New Polynomial Cases of the Weighted Efficient Domination Problem

Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes. In particular, the ED problem remains NP-complete for 2P3-free graphs and thus for P7-free graphs. We show that the weighted version of the problem (abbreviated WED) is solvable in polynomial time on various subclasses of 2P3-free and P7-free graphs, including (P2+P4)-free graphs, P5-free graphs and other classes. Furthermore, we show that a minimum weight e.d. consisting only of vertices of degree at most 2 (if one exists) can be found in polynomial time. This contrasts with our NP-completeness result for the ED problem on planar bipartite graphs with maximum degree 3

Efficient domination through eigenvalues

The paper begins with a new characterization of (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 )-regular set. When τ = 1, this algorithm can be applied to solve the efficient dominating set problem which is known to be NPcomplete. If −1 is not an eigenvalue of the adjacency matrix of the graph, this particular algorithm runs in polynomial time. However, although it doesn’t work in polynomial time in general, we report on its successful application to a vast set of randomly generated graphs