In this paper, we propose a new network reliability measure for some particular kind of service networks, which we refer to as domination reliability. We relate this new reliability measure to the domination polynomial of a graph and the coverage probability of a hypergraph. We derive explicit and recursive formulæ for domination reliability and its associated domination reliability polynomial,...

We introduce a domination polynomial of a graph G. The domination polynomial of a graph G of order n is the polynomial D(G, x) = ∑n i=γ(G) d(G, i)x , where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. We obtain some properties of D(G, x) and its coefficients. Also we compute this polynomial for some specific graphs.

Let G = (V, E) be a simple graph. Hosoya polynomial of G is d(u,v) H(G, x) = {u,v}V(G)x , where, d(u ,v) denotes the distance between vertices u and v. As is the case with other graph polynomials, such as chromatic, independence and domination polynomial, it is natural to study the roots of Hosoya polynomial of a graph. In this paper we study the roots of Hosoya polynomials of some specific g...

A subset of vertices in a graph G is total dominating set if every vertex adjacent to at least one this subset. The domination number the minimum cardinality any and denoted by ?t(G). having nonempty intersection with all independent sets maximum an transversal set. ?tt(G). Based on fact that for tree T, ?t(T) ? ?tt(T) + 1, work we give several relationships between trees T which are leading cl...

Given a graph G = (V, E) of order n and an n-dimensional non-negative vector d = (d(1), d(2), . . . , d(n)), called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum S ⊆ V such that every vertex v in V \S (resp., in V ) has at least d(v) neighbors in S. The (total) vector domination is a generalization of many dominating set type problems,...

A set D ⊆ V of a graph G = (V,E) is called a restrained dominating set of G if every vertex not in D is adjacent to a vertex in D and to a vertex in V \D. The MINIMUM RESTRAINED DOMINATION problem is to find a restrained dominating set of minimum cardinality. Given a graph G, and a positive integer k, the RESTRAINED DOMINATION DECISION problem is to decide whether G has a restrained dominating ...