On the Connected Domination Number of Random Regular Graphs
COCOON '02 Proceedings of the 8th Annual International Conference on Computing and Combinatorics
Note: 2-connected graphs with small 2-connected dominating sets
Discrete Mathematics
Dominating a Family of Graphs with Small Connected Subgraphs
Combinatorics, Probability and Computing
Randomised algorithms for finding small weakly-connected dominating sets of regular graphs
CIAC'03 Proceedings of the 5th Italian conference on Algorithms and complexity
Spanning trees with many leaves in regular bipartite graphs
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Optimal transmission range for topology management in wireless sensor networks
ICOIN'06 Proceedings of the 2006 international conference on Information Networking: advances in Data Communications and Wireless Networks
Bounds on the connected domination number of a graph
Discrete Applied Mathematics
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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.