A graph G is an efficient open domination graph if there exists a subset D of V (G) for which the open neighborhoods centered in vertices of D form a partition of V (G). We completely describe efficient domination graphs among direct, lexicographic and strong products of graphs. For the Cartesian product we give a characterization when one factor is K2 and some partial results for grids, cylind...

In this paper, sharp upper bounds for the domination number, total domination number and connected domination number for the Cayley graph G = Cay(D2n, Ω) constructed on the finite dihedral group D2n, and a specified generating set Ω of D2n. Further efficient dominating sets in G = Cay(D2n, Ω) are also obtained. More specifically, it is proved that some of the proper subgroups of D2n are efficie...

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 a...

Given a simple graph G = (V, E), an edge (u, u) E E is said to dominate itself and any edge (u,x) or (u,x), where x E V. A subset D C E is called an efficient edge dominating set of G if all edges in E are dominated by exactly one edge of D. The efficient edge domination problem is to find an efficient edge dominating set of minimum size in G. Suppose that each edge e E E is associated with a r...

A subset S ⊆ V in a graph G = (V,E) is a [1, 2]-set if for every vertex v ∈ V \ S, 1 ≤ |N(v)∩ S| ≤ 2, that is, every vertex v ∈ V \ S is adjacent to at least one but not more than two vertices in S. In this paper we relate the concept of [1, 2]-sets to a host of other concepts in domination theory, including perfect domination, efficient domination, nearly perfect sets, 2-packings and k-depende...