A function f : V → {−1, 0, 1} is a minus-domination function of a graph G = (V,E) if the values over the vertices in each closed neighborhood sum to a positive number. The weight of f is the sum of f(x) over all vertices x ∈ V. The minus-domination number γ(G) is the minimum weight over all minus-domination functions. The size of a minus domination is the number of vertices that are assigned 1....

We study the complexity of query evaluation of SPARQL queries. We focus on the fundamental fragment of welldesigned SPARQL restricted to the AND, OPTIONAL and UNION operators. Our main result is a structural characterisation of the classes of well-designed queries that can be evaluated in polynomial time. In particular, we introduce a new notion of width called domination width, which relies on...

Let P be an optimization problem, and let A be an approximation algorithm for P . The domination ratio domr(A,n) is the maximum real q such that the solution x(I) obtained by A for any instance I of P of size n is not worse than at least a fraction q of the feasible solutions of I. We describe a deterministic, polynomial time algorithm with domination ratio 1−o(1) for the partition problem, and...

Let G be a simple graph of order n. A dominating set of G is a set S of vertices of G so that every vertex of G is either in S or adjacent to a vertex in S. The domination polynomial of G is the polynomial D(G,x) = ∑n i=1 d(G, i)x , where d(G, i) is the number of dominating sets of G of size i. In this paper we show that cycles are determined by their domination polynomials. AMS Classification:...

An independent dominating set of the simple graph G=(V,E) is a vertex subset that both and in G. The domination polynomial G Di(G,x)=∑Ax|A|, summed over all subsets A⊆V. A root Di(G,x) called an independence root. We investigate polynomials some generalized compound graphs. As consequence, we construct graphs whose roots are real. Also, consider related to paths study number their sets.

We consider new variants of the vertex/edge domination problems on graphs. A vertex is said to dominate itself and its all adjacent vertices, and similarly an edge is said to dominate itself and its all adjacent edges. Given an input graph G = (V,E) and an integer k, the k-Vertex (k-Edge) Maximum Domination (k-MaxVD and k-MaxED, respectively) is to find a subset DV ⊆ V of vertices (resp., DE ⊆ ...

Let γ(G) and ι(G) be the domination and independent domination numbers of a graph G, respectively. In this paper, we define the Price of Independence of a graph G as the ratio ι(G) γ(G) . Firstly, we bound the Price of Independence by values depending on the number of vertices. Secondly, we consider the question of computing the Price of Independence of a given graph. Unfortunately, the decisio...