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?