Let G = (V (G), E(G)) be a simple graph, and let k be a positive integer. A subset D of V (G) is a k-dominating set if every vertex of V (G) − D is dominated at least k times by D. The k-domination number γk(G) is the minimum cardinality of a k-dominating set of G. In [5] Volkmann showed that for every nontrivial tree T, γ2(T ) ≥ γ1(T ) + 1 and characterized extremal trees attaining this bound....

In this paper, we prove strong convergence results for a modified Mann iterative process for a new class of I- nearly weak uniformly L-Lipschitzian mappings in a real Banach space. The class of I-nearly weak uniformly L-Lipschitzian mappings is an interesting generalization of the class of nearly weak uniformly L-Lipschitzian mappings which inturn is a generalization of the class of nearly unif...

Let G = (V, E) be a graph. A subset S ⊆ V is a dominating set of G, if every vertex u ∈ V − S is dominated by some vertex v ∈ S. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. For the generalized Petersen graph G(n), Behzad et al. [A. Behzad, M. Behzad, C.E. Praeger, On the domination number of the generalized Petersen graphs, Discrete Mathematics 308 (2...

Let p be a positive integer and G = (V;E) a graph. A subset S of V is a p-dominating set if every vertex of V S is dominated at least p times. The minimum cardinality of a p-dominating set a of G is the p-domination number p(G): It is proved for a cactus graph G that p(G) 6 (jV j+ jLp(G)j+c(G))=2; for every positive integer p > 2; where Lp(G) is the set of vertices of G of degree at most p 1 an...

Given a simple graph G = (V, E) and a fixed positive integer k. In a graph G, a vertex is said to dominate itself and all of its neighbors. A set D ⊆ V is called a k-tuple dominating set if every vertex in V is dominated by at least k vertices of D. The k-tuple domination problem is to find a minimum cardinality k-tuple dominating set. This problem is NP-complete for general graphs. In this pap...

We prove several new sufficient conditions for hamiltonicity and bipancyclicity in balanced bipartite digraphs, terms of sums degrees over dominating or dominated pairs vertices.

ABSTRACT We present an analytical model to identify thin discs in galaxies, and apply this a sample of SDSS MaNGA galaxies. This fits the velocity dispersion fields galaxies with regular kinematics. By introducing two parameters ? related comparison model’s asymmetric drift correction observed gas kinematics ? dominant component galaxy, we classify as ‘disc-dominated, ‘non-disc-dominated’, or ‘...

-In wireless sensor network, virtual backbone formulation is a cost effective method to complete the broadcasting. Minimum connected dominating set is an outstanding candidate of virtual backbone. However, it is NP-Hard to find a minimum connected dominating set in an arbitrary graph. In this paper, we propose a novel backbone formation algorithm to construct a connected dominating set. In the ...

A subset S ⊆ V in a graph G = (V,E) is a k-quasiperfect dominating set (for k ≥ 1) if every vertex not in S is adjacent to at least one and at most k vertices in S. The cardinality of a minimum k-quasiperfect dominating set in G is denoted by γ 1k (G). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a dec...

For each vertex v in a graph G, let there be associated a subgraph Hv of G. The vertex v is said to dominate Hv as well as dominate each vertex and edge of Hv. A set S of vertices of G is called a full dominating set if every vertex of G is dominated by some vertex of S, as is every edge of G. The minimum cardinality of a full dominating set of G is its full domination number γFH(G). A full dom...