Strong coalitions in graphs

For a graph [Formula: see text], a set [Formula: see text] is a strong dominating set of [Formula: see text], if for every vertex [Formula: see text] there is a vertex [Formula: see text] with [Formula: see text] and [Formula: see text]. A strong coalition consists of two disjoint sets of vertices [Formula: see text] and [Formula: see text], neither of which is a strong dominating set but whose union [Formula: see text] is a strong dominating set. A vertex partition [Formula: see text] of vertices in [Formula: see text] is a strong coalition partition if every set [Formula: see text] either is a strong dominating set consisting of a single vertex of degree [Formula: see text], or is not a strong dominating set but forms a strong coalition with another set [Formula: see text] that is not a strong dominating set. In this paper, we study properties of strong coalitions in graphs.