Let P be a collection of nontrivial simple paths on a host tree T . The edge intersection graph of P , denoted by EPT ( P ) , has vertex set that corresponds to the members of P , and two vertices are joined by an edge if and only if the corresponding members of P share at least one common edge in T . An undirected graph G is called an edge intersection graph of paths in a tree if G = EPT ( P ) for some P and T . The EPT graphs are useful in network applications. Scheduling undirected calls in a tree network or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. An undirected graph G is chordal if every cycle in G of length greater than 3 possesses a chord. Chordal graphs correspond to vertex intersection graphs of subtrees on a tree. An undirected graph G is weakly chordal if every cycle of length greater than 4 in G and in its complement G ‾ possesses a chord. It is known that the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. Moreover, this provides an algorithm to reduce a given EPT representation of a weakly chordal EPT graph to an EPT representation on a degree 4 tree. Finally, we raise a number of intriguing open questions regarding related families of graphs.
Discrete Mathematics – Elsevier
Published: Apr 28, 2008
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