Wednesday, July 8, 2009

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 K1,3. 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 IIIn and IVn 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.

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