Results on set representations of graphs

A set representation of a graph is an assignment of sets to vertices such that two vertices are adjacent if and only if their assigned sets have some specified relationship. We give several results related to set representations of graphs. We show that recognising the overlap and intersection graphs of subtrees in some types of trees is NP-hard. The subtree overlap graphs (SOGs) generalise many other graph classes with set representation characterisations. The complexity of recognising SOGs is open. The complexities of recognising many subclasses of SOGs are known. We consider several subclasses of SOGs by restricting the underlying tree. For a fixed integer k ≥ 3, we consider: • the overlap graphs of subtrees in a tree with k leaves, • the overlap graphs of subtrees in trees that can be derived from a given input tree by subdivision and have at least 3 leaves, • the overlap and intersection graphs of paths in a tree with maximum degree k. We show that the recognition problems of these classes are NP-complete. We give characterisations of several subclasses of overlap graphs of subtrees in a tree in terms of filament representations. List colouring with a fixed colour bound of at least three is NP-complete, even on planar bipartite graphs. We give a polynomial-time algorithm for solving list colouring with a fixed colour bound on permutation and interval graphs, two classes with intersection representations. Finally, we describe a class of impartial combinatorial games on graphs using set representations. In these games, the players antagonistically build a set representation of a graph. We give hardness results for determining the winner of a position of these types of games in general, and give polynomial-time algorithms to solve special cases of these games on trees.