Please use this identifier to cite or link to this item: http://hdl.handle.net/10497/17693
Title: Lower bound on the weakly connected domination number of a cycledisjoint graph
Authors: Koh, Khee Meng
Ting, T. S.
Xu, Z. L.
Dong, Fengming
Issue Date: 2010
Citation: Koh, K. M., Ting, T. S., Xu, Z. L., & Dong, F. (2010). Lower bound on the weakly connected domination number of a cycledisjoint graph. Australasian Journal of Combinatorics, 46, 157-166.
Abstract: For a connected graph G and any non-empty S ⊆ V (G), S is called a weakly connected dominating set of G if the subgraph obtained from G by removing all edges each joining any two vertices in V (G) \ S is connected. The weakly connected domination number γw(G) is defined to be the minimum integer k with |S| = k for some weakly connected dominating set S of G. In this note, we extend a result on the lower bound for the weakly connected domination number γw(G) on trees to cycle-e-disjoint graphs, i.e., graphs in which no cycles share a common edge. More specifically, we show that if G is a connected cycle-e-disjoint graph, then γw(G) ≥ (|V (G)| − v1(G) − nc(G) − noc(G) + 1)/2, where nc(G) is the number of cycles in G, noc(G) is the number of odd cycles in G and v1(G) is the number of vertices of degree 1 in G. The graphs for which equality holds are also characterised.
URI: http://hdl.handle.net/10497/17693
ISSN: 1034-4942
Website: http://ajc.maths.uq.edu.au/pdf/46/ajc_v46_p157.pdf
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