Families of induced trees and their intersection graphs

Abstract

This paper is inspired in the well known characterization of chordal graphs as the intersection graphs of subtrees of a tree. We consider families of induced trees of any graph and we prove that their recognition is NP-Complete. A consequence of this fact is that the concept of clique tree of chordal graphs cannot be widely generalized. Finally, we consider the fact that every graph is the intersection graph of induced trees of a bipartite graph and we characterize some classes that arise when we impose restrictions on the host bipartite graph.