Vertex covers and secure domination in graphs

Burger A.P. ; Henning M.A. ; Van Vuuren J.H. (2008)


Let G = (V, E) be a graph and let S ⊆ V. The set S is a dominating set of G if every vertex in V \ S is adjacent to some vertex in S. The set S is a secure dominating set of G if for each u ∈V \ S, there exists a vertex v ∈ S such that uv ∈ E and (S \ {v}) ∪ {u}is a dominating set of G. The minimum cardinality of a secure dominating set in G is the secure domination number γs(G) of G. We show that if G is a connected graph of order n with minimum degree at least two that is not a 5-cycle, then γs (G) ≤ n/2 and this bound is sharp. Our proof uses a covering of a subset of V(G) by vertex-disjoint copies of subgraphs each of which is isomorphic to K2 or to an odd cycle. © 2008 NISC Pty Ltd.

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