A study on the restrained domination number in diverse families of derived signed graphs

A derived graph is obtained by applying a specific graph operation to a given graph [Formula: see text]. A set [Formula: see text] of vertices in a signed graph [Formula: see text] is a restrained dominating set of [Formula: see text] if [Formula: see text] is a restrained dominating set of the underlying graph [Formula: see text], and every cycle formed by the edges connecting [Formula: see text] to [Formula: see text] and those within [Formula: see text] is balanced. The restrained domination number [Formula: see text] is the minimum cardinality of a restrained dominating set of [Formula: see text]. In this paper, we establish results on the restrained domination number for derived signed graphs, including line signed graphs, semi-total point signed graphs, semi-total line signed graphs, and total signed graphs and we also examine their structural properties in relation to their restrained domination number.