Connected Domination and Spanning Trees with Many Leaves

Abstract

Let G=(V,E) be a connected graph. A connected dominating set $S \subset V$ is a dominating set that induces a connected subgraph of G. The connected domination number of G, denoted $\gamma_c(G)$, is the minimum cardinality of a connected dominating set. Alternatively, $|V|-\gamma_c(G)$ is the maximum number of leaves in a spanning tree of G. Let $\delta$ denote the minimum degree of G. We prove that $\gamma_c(G) \leq |V| \frac{\ln(\delta+1)}{\delta+1}(1+o_\delta(1))$. Two algorithms that construct a set this good are presented. One is a sequential polynomial time algorithm, while the other is a randomized parallel algorithm in RNC.