Induced Subgraphs and Subtournaments
Let H be a graph and let I be the set of all graphs that do not contain H as a minor. Then H is planar if and only if there is an upper bound on the treewidth of the members of I; and there are many other similar theorems that relate properties of H to structural properties of the members of I. Let us call them "structure theorems". What about structure theorems for other containment relations instead of minor containment? For induced subgraph containment, there are virtually no such theorems known, but for subtournament containment there are some quite pretty structure theorems, mostly excluding tournaments that are almost transitive. A transitive tournament is the tournament analogue of both a stable set and a clique, so what happens if we exclude as induced subgraphs TWO graphs instead of one; one almost a stable set, and the other almost a clique? This turns out to be a better problem with some nice answers. Partly joint work with Maria Chudnovsky, Sasha Fradkin, Ilhee Kim, Gaku Liu, Sergei Norin, Bruce Reed.