Showing papers on "Dominating set published in 1986"

On the domination of the products of graphs II: Trees

Abstract: For a graph G, a subset of vertices D is a dominating set if for each vertex X not in D, X is adjacent to at least one vertex of D. The domination number, γ(G), is the order of the smallest such set. An outstanding conjecture in the theory of domination is for any two graph G and H, One result presented in this paper settles this question in the case when at least one of G or H is a tree. We show that for all graphs G and any tree T. Furthermore, we supply a partial characterization for which pairs of trees, T1 and T2, strict inequality occurs. We show for almost all pairs of trees.

Stability, domination and irredundance in a graph

Abstract: In a graph G, a set X is called a stable set if any two vertices of X are nonadjacent. A set X is called a dominating set if every vertex of V – X is joined to at least one vertex of X. A set X is called an irredundant set if every vertex of X, not isolated in X, has at least one proper neighbor, that is a vertex of V – X joined to it but to no other vertex of X. Let α′ and α, γ, and Γ, ir and IR, denote respectively the minimum and maximum cardinalities of a maximal stable set, a minimal dominating set, and a maximal irredundant set. It is known that ir ⩽ γ ⩽ α′ ⩽ α ⩽ Γ ⩽ IR and that if G does not contain any induced subgraph isomorphic to K1,3, then γ = α′. Here we prove that if G contains no induced subgraph isomorphic to K1,3 or to the graph H of figure 1, then ir = γ = α′. We prove also that if G contains no induced subgraph isomorphic to K1,3, to H, or to the graph h of figure 3, then Γ = IR. Finally, we improve a result of Bollobas and Cockayne about sufficient conditions for γ = ir in terms of forbidden subgraphs.

Connected domatic number of a graph

Abstract: All graphs considered in this paper are finite graphs without loops and multiple edges. The domatic number of a graph was defined by E. J. Cockayne and S. T. Hedetniemi [1]. Later some related concepts were introduced. The same authors together with R. M. Dawes [2] have introduced the total domatic number; R. Laskar and S. T. Hedetniemi [3] have introduced the connected domatic number. A dominating set (or a total dominating set) in an undirected graph G is a subset D of the vertex set V(G) of G with the property that to each vertex x e V(G) — D (or to each vertex xeV(G) respectively) there exists a vertex y e D adjacent to x. A connected dominating set of G is a dominating set of G with the property that the subgraph of G induced by it is connected. A domatic (or total domatic, or connected domatic) partition of G is a partition of V(G), all of whose classes are dominating (or total dominating, or connected dominating, respectively) sets of G. The maximum number of classes of a domatic (or total domatic, or connected domatic) partition of G is called the domatic (or total domatic, or connected domatic, respectively) number of G. The domatic number of G is denoted by d(G), its total domatic number by <4(G), its connected domatic number by dc(G). The connected domatic number of a graph is well defined only for connected graphs; in a disconnected graph there exists no connected dominating set and thus no connected domatic partition, while in every connected graph there exists at least one connected domatic partition, namely that which consists of one class. The connected domatic number of G is closely related to the vertex connectivity number of G. If G is a connected graph, then a vertex cut of G is a subset R of V(G) with the property that the subgraph of G induced by V(G) — R is disconnected. If G is not a complete graph, then the vertex connectivity number x(G) is the minimum cardinality of a vertex cut of G. If G is a complete graph (i. e. without vertex cuts) with n vertices, then we put x(G) = n — 1. Lemma. Let G be a connected graph which is not complete, let R be its vertex cut, let D be its connected dominating set. Then DnR^0.

Parallel Independent Set Algorithms for Sparse Graphs

Abstract: In this paper we present algorithms for the independent set problem on n-vertex m-edge graphs using n + m proceBBors and O(n + m) space on the concurrent-read concurrent-write parallel RAM model of computation. If the graph is planar, a maximal independent set with at least niH vertices can be found in O(log'"nlogn) time. If!:i. is a graph's maximum degree, we show how to find in O(2.6.1og· n) time a maximal independent set with at least n/(.6.+1) vertices. All previous work on parallel algorithms for the maximal independent set problem has dealt with showing that the problem is in the class Ne and has produced algorithms which use at least nS / log3 n processors. ·Supported in part by Hewlett-Packard's Faculty Development Program and by the National Science Foundation under grant DCR-8320124.