On double domination numbers of signed complete graphs

Let [Formula: see text] be a signed graph with a vertex set [Formula: see text]. A set [Formula: see text] is said to be a double dominating set of [Formula: see text] if it satisfies the following conditions: (i) [Formula: see text] for each [Formula: see text], and (ii) [Formula: see text] is balanced, where [Formula: see text] denotes the closed neighborhood of [Formula: see text] and [Formula: see text] denotes the subgraph induced by the edges of [Formula: see text] with one end vertex in [Formula: see text] and the other end vertex in [Formula: see text]. The minimum size among all the double dominating sets of [Formula: see text] is the double domination number [Formula: see text] of [Formula: see text]. In this study, we investigated this parameter for signed complete graphs. We prove that, for [Formula: see text], if [Formula: see text] is a signed complete graph, then [Formula: see text] and these bounds are sharp. Moreover, for all signed complete graphs over [Formula: see text] we determined their possible double domination numbers. Finally, we compute the double domination numbers of all signed complete graphs of orders up to six.