Trees with large neighborhood total domination number (original) (raw)

A note on neighborhood total domination in graphs

Proceedings - Mathematical Sciences, 2015

Let G = (V , E) be a graph without isolated vertices. A dominating set S of G is called a neighborhood total dominating set (or just NTDS) if the induced subgraph G[N(S)] has no isolated vertex. The minimum cardinality of a NTDS of G is called the neighborhood total domination number of G and is denoted by γ nt (G).In this paper, we obtain sharp bounds for the neighborhood total domination number of a tree. We also prove that the neighborhood total domination number is equal to the domination number in several classes of graphs including grid graphs.

On trees with total domination number equal to edge-vertex domination number plus one

Proceedings - Mathematical Sciences, 2016

An edge e ∈ E(G) dominates a vertex v ∈ V (G) if e is incident with v or e is incident with a vertex adjacent to v. An edge-vertex dominating set of a graph G is a set D of edges of G such that every vertex of G is edgevertex dominated by an edge of D. The edge-vertex domination number of a graph G is the minimum cardinality of an edge-vertex dominating set of G. A subset D ⊆ V (G) is a total dominating set of G if every vertex of G has a neighbor in D. The total domination number of G is the minimum cardinality of a total dominating set of G. We characterize all trees with total domination number equal to edge-vertex domination number plus one.

A note on the total domination number of a tree

2000

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimumcardinality of a total dominating set is the total dom- ination number t(G). We show that for a nontrivial tree T of order n and withleaves, t(T) > (n +2 `)/2, and we

Super Dominating Sets in Graphs

Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 2015

Let G = (V, E) be a graph. A subset D of V (G) is called a super dominating set if for every v ∈ V (G) − D there exists an external private neighbour of v with respect to V (G) − D. The minimum cardinality of a super dominating set is called the super domination number of G and is denoted by γsp(G). In this paper some results on the super domination number are obtained. We prove that if T is a tree with at least three vertices, then n 2 ≤ γsp(T) ≤ n − s, where s is the number of support vertices in T and we characterize the extremal trees.

Total domination in partitioned trees and partitioned graphs with minimum degree two

Journal of Global Optimization, 2008

Let G = (V, E) be a graph and let S ⊆ V . A set of vertices in G totally dominates S if every vertex in S is adjacent to some vertex of that set. The least number of vertices needed in G to totally dominate S is denoted by γ t (G, S). When S = V , γ t (G, V ) is the well studied total domination number γ t (G). We wish to maximize the sum γ t (G) + γ t (G, V 1 ) + γ t (G, V 2 ) over all possible partitions V 1 , V 2 of V . We call this maximum sum f t (G). For a graph H, we denote by H • P 2 the graph obtained from H by attaching a path of length 2 to each vertex of H so that the resulting paths are vertex-disjoint. We show that if G is a tree of order n ≥ 4 and G / ∈ {P 5 , P 6 , P 7 , P 10 , P 14 }, then f t (G) ≤ 14n/9 with equality if and only if G ∈ {P 9 , P 18 } or G = (T • P 2 ) • P 2 for some tree T . If G is a connected graph of order n with minimum degree at least two, we establish that f t (G) ≤ 3n/2 with equality if and only if G is a cycle of order congruent to zero modulo 4.

On the Domination Number of Some Graphs

2008

Let G = (V,E) be a simple graph. A set S ⊆ V is a dominating set of graph G, if every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set in G. It is well known that if e ∈ E(G), then γ(G−e)−1 ≤ γ(G) ≤ γ(G−e). In this paper, as an application of this inequality, we obtain the domination number of some certain graphs.

Neighbourhood total domination in graphs

Opuscula Mathematica, 2011

Let G = (V, E) be a graph without isolated vertices. A dominating set S of G is called a neighbourhood total dominating set (ntd-set) if the induced subgraph N (S) has no isolated vertices. The minimum cardinality of a ntd-set of G is called the neighbourhood total domination number of G and is denoted by γnt(G). The maximum order of a partition of V into ntd-sets is called the neighbourhood total domatic number of G and is denoted by dnt(G). In this paper we initiate a study of these parameters.

On the Domination Number of Some Families of Special Graphs

A domination in graphs is part of graph theory which has many applications. Its application includes the morphological analysis, computer network communication, social network theory, CCTV installation, and many others. A set D of vertices of a simple graph G, that is a graph without loops and multiple edges, is called a dominating set if every vertex u ∈ V (G) − D is adjacent to some vertex v ∈ D. The domination number of a graph G, denoted by γ(G), is the order of a smallest dominating set of G. A dominating set D with |D| = γ(G) is called a minimum dominating set, see Haynes and Henning [5]. This research aims to find the domination number of some families of special graphs, namely Spider Web graph W b n , Helmet graph H n,m , Parachute graph P c n , and any regular graph. The results shows that the resulting domination numbers meet the lower bound of an obtained lower bound γ(G) of any graphs.

On locating and differetiating-total domination in trees

Discussiones Mathematicae Graph Theory, 2008

A total dominating set of a graph G = (V, E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V − S, N (u) ∩ S = N (v) ∩ S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V , N [u] ∩ S = N [v] ∩ S. Let γ L t (G) and γ D t (G) be the minimum cardinality of a locating-total dominating set and a differentiating-total dominating set of G, respectively. We show that for a nontrivial tree T of order n, with leaves and s support vertices, γ L t (T) max{2(n + − s + 1)/5, (n + 2 − s)/2}, and for a tree of order n ≥ 3, γ D t (T) ≥ 3(n+ −s+1)/7, improving the lower bounds of Haynes, Henning and Howard. Moreover we characterize the trees satisfying γ L t (T) = 2(n + − s + 1)/5 or γ D t (T) = 3(n + − s + 1)/7.