The image-edge intersection graphs of paths in a tree Original Research Article

We consider a generalization of edge intersection graphs of paths in a tree. Let image be a collection of nontrivial simple paths in a tree image. We define the image-edge image intersection graph image, whose vertices correspond to the members of image, and two vertices are joined by an edge if the corresponding members of image share image edges in image. An undirected graph image is called a image-edge intersection graph of paths in a tree, and denoted by image-EPT, if image for some image and image. 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 image-EPT graphs are NP-complete for any fixed image. We show that the problem of finding the largest clique on image-EPT graphs is polynomial, as was the case for 1-EPT graphs, and determine that there are at most image maximal cliques in a image-EPT graph on image vertices. We prove that the family of 1-EPT graphs is contained in, but is not equal to, the family of image-EPT graphs for any fixed image. We also investigate the hierarchical relationships between related classes of graphs, and present an infinite family of graphs that are not image-EPT graphs for every image. 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.