As a generalization of connected domination in a graph G we consider domination by sets having at most k components. The order γ c (G) of such a smallest set we relate to γc(G), the order of a smallest connected dominating set. For a tree T we give bounds on γ c (T ) in terms of minimum valency and diameter. For trees the inequality γ c (T ) ≤ n− k − 1 is known to hold, we determine the class o...

Results about wreath products of circulant graphs are used to construct infinitely many circulant graphswith efficient dominating setswhose elements need not be equally spaced in Zn. It is proved that if a circulant graph of large degree has an efficient dominating set, then either its elements are equally spaced, or the graph is the wreath product of a smaller circulant graph with an efficient...

—In this paper, a Connected Dominating Set (CDS) construction algorithm CSCDS (Connected Subset based CDS) is proposed, which is based on the connected subset concept. The CSCDS contains two main stages, which are dominating set construction stage and connected dominating set construction stage respectively. In the first stage, the dominators are proposed based on the one hop white neighbor in...

A connected dominating set of a graph G = (V, E) is a subset of vertices CD ⊆ V such that every vertex not in CD is adjacent to at least one vertex in CD, and the subgraph induced by CD is connected. We show that, given an arc family F with endpoints sorted, a minimum-cardinality connected dominating set of the circular-arc graph constructed from F can be computed in O(|F |) time.

Given a graph $$G=(V,E)$$ and an integer k, the Minimum Membership Dominating Set (MMDS) problem seeks to find dominating set $$S \subseteq V$$ of G such that for each $$v \in , $$\vert N[v] \cap S\vert $$ is at most k. We investigate parameterized complexity obtain following results MMDS problem. First, we show NP-hard even on planar bipartite graphs. Next, W[1]-hard parameter pathwidth (and t...