Domination Numbers of Trees (original) (raw)

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

A Survey on Characterizing Trees Using Domination Number

Mathematics

Ever since the discovery of domination numbers by Claude Berge in the year 1958, graph domination has become an important domain in graph theory that has strengthened itself as a theory and has extended its contributions to various applications. Tree characterization is an important problem in graph domination. This survey focuses on presenting a collection of results on characterizing trees using domination number.

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 trees with double domination number equal to 2-domination number plus one

Houston Journal of Mathematics, 2013

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.

Characterization of trees with equal 2-domination number and domination number plus two

Discussiones Mathematicae Graph Theory, 2011

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 γ k (G) is the minimum cardinality of a k-dominating set of G. In [5] Volkmann showed that for every nontrivial tree T, γ 2 (T) ≥ γ 1 (T) + 1 and characterized extremal trees attaining this bound. In this paper we characterize all trees T with γ 2 (T) = γ 1 (T) + 2.

On trees with double domination number equal to the 2-outer-independent domination number plus one

Chinese Annals of Mathematics, Series B, 2012

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 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.

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)

A note on complementary tree domination number of a tree

Proyecciones (Antofagasta), 2015

A complementary tree dominating set of a graph G, is a set D of vertices of G such that D is a dominating set and the induced sub graph hV \ Di is a tree. The complementary tree domination number of a graph G, denoted by γ ctd (G), is the minimum cardinality of a complementary tree dominating set of G. An edge-vertex dominating set of a graph G is a set D of edges of G such that every vertex of G is incident with an edge of D or incident with an edge adjacent to an edge of D. The edge-vertex domination number of a graph, denoted by γ ev (G), is the minimum cardinality of an edge-vertex dominating set of G. We characterize trees for which γ(T) = γ ctd (T) and γ ctd (T) = γ ev (T) + 1.