Tolerance intersection graphs on binary trees with constant tolerance
Discrete Mathematics
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Graphs and Hypergraphs
Equivalences and the complete hierarchy of intersection graphs of paths in a tree
Discrete Applied Mathematics
Intersection models of weakly chordal graphs
Discrete Applied Mathematics
Edge-Intersection Graphs of k-Bend Paths in Grids
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
Path Partitions, Cycle Covers and Integer Decomposition
Graph Theory, Computational Intelligence and Thought
Towards a comprehensive theory of conflict-tolerance graphs
Discrete Applied Mathematics
Recognizing vertex intersection graphs of paths on bounded degree trees
Discrete Applied Mathematics
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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 @C"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=@C"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.