On some variations of dominating identification in graphs

In this study, we introduce the locating-dominating value and the location-domination polynomial of graphs and location-domination polynomials of some families of graphs were identified. Locatingdominating set of graph G is defined as the dominating set which locates all the vertices of G. And, location-domination number G is the minimum cardinality of a locating-dominating set in G.

On the domination polynomial of a digraph: a generation function approach

Let G be a directed graph on n vertices. The domination polynomial of G is the polynomial D(G, x) =∑ni=0 d(G, i)xi, where d(G, i) is the number of dominating sets of G with i vertices. In this paper, we prove that the domination polynomial of G can be obtained by using an ordinary generating function. Besides, we show that our method is useful to obtain the minimum-weighted dominating set of a graph.

More on the unimodality of domination polynomial of a graph

Let [Formula: see text] be a graph of order [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a dominating set of [Formula: see text] if every vertex in [Formula: see text] is adjacent to at least one vertex of [Formula: see text]. The domination polynomial of [Formula: see text] is the polynomial [Formula: see text], where [Formula: see text] is the number of dominating sets of [Formula: see text] of size [Formula: see text], and [Formula: see text] is the size of a smallest dominating set of [Formula: see text], called the domination number of [Formula: see text]. Motivated by a conjecture in [S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, ARS Combin. 114 (2014) 257–266] which states that the domination polynomial of any graph is unimodal, we obtain sufficient conditions for this conjecture to hold. Also we study the unimodality of graph [Formula: see text] with [Formula: see text], where [Formula: see text] is an integer.