Thursday, July 9, 2009

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.

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