Abstract
Graph Theory The notion of graph powers is a well-studied topic in graph theory and its applications. In this paper, we investigate a bipartite analogue of graph powers, which we call bipartite powers of bigraphs. We show that the classes of bipartite permutation graphs and interval bigraphs are closed under taking bipartite power. We also show that the problem of recognizing bipartite powers is NP-complete in general.
Highlights
For a positive integer k, the kth power Gk of a graph G has the same vertex set as G, and two vertices are adjacent in Gk if and only if their distance in G is at most k
If Hk = G, H is called a kth root of G. (In general, kth roots of a graph are not unique and do not necessarily exist.) The concept of graph powers has been extensively studied in graph theory and its algorithmic applications (see Prisner (1995) and the references therein)
It is known that several important graph classes are closed under the power operation
Summary
For a positive integer k, the kth power Gk of a graph G has the same vertex set as G, and two vertices are adjacent in Gk if and only if their distance in G is at most k. It is known that several important graph classes are closed under the power operation. For odd k ∈ Z+, the kth bipartite power G[k] of a bigraph G has the same vertex set as G, and two vertices are adjacent in G[k] if and only if their distance in G is at most k and odd. Chandran and Mathew (2012) have strengthened the closure-property result by showing that the class of chordal bipartite graphs is closed under the bipartite power operation. We complement the known closure-property results by proving that some subclasses of the class of chordal bipartite graphs are closed under the bipartite power operation. Note that the closure property of a graph class C under the bipartite power operation is not implied by the closure property of a superclass of C, and vice versa.
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