Efficient edge domination in regular graphs

An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p ≥ 3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete. © 2008 Elsevier B.V. All rights reserved.

Efficient domination through eigenvalues

The paper begins with a new characterization of (k,τ)(k,τ)-regular sets. Then, using this result as well as the theory of star complements, we derive a simplex-like algorithm for determining whether or not a graph contains a (0,τ)(0,τ)-regular set. When τ=1τ=1, this algorithm can be applied to solve the efficient dominating set problem which is known to be NP-complete. If −1−1 is not an eigenvalue of the adjacency matrix of the graph, this particular algorithm runs in polynomial time. However, although it does not work in polynomial time in general, we report on its successful application to a vast set of randomly generated graphs.