Handbook of combinatorics (vol. 1)
On weakly connected domination in graphs
Discrete Mathematics
Approximating minimum size weakly-connected dominating sets for clustering mobile ad hoc networks
Proceedings of the 3rd ACM international symposium on Mobile ad hoc networking & computing
On the number of edges in graphs with a given weakly connected domination number
Discrete Mathematics
Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Weakly-Connected Dominating Sets and Sparse Spanners in Wireless Ad Hoc Networks
ICDCS '03 Proceedings of the 23rd International Conference on Distributed Computing Systems
Randomised algorithms for finding small weakly-connected dominating sets of regular graphs
CIAC'03 Proceedings of the 5th Italian conference on Algorithms and complexity
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Let G=(V,E) be a connected graph. A dominating set S of G is a weakly connected dominating set of G if the subgraph (V,E@?(SxV)) of G with vertex set V that consists of all edges of G incident with at least one vertex of S is connected. The minimum cardinality of a weakly connected dominating set of G is the weakly connected domination number, denoted @c"w"c(G). A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number @c"t(G) of G. In this paper, we show that 12(@c"t(G)+1)@?@c"w"c(G)@?32@c"t(G)-1. Properties of connected graphs that achieve equality in these bounds are presented. We characterize bipartite graphs as well as the family of graphs of large girth that achieve equality in the lower bound, and we characterize the trees achieving equality in the upper bound. The number of edges in a maximum matching of G is called the matching number of G, denoted @a^'(G). We also establish that @c"w"c(G)@?@a^'(G), and show that @c"w"c(T)=@a^'(T) for every tree T.