Distance and related domination parameters in the edge corona product of graphs

For [Formula: see text], in a graph [Formula: see text] a set of vertices [Formula: see text] is a distance [Formula: see text]-dominating set of [Formula: see text] if all vertices in [Formula: see text] are within distance [Formula: see text] from a vertex in [Formula: see text]. The distance [Formula: see text]-domination number, denoted by [Formula: see text], is the smallest cardinality overall distance [Formula: see text]-dominating sets of [Formula: see text]. In this paper, we study some distance parameters and the distance [Formula: see text]-domination in the edge corona product of graphs. Let [Formula: see text] and [Formula: see text] be two simple graphs. The edge corona product of [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is the graph obtained from [Formula: see text] by adding a copy of [Formula: see text] for each edge [Formula: see text] of [Formula: see text] and joining the end-vertices of [Formula: see text] to every vertex in its corresponding copy of [Formula: see text]. First, we prove that the diameter of [Formula: see text] is equal to the diameter of the line graph of [Formula: see text] plus one. This allows us to show that [Formula: see text], then we characterize the graphs achieving both bounds. Also, we determine the radius of [Formula: see text] in terms of the radius of [Formula: see text]. Afterward, we focus on the distance [Formula: see text]-domination of [Formula: see text]. We show that [Formula: see text]. Moreover, we give a criterion for graphs achieving both bounds. Then, we show that [Formula: see text], where [Formula: see text] is the vertex-edge domination number of [Formula: see text]. Later, for [Formula: see text], we show that [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th power graph of [Formula: see text]. Then, we characterize the graphs achieving both of these bounds. Finally, for all [Formula: see text], we provide [Formula: see text] when [Formula: see text] is a complete graph, path, cycle or some other diameter [Formula: see text] graph family where [Formula: see text] is any graph.