Connected domination in random graphs

Abstract

Given a graph $G=(V,E)$, a dominating set is a subset $D\subseteq V$ such that every vertex in $V\setminus D$ is adjacent with at least one vertex in D. The domination number of G, denoted by $\gamma (G)$, is the minimum cardinality of a dominating set in G. Assuming that the graph $G=(V,E)$ is connected, a subset $D\subseteq V$ is said to be a connected dominating set if it is a dominating set and the subgraph G[D] induced by D is connected. The minimum cardinality of a connected dominating set is termed the connected domination number, denoted by $\gamma _c(G)$. Comparing $\gamma (G)$ and $\gamma _c(G)$ for a random graph with constant edge probability p, we obtain that the two parameters are asymptotically equal with probability tending to 1 as the number of vertices gets large. We also consider nonconstant edge probability $p_n$ tending to zero (where n is the number of vertices). Among other results, we extend an asymptotic formula of Gilbert on the probability of connectivity.