We consider a generalization of edge intersection graphs of paths in a tree. Let P be a collection of nontrivial simple paths in a tree T. We define the k-edge (k >= 1) intersection graph Gamma(k) (P), whose vertices correspond to the members of P, and two vertices are joined by an edge if the corresponding members of P share k edges in T. An undirected graph G is called a k-edge intersection graph of paths in a tree, and denoted by k-EPT, if G = Gamma(k)(P) for some P and T. It is known that the recognition and the coloring of the 1-EPT graphs are NP-complete. We extend this result and prove that the recognition and the coloring of the k-EPT graphs are NP-complete for any fixed k >= 1. We show that the problem of finding the largest clique on k-EPT graphs is polynomial, as was the case for 1-EPT graphs, and determine that there are at most O(n(3)) maximal cliques in a k-EPT graph on n vertices. We prove that the family of 1-EPT graphs is contained in, but is not equal to, the family of k-EPT graphs for any fixed k >= 2. We also investigate the hierarchical relationships between related classes of graphs, and present an infinite family of graphs that are not k-EPT graphs for every k >= 2. The edge intersection graphs are used in network applications. Scheduling undirected calls in a tree is equivalent to coloring an edge intersection graph of paths in a tree. Also assigning wavelengths to virtual connections in an optical network is equivalent to coloring an edge intersection graph of paths in a tree. (C) 2007 Published by Elsevier B.V.
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