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

Let ν be some graph parameter and let G be a class of graphs for which ν can be computed in polynomial time. In this situation it is often possible to devise a strategy to decide in polynomial time whether ν has a unique realization for some graph in G. We first give an informal description of the conditions that allow one to devise such a strategy, and then we demonstrate our approach for thre...

Let G V, E be a simple graph. A set S ⊆ V is a dominating set of G, if every vertex in V \S is adjacent to at least one vertex in S. Let Pi n be the family of all dominating sets of a path Pn with cardinality i, and let d Pn, j |P n|. In this paper, we construct Pi n, and obtain a recursive formula for d Pn, i . Using this recursive formula, we consider the polynomialD Pn, x ∑n i n/3 d Pn, i x ...

Fomin and Villanger ([14], STACS 2010) proved that Maximum Independent Set, Feedback Vertex Set, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant t, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let Gpoly be the class of graphs with at most poly(n) minimal...

a roman dominating function (rdf) on a graph g = (v,e) is defined to be a function satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. a set s v is a restrained dominating set if every vertex not in s is adjacent to a vertex in s and to a vertex in . we define a restrained roman dominating function on a graph g = (v,e) to be ...