vertices adjacent to all of those neighbors (I refer to such a path as heavy). More on heavy paths and middle vertices later; tomorrow, I'll talk about suns and strongly chordal graphs.

Deep Impact

So, on to the promised new content. I'll put up some trivia this week, too, but for now, I'm beginning a little lecture on graph theory. For those unfamiliar with the topic, one can find some basic definitions on Wikipedia, or Mathworld, or in this nice little primer.

In my final post of 2008, I defined a strongly asteroidal graph as one which contains three distinct vertices a, b, and c, such that there is a path between each pair of these vertices which

1. contains no neighbor of the third vertex, and which
2. does not contain two consecutive vertices adjacent to every neighbor of the third vertex.

The vertices a, b, and c are then referred to as a strongly asteroidal triple. This will often be referred to as a sAT in the later posts on this topic.

All very well and good, but why am I telling you this? In my dissertation, I showed that the class of graphs known as co-TT graphs (which are the complements of threshold tolerance graphs, defined in this paper) are precisely the graphs which are chordal and sAT-free. This is a major step towards finding the minimal forbidden subgraphs for this class. It is, therefore, of some interest to find the minimal graphs (a graph minimally satisfies a condition if the removal of any vertex leaves a graph which no longer satisfies that condition) which are both chordal and contain a sAT. There are three infinite families of minimal (chordal) strongly asteroidal graphs, plus at least 54 sporadic cases. Finding these cases is tedious, and does not lend itself well towards publishing in a journal. I want this work out there, though, so that there is actually a record of what the minimal forbidden subgraphs for this class are. So, for the next few weeks, I'll be working through the minimal strongly asteroidal graphs. There will be pictures, there will be words, there may be mistakes (which is one reason I hope people start following this). If you stick with me, you might learn something, or you might teach me something, or you might be really bored.

A rough outline of where we're going:

1. Intro (that's this post)

2. Astral and asteroidal triples

3. Suns and strongly chordal graphs

4. Constructing strongly asteroidal graphs from asteroidal graphs

a. Type A constructions

b. Type B constructions

c. Type C constructions

Like School In Summer

I hope everyone is enjoying their Summer Break. Here at the Preschool, we're undergoing the long-threatened redesign, which involves more than just a layout change. There will still be trivia (I've been saving some up), but I've got something planned for after Independence Day weekend. If you're not a mathematician, you probably won't find the first series interesting. Or maybe you will; it's graph theory, which means there might be pictures. Happy weekend!

Don't Be Cruel, Just Beat It

The L.A. Times reports that Michael Jackson has died. Someone will come up with a conspiracy theory fairly quickly; this is my attempt to beat everyone else to it. The following statement is entirely fabricated.

As everyone knows, Elvis faked his death back in 1977, and has since been living a comfortable retirement on a South Pacific island. What you may not know, however, is that Presley actually passed away last month at the age of 74. In an auction attended only by members of the Pentavirate (Jacko joined shortly after the release of Bad, filling the void left by Colonel Sanders a decade earlier), Jackson purchased Presley's island, and has faked his own death to retire to same.

The members of Duke Phillips Preschool wishes to send their condolences and prayers to Jackson's family.

I'm gonna be a supermodel

Everyone loved Suzanne Vega so much, now I give you:

"Ordinary Girl"

Go ahead, try to get it out of your head.

Slouching Towards Oblivion

I am contemplating (again) a site redesign, but in the meantime, I give you the misses from last week's trivia night. I've said this before, but this was almost certainly my last appearance at this particular trivia night. This time, it's not the general crappiness of the questions, but the fact that I no longer have any reason to be in Clemson on a Thursday.

Last week was the restaurant's 10th anniversary, so all questions are about events that took place in 1999. We had first place wrapped up, until we declined to guess which day of the month the restaurant opened. It was, apparently, an easy thing to guess, as we did not place, despite a healthy lead.

1. One of the most devastating earthquakes of the 20th Century struck in August of 1999, killing over 15,000 people in what country?

2. What popular Nickelodeon cartoon debuted in May 1999, with an episode entitled "Help Wanted/Reef Blower/Tea at the Treedome?"

3. Halftime: Name the three countries which joined NATO on March 12, 1999.

4. Which two oil companies merged in November of 1999, creating what was at the time the largest company on the planet?

5. Which Royal couple was featured on a postage stamp prior to their June 1999 wedding?

Guaranteeing me 25 extra minutes in Purgatory

Internet Meme Of The Week!

Inspired by a comment thread over at ALOTT5MA, I hereby stick in your head Suzanne Vega's "Tom's Diner."

Original version of the song is here.

The version you probably know is here.

(edited to remove the wacky suggestion that you reference Vega as often as possible for the rest of the week)

Return of the Son of Trivia Night

Bad questions abound! Bonus points for explaining why they're bad questions.

1. Which Spice Girl was first to have a baby?

2. What famous five-word phrase, spoken in Apollo 13 by Ed Harris, was never actually uttered by Gene Kranz?

3. Which Yankee hurler threw the 15th perfect game in Major League Baseball history?

Final:

4. The Sonny Bono Act protects an artist's copyrighted work for how many years after their death?

The Jailhouse Queen

As a combination late-birthday/graduation present, the w and I got backstage passes to last week's Death Cab For Cutie concert at Davidson College. Here follows my largely stream-of-consciousness review. Full disclosure at this point: I've been going to Death Cab shows since 1999, when they played places like the Go! Rehearsal Room in Carrboro and T.T. The Bear's in Cambridge. I've only met (drummer) Jason McGerr once, but I do know the other members of the band.

Parking was... a challenge. The parking lots looked a lot bigger on the map than they actually were. By the time we found a place across campus and made it back to the arena, it was 8:30, and we managed to hear a song and a half from Ra Ra Riot. I did like the tune I heard, and I definitely dig a band with a violinist and cellist on stage.

When their set ended, we figured out how to get backstage (people are very friendly when you're wearing a VIP sticker), and Ben Gibbard happened to be hanging out just outside the dressing room. We talked about the Carolinas, and whether or not Western Washington would be willing to give Ben an honorary doctorate someday. I got to introduce the w to Chris Walla, and I said hi to Nick Harmer, but I didn't get to talk to them. Back in the days when the band was still schlepping their own instruments, they used to come out into the crowd to watch the openers, and we'd have conversations, but that hasn't happened in about five years. I kind of miss that.

We went up to the floor to see Cold War Kids. They're the sort of band that I feel like I've heard before, but probably haven't - I had the same experience the first time I heard a Soul Coughing album. Not that CWK sound like Soul Coughing, they're more like White Stripes + honky-tonk piano + extra guitar. Anyway, they started out great, but they kind of bogged down after a while; the middle of their set had what I considered an unreasonable amount of falsetto, plus one thing with a constant vibrating bassline that just felt like something Beck decided not to put on "Midnite Vultures." Still, they finished strong.

It occurs to me that I'm not as enamored with being on the main floor as I used to be. When DCFC played Clemson (my first "date" with the w), we had floor passes, which was nice, but I just don't feel like standing up for that long anymore. Plus, people on the floor are irritating, like they feel like the communal atmosphere gives them license to push social boundaries. Our favorites tonight were the three girls who spent most of the set holding shouted conversations and texting their friends and started their own mini-mosh pit and generally ignored the band, then got annoyed with us when we decided to move in front of them because they couldn't see anymore.

After CWK, we decide to find seats in the upper area. We make the mistake of going to the student section, which are bleachers as opposed to actual seats, but at least we're sitting. They start with "The Employment Pages." A song from Facts (five albums and several eps ago)? Here's a question for your favorite bands: do they prefer tours when they're supporting an album, and thus playing mostly new stuff, or the tweener tours, when they have two or three new songs they're toying with, and can otherwise play anything they want? Personally, I prefer the non-album tours.

Davidson's arena, by the way, has a fantastic sound system. The w told me she had earplugs, but she meant *she* had earplugs, not several pairs of them. But the sound hear manages to be clear without being ear-splittingly loud. It isn't even distorted up in the bleachers.

The lighting for this tour has some good and bad things. I particularly liked the LED "tears" during at least one song. But the bits with a couple of spotlights expanding and contracting in patterns were, at best, distracting.

The set is mostly tight, except Ben gets lost in the middle of "Movie Script Ending." They have a chuckle on stage, and then just jump right back into the song.

More annoying people: some guy starts shouting "You're awesome! Woo!" during the quiet part of "I Will Follow You Into the Dark". Several times. You, sir, are a douchebag. Sit down.

They move on to "I Will Possess Your Heart." This is the new "Transatlantacism," with the long buildup and kickass bassline. I have to think they wrote this song thinking what a blast it would be to play live. Too bad this means they probably won't play "Transatlantacism." They do, however, play "Fake Frowns," which I haven't heard live in a long, long time. Awesome.

The set ends, they leave the stage, everybody keeps cheering, of course there's going to be an encore. Still, that doesn't stop the two coeds behind us from completely losing their nut when the band comes back out on stage. I can only assume that this is their first show, ever. They start the encore with "What Sarah Said," and there's even more screaming. Seriously? The most depressing song in, let's face it, a catalog of depressing songs, and folks are bouncing off the walls. Then there's a song off the ep - I really want the ep now - then "Soul Meets Body," and then "Transatlantacism!" This is my favorite Death Cab live song, mainly because I'm waiting for the time when Nick simply snaps the neck off his bass. The song ends, we wander back across campus to the car, and find our way to the nearest Waffle House.

I could also play one on TV

When I went off to college, in a far-away city, the folks and I decided it would be a good idea if I signed up for some frequent-flyer programs. You know how the applications have the boxes to check for your salutation (Mr., Mrs., Ms., etc.)? Well, I checked "Dr." At 20, I found this to be hilarious. Whenever I have called customer service, I have been referred to as "Doctor." Again, this was much more amusing to me at 24 then at 34.

The official ceremony is not until May, but, as of yesterday, I am pleased to report that I am no longer living a lie. Now, if only the advanced degree actually came with a job.

Another vague post

Let's say you have a plan. The goal of this plan is not relevant; maybe it's to colonize Mars, maybe it's to ensure the Detroit Lions win the Super Bowl. Part of this plan, though, involves poisoning a number of people. Not a large number of people, just a few. This is done covertly, and even if it is discovered that these people have been poisoned, the cause will be attributed to asbestos or unwashed spinach or something.

Would you make a list of these people? Would you leave this list in a place where anyone, at all, could find it? Would there be any possible reason to have a paper trail of this particular deed?

No, YOU are!

When I was in college, I was in a fraternity. This is apropos of very little, except that I'm about to tell a story involving my little brother, which is a frat thing; I do not actually have a younger brother.

Anyway, one night, my little brother, aided by some goading from his housemates, made a phone call which... well, let's just say he probably still regrets it. Before we go on, I will point out that he was not much of a drinker, and in particular was not drinking that evening, in my recollection. I don't know if that makes it better, or worse. But I digress. The next day, I talked with him about how to make amends. Our house President talked with him. His pledge trainer --- whoops! We don't have those, uh, his "New Member Liaison" --- talked with him.  At that point, he trudged off to some afternoon activity. However, that's when the fallout really began. As brothers began getting home from class, they stopped by our room.

"Is Skippy (note: not his real name, nor, despite my efforts, his nickname) around?"

"No."

"Well, when you see him, tell him I want to talk to him."

This conversation was repeated over a dozen times. Most of the brothers wanted to talk to him. Several of their girlfriends wanted to talk to him. Some guy I had never met before wanted to talk to him. Finally, it occurred to me to start getting annoyed, and I asked one person why he wanted to talk to Skippy.

Person:"You heard what happened?"

Me:"Yes. I talked to him about it. [President] talked to him. [N.M.L.] talked to him. Why do you need to talk to him?"

P:"Well, a lot of people are angry."

M:"Yes, three people have mentioned that to him. He feels bad. Why do you need to talk to him?"

P:"To let him know that he's made a lot of people angry."

M:"Oh. I never would have thought to do that."

In the end, what really bothered me was not the staggering number of people who thought that they were going to offer some unique perspective on the situation. It was the readiness with which they were willing to dispense their bile, without really knowing what had happened (none of the people present at the incident, it should be noted, came by to talk).

I think about this incident whenver I see things like this note, in which one participant of Top Chef gets, well, a bit catty about another contestant. Certainly, she probably shouldn't have said what she said. Probably, she regrets whatever she said, whether or not the quote is accurate. But the followup is inexcusable; the 200+ comments read like a contest to see who can come up with the meanest, most vitriolic response. Very few of the commenters claim to know Casey Thompson, or to have any knowledge of the actual goings-on during the final episode, and those who do are shouted down.

I stopped participating in most forms of Internet discussion a while ago, because I just had no stomach for the folks who came in determined to ruin someone else's day. There are forums where people participate in reasoned, respectful discussion, to be sure. But this kind of anger just seems like the norm, and makes me wonder why it's tolerated.

Oscars Trivia!

Something to do while you're waiting for the Barbara Walters special to end:

1. When was the first televised Academy Awards program?
2. What was the first Best Picture winner?
3. Who has the most Academy Awards for directing?
4. How many times have all four of the major acting Oscars gone to non-Americans?
5. What individual has won the most Academy Awards?
6. Name the only two men to have won three Academy Awards for acting.
7. Name all three films to have won the "Big Five" awards (Best Picture, Best Adapted Screenplay, Best Actor, Best Actress, Best Director).

Final: Who holds the record for most Academy Award nominations without a win?

(This may or may not be a hint: The answer to this question does not depend on tonight's results.)

Working Title

I'm not sure what to do about trivia this week. It was "movie trivia" night, which we figured was in honor of the Academy Awards, but instead of asking questions about, you know, Oscar-winning films, we got a bunch of crap along the lines of "what was the name of the dog in Next Friday?"

Maybe I'll post some of our misses later, but more likely I'll come up with some better movie trivia before Oscar Night. Stay tuned.

He'll tell you what he wants, what he really really wants

This week's trivia ended badly, as we decided to guess on the final question, and missed, costing us a tie for second. We never get the tiebreaker questions, anyway. Irritatingly, a fairly clear error on one question is what put us in the tie in the first place. Our "misses:" 

1. How many toes does an ostrich have?

2. Which President is on record as having given the longest inaugural speech?

3. Who was the first male tennis player to win Grand Slam titles on all three playing surfaces?

4. Which diva's "Reach" was the theme song of the 1996 Summer Olympics?

5. What Nobel Peace Prize winner referred to meeting the Spice Girls as one of the greatest moments of his life?

Useless Information

Because I have no other way of generating content: Trivia returns!

This may only be for a week: it happened that I was in town on a Thursday, I decided to go, and after two hours of recycled and otherwise easy questions, I'm in no particular mood to go back. The current writing crew is, apparently, completely out of ideas, as there were rounds (plural) of questions that I've heard before. In fact, I'm pretty sure they repeated the final question (ordering Kevin Costner movies). Yes, I've been doing this thing for two-plus years, and you're bound to hear the same question once in a while, but it isn't like it's impossible to come up with a good trivia question, or at least a new one. Anyway, here's my miss, and a couple others I made up.

1. In what country was roulette invented?

2. During the Olympic Parade of Nations, which country almost always enters the venue first? Also, what is the exception to this rule?

3. We all know Johnny Depp's first film was A Nightmare on Elm Street, and Kevin Bacon was in the original Friday the 13th. But name the mega-stars whose early work includes these 80's horror films:

a) Leprechaun

b) Cutting Class

c) Return to Horror High

4. Who was the last member of the Whig party to be elected President of the United States?

Happy Almost-new Year

Well, every year I get started on the posts later and later.

All in all, 2009 looks to be a banner year for the Preschool. I'm living in a new town (well, I'm sort of living in two towns right now), with a new person, and, with any luck, will soon have a new job. "Luck," of course, being the operative word. All this excitement is not without downsides, though. The new town meant giving up the old job. Chances are, the new job will be in yet another new town, which means the w might also get to experience the joy of a job search in what might be charitably termed a "tight" job market. And, most importantly, I'm rarely around for trivia night anymore, thus dealing a devastating blow to the major source of content around here. And, sadly, no replacement seems in the offing; the w did find an ad for trivia night at a local pub, but when we went there for a drink on Saturday, we found the place shuttered and "For Lease." Not a good sign.

There are content possibilities: No way am I detailing the academic job search, but I could always bring back the cooking segment. In the meantime, I'll fill space by looking at how I did on last year's resolutions.

1. Get the PhD. Well, oops. I'll have it in May, though, I swear.
2. Brown-bag it more. I was helped tremendously by the 12:20 class I taught in the fall, which kept me from even eating lunch most days.
3. Less red meat. Less butter. Less sugar. Definitely less red meat - it helps a lot that the w doesn't like it much - but I do like me some cookies.
4. Way less pizza. I was doing better on this the first half of the year. I can, however, count on one hand the times I hit Todaro's AYCE buffet.
5. Stop using sponges to clean the dishes, as they are, apparently, nothing more than slightly soapy bacteria farms. We do have scrub brushes in the kitchen now, and I try to use them for most things. Still, the sponge is terribly convenient for cleaning knives. I'm still working on this.
6. Reach 400 on the 1001 Movies You Must See list. I was well ahead of the pace through the summer, but then the getting-married and writing-the-dissertation bits got in the way, as did the seemingly interminable sequence of 1950s Japanese dramas in my Netflix queue. I ended the year having seen 104 movies on the list, giving me a total of 370. Of course, by now, the list has likely been updated, so who knows where I am.
7. Ride the bus more. I definitely pulled this one off, even if it occasionally involved driving halfway to campus, parking at the grocery store, and riding the bus from there. I usually went shopping on my way home.

A Post Few of My Readers Are Likely to Understand

A graph G is strongly asteroidal if it contains three vertices a, b, c such that there is a path between any two of these vertices that

1. Does not contain the third vertex, nor any neighbor of the third vertex.
2. Does not contain two consecutive vertices that are adjacent to every neighbor of the third vertex.

This? Is my job.

Fun Fact: There's only one Stephentown on Earth

Okay, so last week really was my last trivia night. Maybe. I'll have to spend some time in Clemson next semester, and the w is convinced I'll be arranging things so that I'm there on Thursday nights. We'll see. For now, savor the misses from last week's victory.

1. In what southeastern state might you find the towns of Frog Jump, Only, and Sweetlips?
2. In what southwestern state would you find the town of Ding Dong?
3. What freshwater mollusk produces red or pink pearls?
4. Which family of trees includes both the largest and fastest-growing living thing on earth?
5. What is the more common name of herpes zoster?

Where have all the tan M&M's gone?

Because I ran across this while I was cleaning out my bookbag: trivia returns!

1. What original Lifesavers flavor was nearly removed from the roll in 1999, after Nabisco announced that it was "not Y2K compliant?"

2. How many times did Marshal Mathers, a.k.a. Eminem, flunk the ninth grade?

3. What country's first nuclear weapons research facility is now a tourist attraction known as "Nuclear City?"

4. What film, starring Tupac Shakur and Janet Jackson, was banned from the Cineplex Odeon Universal City in Los Angeles, out of fears that it might cause trouble in the trendy City Walk complex nearby?

5. What part of the Britney Spears doll's anatomy do you push to get her to sing?

Tomorrow will likely be my last trivia night in this town. Goodness knows what direction this blog will take afterwards. Will I find a new trivia night in Winston-Salem? Start writing my own? Reprint all the Ken Jennings questions I miss? Stay tuned.

Been a Long, Lonely, Lonely, Lonely, Lonely, Lonely Time

Okay, so, Fall Break has stretched into an extended hiatus, as wedding and honeymoon time have been supplanted by research time and job-application time and write-the-dissertation-already time. I have a brief moment of blogging time tonight, though, so I return with a question:

Curriculum Vita or Curriculum Vitae? Or does it matter? My understanding was that "Vitae" was plural, but still acceptable usage (similar to "criteria").

Se Habla Ingles!

In case you haven't noticed, the Preschool has been on Fall Break. There was no trivia last week, and there probably won't be any this week. I am pleased to report, however, that on October 18, the f officially became the w. In the next few days, there will be a recap of our honeymoon in Mexico, or at least a recipe for fried bananas.

I thought it was Jimmy Durante!

Trivia night got off to a rocky start, when I arrived to find that the rest of my team was absent. A couple of early misses had me thinking about leaving at halftime, but, ninety minutes and a couple of lucky guesses later, an improbable victory gave a much-needed infusion of cash to the Preschool's coffers.

The difficulty level of trivia has been a lot higher lately, but I feel like the ones I missed were actually pretty easy. Judge for yourself.

1. What singer was the subject of the Hello Gorgeous Museum in San Francisco, which gave out wigs and fake noses to visitors wishing to impersonate her?
2. What multi-faceted character did comic Don Rickles voice in the movie Toy Story?
3. What medical procedure, applied to humans, did the World Health Organization decry as "against human integrity and morality?"
4. What clothing line, a spin-off of Phat Farm, is designed for urban women?
5. Who was the first athlete to be named Sports Illustrated's Sportsman of the Year twice?
6. Final: What Steven Spielberg movie did Senator Tom Coburn bash NBC for airing, citing its "full-frontal nudity, violence, and profanity?"

Buckaroo Banzai is not the answer to any question

We missed a lot of questions this week, and many of them were good questions. Unfortunately, I misplaced the question list somewhere between the bar and the car, and I can only remember one of our misses. Such is the risk of bar trivia. Happily, it turns out I did back up my extra trivia document after all, so I'll add a couple extras of my own.

1. What Grammy-winning singer made tabloid headlines in 1996 when it was reported (erroneously) that he had asked his wife for a divorce via fax?

2. Who is the only driver to complete the Indianapolis 500 and the Coca-Cola 600 on the same day?

3. What dubious achievement did the Philadelphia Phillies accomplish on July 15, 2008, against the St. Louis Cardinals?

4. Which amendment to the Constitution provided for separate election of the President and Vice-President?

You Enjoy Themselves

Phish is touring.

You Shall Never Be Hungry Again!

Fantastic week at trivia this week. Not because we won - we didn't - but because trivia was actually hard. At least, we thought it was hard. We have a veritable smorgasbord of missed questions for you, my trivia-hungry readers. Dig in!

1. Who won ESPN's award for "Wide Receiver of the '80s?"
2. What was the first animated film to be nominated for Best Picture?
3. What was the title of Alexandra Ripley's much-anticipated sequel to Margaret Mitchell's Gone With the Wind?
4. What superstar lamely told Madonna that her show was "neat," during the documentary Truth or Dare?
5. Which hip-hop heroes collaborated with heavy metal band Anthrax to remake the hit "Bring the Noise?"
6. What fast food chain has begun selling Brazil nut chocolate chip cookies, in an effort to benefit rainforest communities?
7. What self-described "Architect of Rock and Roll" helped cut the ribbon at the opening ceremonies of the Rock and Roll Hall of Fame?

Note - we actually missed eight questions, but I am unable to verify the answer to the eighth, or even to find a reference to the incident the question is about. So, out it goes.

Where'd you go?

Three reasons for last week's lack of posts:

1. My laptop died, thus depriving me of my usual entry to the Internets.
2. We missed zero questions at trivia. I could have taken some from the stash I save for just such an occasion, but those were on my laptop, in a folder I don't usually back up - see reason #1.
3. Heroes premiere + no TiVo = no more watching OTH on Mondays.

I hereby promise to miss trivia questions this week.

Trivia: Still More Fun Than Studying

This week did not go nearly as well for us as last week. But, as usual, our loss is your gain, because more misses for us is more questions for you. I know there's a subset of my tiny tiny readership that will have no trouble with the halftime question...

1. What does the "W" stand for on a WAP phone?
2. Who was the first African-American to grace the cover of the Sports Illustrated Swimsuit Issue?
3. Halftime: Name the last seven films to win the Academy Award for Best Picture.
4. In what year was the original Playstation released?
5. Which Peanuts character was dressed in clothes designed by over three hundred fashion designers in a 1990 retrospective at the Louvre?
Final:
6. Which English-speaking nation stopped minting one- and two-cent coins in 1991, when the cost to produce them exceeded their value?

Is "Oedipal" the word I'm looking for here?

Okay, so going into episode two, let's recap where everyone is:
Lucas is married... to Brooke! No wait, it's Peyton! Um, it's Lindsey? No, it's a dream sequence, and he's in the airport, waiting for... Peyton, which is exactly what we expected all along. They're in Vegas, getting married, or maybe they're not, I'm not entirely sure, because I spent a lot of the episode watching on mute because I was on the phone with the f.
Dan is not dead after he was hit by a car... but he's in the hospital. No, wait, he's in a room made up to look like a hospital room, and his fake nurse is - wait for it - Fran, the Nutjob Nanny! This is her elaborate revenge scheme to get Jamie back. I get the feeling that this week, she will make Dan re-write his last book so that her favorite character doesn't die.
Nathan's back hurts.
Haley is doing something.
Skills is sleeping with Nathan's mom. Yeah.
Mouth is leaving for Omaha, and wants Millie to go with him. Millie decides to go, but changes her mind, because Brooke is going to war with her mom for control of the company. How exciting! Brooke, ever the martyr, tells Millie to go, and the show's two "unattractive" people ride off into the sunset. Even more exciting, Brooke is attacked by a masked man as she locks up the store for the night. Is it an Oedipal plot? Find out in two minutes!

You Can Do What You Want to Do

I'm disappointed in the difficulty level of the trivia lately. It's one thing when a team gets a ridiculously high score for the week; any given set of questions could be right in one team's wheelhouse. But, and I know I've harped on this before, when the top three teams are within two points of a perfect score, that's just no leeway for teams to get anything outside their areas of expertise.
Anyway, on to a story of our only miss: Each round, we're given the categories for each question in advance. This, in theory, helps us decide how many points to bid on the first two questions. To pass the time between questions, our team usually tries to come up with possible questions for the later categories. On occasion, we have been able to come up with the exact question - last week, I correctly predicted that the Just Shoot Me question would ask for the name of the magazine (sadly, this clairvoyance did not help us correctly answer the question). On rarer occasions, the category is so obvious that we have come up with the correct answer without a question. Tonight, the category was "backup dancers," and we figured the answer simply had to be K-Fed. When it wasn't, we were so stuck that we were unable to come up with anything. So that's question one, question two is the only other question we really had to think about, and then there's two I made up. Please to enjoy.

1. What future superstar was a backup dancer in Janet Jackson's video for "That's the Way Love Goes?"
2. What rap star suavely blamed his teammates after he was cut from a tryout for the Toronto Raptors?
3. What is the oldest continuously used auto raceway in the U.S.?
4. In Calvin and Hobbes, what are Calvin's parents' names?

Or, you could watch the RNC

We're just over an hour from the premiere of One Tree Hill, and when we last saw the gang, Lucas was sitting in the airport with two tickets to Vegas, proposing to his phone. We also saw Brooke, Peyton, and Lindsey on their respective phones. Who is on the receiving end of the proposal? Well, if it's Brooke, that would just be weird. As for Lindsey, she just got done telling Lucas that, when she told him she was seeing someone else - thus causing him to flip out and assault a high-schooler, possibly costing him his coaching job - she was lying. "Hey, sorry I f%$@ed with your mind and screwed up your life" doesn't strike me as the sort of thing that would lead to a proposal. Seriously, if it's Lindsey, this is going to be a short season of recaps.
Also to be answered: Will Mouth really quit his job and move to Omaha? Will Millie the Absurdly Gorgeous Frumpy Girl go with him? Is Dan finally dead? Will everything this season be Nathan's fault? Will Skills have a meaningful plot line? Tune in tonight and see!

It's all right, 'cause there's no Saved By The Bell!

This week was "90's sitcom trivia," and, despite having watched a great deal of television in that decade, I learned just how much I don't know about 90's sitcoms. For example, I did not know that JAG, Melrose Place, Dawson's Creek, and Beverly Hills, 90210 were sitcoms. I'm on the record as being against "theme nights," and the inability of the organizers to stay inside their announced themes is one reason. Having to type "On [show], [question]" over and over is another.

1. On The Fresh Prince of Bel-Air, what is Will's father's occupation?
2. And where did Will and Carlton attend college?
3. On Full House, what was Jesse's original last name?
4. On Everybody Loves Raymond, what unusual quirk does Raymond's brother Robert have when he eats?
5. What is the name of the magazine where the cast of Just Shoot Me works?
Final:
6. On Seinfeld, what food caused Jerry to end his fourteen-year non-vomit streak?

Like the Ken Jennings Trivia Email, only easier. And not in email form.

The trivia team got the term off to a solid start, cruising to an easy victory at an easy trivia night. We only missed two questions (well, more, because we could only name six of the eight events that Michael Phelps won last week), so I picked out two extras.

1. Whose last words were reported as "I love you, Jesus," by a nun, and, "I can't breathe," by a doctor?
2. Which newspaper first received the tip that led to the man who admitted to killing Bill Cosby's son?
3. Which U.S. presidential candidate said, "The Internet is a great way to get on the 'Net?"
Final:
4. What international event prompted USA Today's first weekend edition, in 1996?

60 is the new 55

I'd like to take this opportunity to assure my mother that I did, indeed, send her a card for her birthday, and to thank her for making this all possible.

Back To School

School is back in session, and with it, new trivia. We only missed three this week, but missing three questions was enough to keep us out of the money. Especially galling was the fact that we probably would have gotten the first question if we had heard it correctly - it wasn't a question about the Pharaohs.

1. In what ancient Egyptian city did the Pharos reside?
2. What is the average lifespan of a human red blood cell?
3. Which automaker purchased Land Rover (at the time, the last major British-owned car company) in 1994?

Anything You Can Do, I Can Do Better

Including get in a fight at a basketball game, apparently.
The truly brilliant part of this? Bill Laimbeer and Rick Mahorn as the voices of reason.

Summertime, and the Livin's Lazy

We're on Summer Break again at the Preschool, so I'm spacing things out. We didn't go to trivia last week, and probably won't go next week, so this will have to tide you over for a while.

1. What product's advertising first used the catchphrase "Good to the Last Drop," in 1907?
2. Who was the first Playboy centerfold, in 1953?
3. What is the name of Smokey the Bear's mate?
4. What organization's motto is "Blood and Fire?"
5. How many days did the 1968 march from Selma to Montgomery take?
6. Which country is the world's largest producer of soybeans?
7. What breed of dog was Winston Churchill's favorite dog, Rufus?
Final:
8. How many grooves are on the edge of a quarter?

Link it up!

Today is Wine Library TV's 499th episode, and host Gary Vay...Ner...Chuk promises that this episode will "rock your face off." Tomorrow will be a "live" (they're taping it live, but it's happening today) broadcast from their 500th episode celebration in San Francisco.
Vaynerchuk's almost-daily podcast discusses wines from around the globe. Most days there's a theme - only 100% Petit Verdot, wines from Oklahoma (go find that one in the archives, it's worth it), even a two-buck-Chuck tasting. He tastes a few wines every show, and gives tasting notes and a score for each wine, but he stresses that these things - the score, in particular - are just his feelings about a wine, and that you need to trust your own palate. That, I think, is the reason for the popularity of the show - Vaynerchuk brings in varietals and regions that most casual wine drinkers have never heard of, and urges you to expand your tasting experience, because there's a lot of good wine out there, at all different price points, and it's a shame to always spend $12 on Yellow Tail just because you already know what it tastes like*.
Anyway, check out today's show, either at the link above, or by grabbing the podcast at the iTunes Store.

A Not-Necessarily-Musical Question

Pretend, for the moment, that you are a musical artist. Someone compares your work - at this point, you are free to imagine whether it is the song you just sang, your latest album, your entire ouvre - with someone else's work. Are you upset that someone has just suggested your work is derivative, or do you welcome the comparison to other artists? Discuss.

I'm thinking of a number between 20 and 50. What is it?

There actually was trivia this week, and it was actually good, except for what is now disturbing trend of "guess a number" final questions. It's one thing to ask about some obscure fact - that's trivia, after all - but I really feel like there should be some way to come up with an educated guess when you don't know the answer to a question, and in this case, I just don't see how one could reason out the answer.
Anywho:

1. What is Scarlett O'Hara's actual first name?
2. In which state was the Confederate flag first designed and flown?
3. What legendary baseball player was born in Mobile, Alabama, in 1934?
4. What university counts the greatest number of U.S. Presidents among its alumni?
5. How many laps in the Daytona 500?
Final:
6. How many restrooms are in the White House?

It doesn't pay to be a Gentile at trivia night

Thursday trivia was no fun - we didn't miss anything that made an interesting trivia question, and the final was another "guess a number and we'll tell you if you're close" question - so you get Saturday trivia, which I continue to insist is better anyway. There is a significant probability that there will not be trivia this week, in which case, I'll just make something up.

1. In which EU nation is the primary National Language spoken fluently by less than 10% of the population?
2. What is the only stadium to host consecutive Super Bowls?
3. Street lingo: What is "Adam's Ale?"
4. What are the capitals of a) Somalia b) Australia?
5. A Jewish boy has his Bar Mitzvah at 13. At what age does a girl have her Bat Mitzvah?
6. In the iconic photo, how many soldiers are raising the American flag at Iwo Jima?
7. What African nation is home to the militia known as the Janjaweed?

Back in the saddle again

The nasty throat/chest virus seems to be gone, so I was back at trivia this week. Things looked bad when we started blowing our five-pointers in the middle rounds, but then we wound up winning the thing at the end, by being the only team to correctly guess how many tons of jellybeans the White House purchased during the Reagan Administration. Seriously, I can't stand that type of question. They said to get within two, so we figured it couldn't be very high, but unless you happen to have just read 101 Completely Random Facts about Ronald Reagan, there's no further reasoning that can get you any closer to the correct answer; the question might just as well have been, "I'm thinking of a number between 2 and 30, what is it?"

The rest of our misses:
1. What television personality claimed that he was fired as a local weatherman after joking that a storm had hail stones "the size of canned hams?"
2. What is the name of Felix the Cat's girlfriend?
3. What was the name of Melissa Etheridge's first album after announcing her homosexuality?
4. Which country had a record turnout of 315 million voters during its 1991 elections?
5. Who wore a suit, instead of his standard military fatigues, to celebrate the 50th anniversary of the founding of the United Nations?

Why there was no trivia this week

I have spent the last week waylaid by a virus, which started in my throat, moved to my chest once I started taking prednisone for the throat, then moved back into my throat once I ran out of prednisone and started taking cough syrup. Today, I feel well enough to move around, and can even talk for several minutes without straining my voice. But, obviously, I had to skip both Thursday and Saturday trivia this week. Maybe I'll get to alternate content (I do have a nice onion spaghetti recipe I could share), maybe not.

Why I prefer Saturday Trivia

As I have mentioned before, the hard-and-fast rule of Thursday trivia is, "Don't miss your five-point questions." So when we managed to blow a five-pointer on the very first question (mis-counting the number of molars in the human mouth - incidentally, I only have 25 teeth in my mouth, which is somewhat disconcerting), and then missed another five-pointer in the second round, we knew this would not be our night. We were dead last at halftime, made a big comeback in the second half, but would not have placed even if we had gotten the final question right. That's okay, because, as is too often the case, the trivia writers asked an ambiguous question with a difficult-to-verify answer, which a brief Google search would indicate they got wrong.

1. What is the more common name for the chemical compound epinephrine?
2. Which planet takes 84 Earth years to complete its orbit?
3. Which golfer bid a tearful farewell to fans after playing his last U.S. Open in 1994?
4. To whom did Herman Melville dedicate Moby Dick?
5. For which Biblical figure did the Sun and Moon stand still? (sorry if this one is worded badly)
6. Which country is home to the oldest active brewery (which has been running since 1040 A.D.)?
Final:
7. What is the average length of an African elephant's trunk?

Good luck.

Back from hiatus!

Like most schools, DPP has been taking a short break before starting summer classes. I was planning to go elsewhere with content today, but, due to a minor incident with the food processor, it's a little hard to type. As a result, you're just getting this week's trivia. Again. I hate being so one-note, but it turns out being a full-time researcher is significantly more time-consuming than taking classes ever was.

1. What is the hardest (on the Mohs scale, I assume) part of the human body?
2. Which successful pop duo was originally known as "Caesar and Cleo?"
3. Before Tokyo, which city was the capital of Japan?
4. Who was the oldest actress on The Golden Girls: Bea Arthur, Estelle Getty, Rue McLanahan, or Betty White?
Final:
5. How many different shapes are currently in the Barnum's Animals (TM) cracker menagerie?

Theme Night!

I have to make up questions on the fly this week, so here goes:

1. What song's copyright is held by the estates of Mildred J. Smith and Patty Hill Smith until 2030?
2. What is the street address of NBC's headquarters in Manhattan?
3. What 90s television show was the debut of twins Tia and Tamera Mowry?
4. What card game is usually begun by removing one queen from a 52-card deck, then dealing all the remaining cards to the players?

Trick Question! Tupac faked his death!

All of the calculus classes have their exams on Thursday night of exam week, so we missed Thursday trivia. However, rather than leave my readers in the lurch, I made the journey to Saturday trivia. Things did not look good early, when I could not convince anyone that a Reuben is made with pastrami (possibly because it is actually made with corned beef), and I was not up to snuff on my celebrity death locations. I placed by bidding zero points on the final question, and letting the other teams fall back to me.

1. Which actor played the Dark Knight for the only time in Batman Forever?
2. Which of the four gospels (Matthew, Mark, Luke, John) was written by a former tax collector?
3. Where were the following people killed?
a) Tupac Shakur
b) Martin Luther King, Jr.
4. What 1990 dance craze gave The Digital Underground its biggest hit?
5. Place the following movies in order of their 2005 DVD rentals, from most to least:
a) Meet The Fockers b) Hitch c) National Treasure d) The Notebook

My mom loves your shows, Shonda!

It's a good 24-hour period for me. We won trivia last night, I'll be receiving a Major University Award this afternoon, and when I snarked on Grey's Anatomy, Shonda Rhimes snarked right back. Plus, it's the last day of classes, which is always good.
Speaking of the last day of classes, non-trivia content has been light (to say the least), but I expect that to change now that my days are a little more free. In particular, OTH recaps should be back soon.

1. What country produces 2/3 of the world's vanilla?
2. What mollusk shares its name with the first waterproof watch?
3. Which Spaniard won two Masters titles in the 1980's?
4. What nation has the world's second-highest level of gold production, behind South Africa?
Final:
5. Which novel contains the longest sentence in literature?

What if it was a hiphopopotamus?

Thursday was Sports Trivia, and I didn't bother to write down our misses; if I remember any, I'll add them. Instead, you get Saturday's trivia, which has consistently been of higher quality than the Thursday night version this semester (and easier to win, as fewer teams show up). The downside, of course, is that it requires me to spend my Saturday night in this particular bar, which means the f has to be visiting me (as opposed to the other way around) and feel like going out for trivia (which usually is the case, as she is a fan of trivia, but sometimes we have Flight of the Conchords on DVD and don't feel like going out).
The f actually had to leave town early yesterday, so I was planning to go stag, but I ran into some friends, and our victory enabled us to pay most of our crippling bar tab. The misses:

1. On Scrubs, where did J.D. and Turk attend college?
2. Name the Major League pitchers with the five highest career strikeout totals.
3. Kevin Spacey has won Academy Awards for his work in which two films?
4. What is added to a martini to make it a gimlet?
5. Place the following animals in order of average life expectancy in the wild, from longest to shortest:
Kangaroo, hippopotamus, deer, rat.

No, I don't understand why the baseball questions were twelve years old

Last night was a great night for everyone as far as trivia is concerned. It was great for us, because we won. It's great for you, dear readers, because we still managed to miss a solid selection of questions. As usual, I cannot vouch for the veracity of the answers given to us, but we believed them.

1) What Arab nation has the highest percentage of Christian citizens?
2) Who hit a line-drive single in his first at-bat of the 1996 season, becoming the first Red Sox pitcher to get a base hit in 24 years?
3) Who was the winningest manager in Major League Baseball history winningest active manager in baseball when he retired in 1996?
4) What recording artist was the top-selling artist of the 1970's, according to Billboard?
5) Of flyweight, bantamweight, or featherweight, which is the heaviest weight class?
Final:
6) What mythical beast has the head of a man, the body of a lion, and the tail and feet of a dragon?

I took "Potpourri" for $100, and then my head started to spin

Last night's final question was: What is the maximum number of "answers" that can be read during a game of Jeopardy!?
I think a better question is this: What is the maximum number of possible "questions" (i.e., responses) during a game of Jeopardy!?


Apologies for any confusion caused by the awkward punctuation.

I doubt they actually said "adieu"

When trivia night pays three places, fourth is the most annoying place to be. The new question-writers are still finding their legs as far as difficulty, so it was another night where close to a dozen teams finished with over 70 points (the maximum is 78). The final question, about the number of answers on Jeopardy!, became more difficult when the announcer tried to clarify the difference between an "answer" and a "question." This led to great confusion, but apparently not enough confusion to slip us into the money.

1. Which country bid adieu to the United Kingdom in 1921?
2. How many days can an ant survive underwater?