On double domination numbers of signed complete graphs

Let Sigma=(G,sigma) be a signed graph with a vertex set V(G). A set D subset of V(G) is said to be a double dominating set of Sigma if it satisfies the following conditions: (i) |N[v]boolean AND D|>= 2 for each v is an element of V(G), and (ii) Sigma[D,D (c)] is balanced, where N[v] denotes the closed neighborhood of v and Sigma[D,D (c)] denotes the subgraph induced by the edges of Sigma with one end vertex in D and the other end vertex in D (c). The minimum size among all the double dominating sets of Sigma is the double domination number gamma x2(Sigma) of Sigma. In this study, we investigated this parameter for signed complete graphs. We prove that, for n >= 5, if (K-n,K-sigma) is a signed complete graph, then 2 <=gamma x2(K-n,K-sigma)<= n-1 and these bounds are sharp. Moreover, for all signed complete graphs over Kn we determined their possible double domination numbers. Finally, we compute the double domination numbers of all signed complete graphs of orders up to six.