A note on the total domination number of a tree (original) (raw)

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.

On total domination and support vertices of a tree

The total domination number \gamma_t(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is adjacent to some vertex in D. In this paper we prove two new upper bounds on the total domination number of a tree related to particular support vertices (vertices adjacent to leaves) of the tree. One of these bounds improves a 2004 result of Chellali and Haynes. In addition, we prove some bounds on the total domination ratio of trees.

Trees with large neighborhood total domination number

Discrete Applied Mathematics, 2015

In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [Opuscula Math. 31 (2011), 519-531]. A neighborhood total dominating set, abbreviated NTD-set, in a graph G is a dominating set S in G with the property that the subgraph induced by the open neighborhood of the set S has no isolated vertex. The neighborhood total domination number, denoted by γ nt (G), is the minimum cardinality of a NTD-set of G. Every total dominating set is a NTD-set, implying that γ(G) ≤ γ nt (G) ≤ γ t (G), where γ(G) and γ t (G) denote the domination and total domination numbers of G, respectively. Arumugam and Sivagnanam posed the problem of characterizing the connected graphs G of order n ≥ 3 achieving the largest possible neighborhood total domination number, namely γ nt (G) = ⌈n/2⌉. A partial solution to this problem was presented by Henning and Rad [Discrete Applied Mathematics 161 (2013), 2460-2466] who showed that 5-cycles and subdivided stars are the only such graphs achieving equality in the bound when n is odd. In this paper, we characterize the extremal trees achieving equality in the bound when n is even. As a consequence of this tree characterization, a characterization of the connected graphs achieving equality in the bound when n is even can be obtained noting that every spanning tree of such a graph belongs to our family of extremal trees.

On trees with double domination number equal to total domination number plus one

Ars Combinatoria Waterloo Then Winnipeg, 2011

A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G is a set D of vertices of G, such that every vertex of G is dominated by at least two vertices of D. The double domination number of a graph G is the minimum cardinality of a double dominating set of G. For a graph G = (V, E), a subset D ⊆ V (G) is a 2-dominating set if every vertex of V (G) \ D has at least two neighbors in D, while it is a 2-outer-independent dominating set of G if additionally the set V (G)\D is independent. The 2-outer-independent domination number of G is the minimum cardinality of a 2-outer-independent dominating set of G. This paper characterizes all trees with the double domination number equal to the 2-outer-independent domination number plus one.

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.

On the (2, 2)-domination number of trees

2010

Let γ(G) and γ 2,2 (G) denote the domination number and (2, 2)domination number of a graph G, respectively. In this paper, for any nontrivial tree T , we show that 2(γ(T )+1) 3 ≤ γ 2,2 (T ) ≤ 2γ(T ). Moreover, we characterize all the trees achieving the equalities.

A note on the locating-total domination in trees

Australas. J Comb., 2016

A total dominating set of a graph G = (V,E) with no isolated vertex is a set D ⊆ V (G) such that every vertex is adjacent to a vertex in D. A total dominating set D of G is a locating-total dominating set if for every pair of distinct vertices u and v in V −D, N(u) ∩D = N(v) ∩D. Let γ L(G) be the minimum cardinality of a locating-total dominating set of G. We show that for a nontrivial tree T of order n, with leaves and s support vertices, γ t (T ) ≥ (n + 2 − s + 1)/2, improving some previous bounds presented by Chellali [Discussiones Math. Graph Theory 28 (3) (2008), 383–392] and Chen and Young Sohn [Discrete Appl. Math. 159 (13-14) (2011), 769–773]. We also characterize the extremal trees achieving the above bound.

An upper bound on the double domination number of trees

Kragujevac Journal of Mathematics, 2015

In a graph G, a vertex dominates itself and its neighbors. A set S of vertices in a graph G is a double dominating set if S dominates every vertex of G at least twice. The double domination number γ ×2 (G) is the minimum cardinality of a double dominating set in G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we show that for any tree T of order n ≥ 2, different from P 4 , γ ×2 (T) ≤ 3a(T)

Locating and total dominating sets in trees

Discrete Applied Mathematics, 2006

Locating and Total Dominating Sets in Trees by Jamie Marie Howard A set S of vertices in a graph G = (V, E) is a total dominating set of G if every vertex of V is adjacent to some vertex in S. In this thesis, we consider total dominating sets of minimum cardinality which have the additional property that distinct vertices of V are totally dominated by distinct subsets of the total dominating set.