Intersection graphs of paths in a tree

A superclass of Edge-Path-Tree graphs with few cliques

“…One of the nicest features of EPT graphs is that they generalize line graphs while retaining the property of possessing polynomially many maximal cliques, i.e., O(n 2 ), n being the order of the EPT graph, as showed by Monma and Wey in [7]. This fact implies that in an EPT graph a maximum weight clique can be found in strongly polynomial time by running a polynomial time delay algorithm that generates all maximal cliques [10,6].…”

“…This fact implies that in an EPT graph a maximum weight clique can be found in strongly polynomial time by running a polynomial time delay algorithm that generates all maximal cliques [10,6]. In this paper we extend the above mentioned results in [4,7] to a proper superclass of EPT graphs: the class of intersection graphs of {0, 1} matrices whose submatrices are not isomorphic to anyone in W = {F 7 , F …”

“…Edge-Path-Tree graphs have polynomially many cliques as proved in [4] and [7]. Therefore, the problem of finding a clique of maximum weight in these graphs is solvable in strongly polynomial time.…”

“…In the former case K is an edge-clique while in the latter one K is a claw-clique. Monma and Wey proved in [7] that the number of edgecliques is O(n). Thus, the number of maximal cliques has the order of the number of claw-cliques of G.…”

“…Corollary 1 specializes the result to EPT graphs. We stress here the fact that the extension is possible because it is possible to give vertex-free proofs of the results in [4,7], namely, proofs that do not exploit arguments involving vertices of the underlying tree realization of an EPT graph.…”