Lesson 11: From C To Shining C

Welcome to the penultimate post on strongly asteroidal graphs. We've seen category A constructions, in which a neighbor is added to one of the asteroidal vertices of S₃, and category B constructions, which modify the path between two asteroidal vertices. Today, we will look at category C constructions, which do both.

In a category C construction, a vertex c₁ is added which is adjacent to a₁, b₁, b₂ and b₃, and a category B construction is added to the path X₁. In addition, edges are added between b₁ and every vertex in the category B construction. The modified category B construction still creates an a₁-light path, because there will not be consecutive vertices adjacent to c₁. Equivalently, this can be thought of as adding a regular category B construction, and then making c₁ adjacent to every vertex except a₂ and a₃. However, it is more convenient, in terms of adding later constructions, to modify the category B construction so that all of the vertices are adjacent to b₁. Each category B construction has a corresponding category C construction, so the category C constructions can intuitively be labeled C1-C4. The constructions are in the figure below.

Because c₁ is adjacent to b₂ and b₃, removing b₁ from construction Cn will make a graph isomorphic to one with construction Bn applied. Therefore, in order to create a new minimal sAT, it is necessary for b₁ to be part of another light path.

In construction C1, this means that a category B or C construction must be applied to either a₂ or a₃, but in constructions C2 and C4, b₁ is already part of at least one light path, which includes one vertex from the category B construction (it is difficult to see, but these paths are colored red and blue in the diagram).

Construction C1 does not combine with construction A3 (a parasol is created), but we do get new minimal sATs by applying construction B1 or B2 to the path X₂, and applying constructions A1, A2, or B1 to a₃ (or X₃); these are graphs 37-41. In addition, construction C1 can be combined with constructions B1 and B3 to create graph 42; construction B4 can be used instead of B3 to create graph 44, which we'll get to in a second. In the above figure, note that several of these graphs have been rotated/reflected.

Applying C1 a second time can lead to several new graphs. If the vertices c₁ and c₂ are not adjacent to each other, then the vertices v₁ and v₂ become unnecessary. If c₁ and c₂ are adjacent, but v₁ and c₂ are not, then v₂ is still unnecessary. Finally, we can allow c₁ to be adjacent to c₂ and v₂, and c₂ to be additionally adjacent to v₁.

Adding construction B1 or B2 to any of these combinations, or B3 to either of the latter two combinations, will create a new minimal strongly asteroidal graph. Graphs 43-45 are those that occur using construction B2, graphs 46-48 use B2, and graphs 49 and 50 use B3. The category A constructions and construction B4 do not lead to new minimal graphs.

Finally, there is one way to apply construction C1 to all three asteroidal vertices. The vertices c₁, c₂, and c₃ are adjacent to each other, and only one additional vertex v is necessary; it is adjacent to b₁, b₂, and b₃. This creates graph 51. Other modifications of the adjacencies between c₁, c₂, c₃, and v₁, v₂, and v₃ do not create new minimal graphs; the most common result is a 3-sun.

Only a few minimal graphs arise from the other category C constructions. Construction C2 actually is a minimal sAT by itself, as the adjacencies between b₁ and the B2 portion of the construction create a₂- and a₃-light paths. This is graph 52. Construction C3 does not lead to any new minimal graphs; because the only added vertex in construction B3 is already adjacent to b₁, the addition of vertex c₁ is unnecessary to create an a₁-light path. Finally, construction C4, like construction C2, contains an a₂-light path. Category B constructions that modify the path X₃ do not combine with this, for the same reason they do not combine with construction B4. However, constructions A1 and A2 can be applied to a₃ to make graphs 53 and 54.

These are the final minimal sATs that arise from the graph S₃. Next time, we'll finish up by looking at the infinite families of asteroidal graphs, and seeing how to modify those to create strongly asteroidal graphs.

Zoot Suit Riotous

WARNING: Minor Glee spoliers ahead. If you haven't watched this week's episode, you may want to come back tomorrow.

It's rehearsal time at McKinley, and we acknowledge the fact that we've had Sue as co-director for a week, but now that's over, and we're back on track for Sectionals. Apparently, Sectionals have been postponed, because they should have been this past weekend. Remember that, writers? When you said that Sectionals were "in two weeks," and then you packed two weeks' worth of "show" time into the next two episodes?

That doesn't matter, though, because now there's trouble brewing at football practice. Coach Tanaka has his knickers in a twist, and has declared that the team must practice this Thursday, when the glee club normally rehearses. The issue of why the team is still practicing at all, given that the football season ended three weeks ago, is not brought up. Remember that, writers? When you said the football team was 0-6, and then ran through two months without showing any more football? High school teams don't play into December. We will note that the students are beginning to wear cold-weather clothing.

Anyway, back to the stated premise of these posts. Let's assume that the original rehearsal is on the Monday following last week's episode (so, November 16). The Thursday practice happens. Let's further assume that the Rachel-Puck scene occurs at the end of that particular practice (even though that means Rachel changed outfits post-rehearsal), which will allow Finn to return to glee club on Friday. This packs a heck of a lot of subplot into one week, but as usual, we are generous here at the Preschool. The date is Friday, November 20. We've missed Halloween, and are getting ready for Thanksgiving break.

And that's what you've missed on Glee. It sounds like there will be a few weeks without new episodes, which might allow real time to catch up with show time. Sectionals, no doubt, will be postponed without explanation again. See you in a month.

Eight Days a Week?

WARNING: Minor Glee spoilers ahead. If you haven't seen this week's episode, you may want to stop reading now. Of course, if you have normal human eyeballs, it's probably already too late. Sorry.

So... was "Keep Holding On" the sectionals song?

As you'll recall, last week's episode began with sectionals two weeks away, and ended a week (plus a few days) later. So, presumably, sectionals will be the next weekend.

This week, we begin with Sue and Schu fighting like Mom and Dad on payday, and then there's a flashback. A lot happens: Sue splits up the Glee Club, there's practices for a singing competition, Will flunks all the Cheerios (except for Quinn and Santana and Brittany, apparently), we find out that Puck is one of the Chosen People (shalom!), there's a big fight in Principal Figgins' office.

We could try to figure out dates for all of these. For example, assuming the previous episode ended on Thursday, the original meeting with Figgins has to be, at the very earliest, Monday - surely, he's not trying to "take the temperature" after a single practice? Fortunately, trying to figure out how all of these events shoehorn into a few days is totally unnecessary, thanks (as usual) to the B-plot - Terri's increasingly absurd fake pregnancy. Without going into any additional commentary on the ideas, writing, and character development surrounding this donnybrook, I'll just point out that the ultrasound appointment takes place (per Will) on Friday, just after the big shouting match with Sue. Assuming everything else did take place in the course of a single week (and, since there is no mention of sectionals, we have to assume that), we have a date for these events - it's Friday, November 13.

And this is where the show (finally) begins to lose date continuity. Will comes into Sue's office, and they apologize, and talk about the set list. We still haven't seen sectionals, so this scene takes place... when? Saturday morning? The sectionals were, per previous exposition, scheduled for this particular weekend. The gang sings us out, but it's unclear if this is the competition, or a practice, or a daydream. So, Will and Sue either make up Saturday, before sectionals, or Monday after sectionals which we didn't see (given the competition, plausible), or sectionals got postponed and no one told us, or most of the McKinley students and faculty own time-turners.

And that's what you missed on Glee. Tune in next week for football three weeks after the end of football season, and perhaps Halloween decorations just in time for Thanksgiving.

I also make a better door than a window

A while back, our regular trivia host took sick, and was missing for two weeks. One of the regulars got in touch with him, picked up the clues, and read them in his stead. In an effort to help out, I offered to write and read a round of trivia, and was allowed to do so the second week. I came up with ten questions that I thought were great. Unfortunately, it turns out I'm a better trivia-taker than trivia writer; the winning group in my round got one (1) question right, and they were the only ones to get that right. To be sure, I did a poor job. Hard questions are fine, but they're for separating the wheat from the chaff; you have to throw a couple of softballs out to keep people playing. Surely, though, you can do better than one.

1. Name the Pep Boys.
2. In Calvin and Hobbes, what is Calvin's favorite bedtime story?
3. Who was the 23rd President of the United States, serving between Grover Cleveland's non-consecutive terms?
4. What dog won the first three Frisbee World Championships, which are now named after him?
5. In the Velvet Underground song, "Waiting for the Man," how much money is in the singer's hand?
6. How many Canadian provinces border at least one Great Lake?
7. Who was the first African-American woman awarded the Nobel Prize in Literature?
8. On which island did Napoleon die, on May 5, 1821?
9. What is the name of the theme music for The Daily Show?
10. What Dutch driver holds the Indy 500 records for fastest practice lap, qualifying lap, four-lap qualifying time, and 200-lap race?

Lesson 10: All This Has Happened B4

It has been two months since I posted lesson 9. There are a few reasons for the delay, but one reason is that the final category B construction has been a thorn in my side for months. I originally thought that it gave rise to no new minimal graphs. Then, I found one new sAT, then two, then three, then went back down to two, and so on. Every time I thought that I had thoroughly examined the construction, another combination occurred to me. Over the last month, I have examined each combination of constructions that included B4, one by one, and have officially convinced myself whether each combination does or does not produce a new minimal sAT.

Construction B4: The vertex v₁ is adjacent to a₂. In addition, a vertex u₁ is added, which is adjacent to v₁ and b₃. This creates the a₁-light path a₂-v₁-u₁-b₃-a₃.

The great difficulty in construction B4 lies in the fact that it lacks symmetry. With the previous category B constructions, adding a construction to a₂ was no different than adding it to a₃. For example, two combinations produce graph 16. With B4, this is no longer the case; when we apply other constructions, it is important whether they are applied to a₂ or a₃. In addition, if we apply B4 a second time, it matters which of the asteroidal vertices is adjacent to the vertex v₂.

One important thing to note is that applying this construction, and then removing the vertex b₂, is equivalent to just applying construction B1. To create a minimal sAT, the other two constructions must be applied in such a way that either b₂ is part of every a₃-light path, or that every a₂-light path would be heavy if b₂ were removed. This means that the construction applied to a₂ cannot be a category A construction, or construction B3. In addition, applying construction B1 or B2 to the path X₃ creates either a graph in which b₂ can be removed, or one with a chordless cycle. Applying construction A3 to either a₂ or a₃ creates a copy of the parasol.

If we apply a category B construction to the path X₂, the added vertices must be adjacent to v₁, in order to avoid suns and parasols. Construction B1 will create new minimal graphs when constructions A1, A2, or B3 are applied to a3; these are graphs 33-35 below. Construction B2 results in a graph in which u₁ is removable; in fact, it creates a category C construction, which we'll get to next time.

Attempting to apply B4 a second time creates problems. If it is applied to the path X₂, we must decide whether the vertex v₂ is adajcent to a₁ or a₃. If it is adjacent to a₁, then we do not get a minimal sAT, as (depending on the adjacencies) the resulting graph will contain suns, chordless cycles, or a category C construction. Suppose, then, that v₂ is adjacent to a₃. Then applying a category A construction to a₃, or construction B3 to the path X₃ is no longer possible (for the same reasons they could not be applied to a₂ or X₂ earlier). However, allowing v₁ to be adjacent to u₂ and v₂, and v₂ to additionally be adjacent to u₁, creates the a₃-light path a₁-b₁-u₂-v₁-a₂, while also preventing the removal of b₂ or b₃. This is, therefore, a minimal sAT; this is graph 36 at left.

Applying construction B4 to path X3 does not create new strongly asteroidal graphs; in any strongly chordal graph where this is applied, either a1 or b2 will be removable.

Next time, I'll start on the category C constructions. It's all downhill from here, kids. Two more posts max, and then we can move to a new topic.

What Time Is It, Mr. Fox?

If you're like me, you spend your Wednesdays from 9-10 p.m. watching Glee (if you're not like me, you still watch it, but you have a DVR, so you watch it later and without commercials). I love the singing, love the one-liners, and I'd watch Jane Lynch read "Sinners in the Hands of an Angry God," because I think she'd read it in a sardonic tone that would just crack me up the whole time. The relationships (all of them) are AWK-ward, but I prefer to view that as intentional, like it's a whole school of Michael Scotts. One thing has been bothering me, though. A couple weeks ago, I made the following comment over at ALOTT5MA:

I'm having trouble with the tiimeline for this show. If there's enough time passing that it's noticeable that Mr. Shuster only shows up for rehearsal once a week, where are we in the semester?

(This was referring to the "Acafellas" episode. And yes, I've misspelled "Schuester.")

Fortunately, the next week, we got an answer to exactly where we were: towards the beginning of the episode, Ken Tanaka made reference to the football team being 0-6. Assuming (generously) that it took one week for the following:

1. Kurt joins the football team as its kicker (Monday)
2. Kurt and Finn convince the entire football team to take dance lessons (Tuesday)
3. The team choreographs and rehearses a dance routine to "Single Ladies" to the point of presentability (Wednesday, Thursday)
4. The football team, with the help of their newfound dancing skills, wins their game on Friday.

Then the team has run its record to 1-6. High school football in the North generally starts the last Friday in August, so, assuming there were no bye weeks, the game took place on October 9. Finally, we have pinned down a date!

(NOTE: At ALOTT5MA, I said this game would have taken place on October 16, based on the fact that football starts the week before Labor Day. However, with the late Labor Day this year, that was not the case. This jibes with my own high school's football schedule.)

Having figured out what day it is for a specific episode, we can keep track of the chronology by estimating how long each new episode takes. Again, I want to be generous with these estimates - a lot more happens in a typical high schooler's day than most of us remember. Anyway, "The Rhodes Not Taken" begins with Rachel having quit the Glee Club to join the school play (this happened at the end of the previous episode). Let's assume that this happened on Saturday, at a weekend rehearsal. Then the episode opens at Monday's reharsal. In the course of this episode:

1. Will talks to Emma about having found April's transcripts, then gets in touch with April.
2. April returns to high school and meets the gang. This, obviously, couldn't happen until Tuesday.
3. The Glee Club rehearses two songs. Along the way, April wins over the teenagers by contributing to their delinquency. This, plausibly, takes place over the course of the rest of the week. Given the number of outfit changes, this can't just be Tuesday and Wednesday. This is important, because:
4. They perform two songs at their invitational. This, presumably, takes place on Saturday - given the previous assumption, it can't only be Thursday, and the football team (and Cheerios) would have a game on Friday. April leaves after the first song. Fortunately, Rachel has gotten fed up with Sandy Ryerson (after, what, two rehearsals? Three?), and re-joins Glee for the second song.

After this episode, I estimate the date as being October 17. We're now caught up to this week's episode, which begins with Mr. Schuester saying that sectionals are in two weeks, and that the kids have been coasting since last week, when they got the notice that their only competition would be a school for the deaf and an all-girls alternative school. I had originally given the benefit of the doubt and decided this announcement took place before the invitational, despite April's absence, but Rachel's presence (and outfit - she did re-join Glee for a day in the last episode, but quit again partway through the rehearsal) indicates that this is post-invitational, and thus we begin on October 26 at the earliest. Will then pits the boys and girls against each other in a singing competition, to be started, and I quote, "one week from today." The boys sing on Tuesday (so it was actually the 27th at the beginning of the episode), the girls sing on Wednesday, a drug scandal hits the program on Thursday. Sue Sylvester is now co-coach of the Glee Club as of, at the earliest, November 5. Two things to note from this:

1. It is still sunny and warm in Lima, Ohio - as evidenced by none of the Cheerios wearing tights or long-sleeved shirts under their uniforms.
2. The Ohio high school football playoffs begin on November 6. Even if the football team has won out to finish their season, they are, at best, 4-6, and not making the playoffs. Football season is over.

And that's what you've missed on Glee. Stay tuned in the following weeks, as I keep everyone up to date on the date!

Back to Bedlam

We're back, after a two-month hiatus. I hope that I have a little more energy to keep this thing going this time.

Anyway, the last couple of graph theory lessons are ready (oh joy!), I've got a stockpile of trivia questions, and there's one new piece that needs to be somewhere other than ALOTT5MA comment threads. Stick around.