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.