Domination, radius, and minimum degree

Graphs and Combinatorics, 2013

Let G be a graph with vertex set V . A set D ⊆ V is a total restrained dominating set of G if every vertex in V has a neighbor in D and every vertex in V \ D has a neighbor in V \ D. The minimum cardinality of a total restrained dominating set of G is called the total restrained domination number of G, and is denoted by γ tr (G). In this paper, we prove that if G is a connected graph of order n ≥ 4 and minimum degree at least two, then γ tr (G) ≤ n − 3 √ n 4 .

Discussiones Mathematicae Graph Theory, 2010

In this note, we prove several lower bounds on the domination number of simple connected graphs. Among these are the following: the domination number is at least two-thirds of the radius of the graph, three times the domination number is at least two more than the number of cut-vertices in the graph, and the domination number of a tree is at least as large as the minimum order of a maximal matching.

1997

Let γ(n, δ) denote the largest possible domination number for a graph of order n and minimum degree δ. This paper is concerned with the behavior of the right side of the sequence n = γ(n,0) ≥ γ(n,1) ≥·· · ≥γ(n, n − 1) = 1. We set δk(n) = max{δ | γ(n, δ) ≥ k}, k ≥ 1. Our main result is that for any fixed k ≥ 2 there is a constant ck such that for sufficiently large n, n − ckn (k−1)/k ≤ δk+1(n) ≤ n − n (k−1)/k. The lower bound is obtained by use of circulant graphs. We also show that for n sufficiently large relative to k, γ(n, δk(n)) = k. Thecase k = 3 is examined in further detail. The existence of circulant graphs with domination number greater than 2 is related to a kind of difference set in Zn.

Discrete Applied Mathematics, 2006

A subset S of the vertex set of a graph G is called acyclic if the subgraph it induces in G contains no cycles. S is called an acyclic dominating set of G if it is both acyclic and dominating. The minimum cardinality of an acyclic dominating set, denoted by γ a (G), is called the acyclic domination number of G. Hedetniemi et al. in [5] introduced the concept of acyclic domination and posed the following open problem: If δ(G)

Journal of Graph Theory, 2013

The total domination number γ t (G) of a graph G is the minimum cardinality of a set S of vertices, so that every vertex of G is adjacent to a vertex in S. In this article, we determine an optimal upper bound on the total domination number of a graph with diameter 2. We show that for every graph G on n vertices with diameter 2, γ t (G) ≤ 1 + n ln(n). This bound is optimal in the sense that given any > 0, there exist graphs G with diameter 2 of all sufficiently large even orders n such that γ t (G) > (1 4+) n ln(n).

Graphs and Combinatorics

A set S of vertices in a graph G without isolated vertices is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ t (G) of G. The girth of G is the length of a shortest cycle in G. Let G be a connected graph with minimum degree at least 2, order n and girth g ≥ 3. It was shown in an earlier manuscript (Henning and Yeo in Graphs Combin 24:333–348, 2008) that $${\gamma_t(G)\le(\frac{1}{2}+\frac{1}{g})n}$$, and this bound is sharp for cycles of length congruent to two modulo four. In this paper we show that $${\gamma_t(G)\le\frac{n}{2}+\max(1,\frac{n}{2(g+1)})}$$, and this bound is sharp.

Discrete Mathematics, 1993

Topp, J. and L. Volkmann, Some upper bounds for the product of the domination number and the chromatic number of a graph, Discrete Mathematics 118 (1993) 2899292. Some new upper bounds for yx are proved, where y is the domination number and x is the chromatic number of a graph. All graphs considered in this note are finite, undirected, and without loops or multiple edges. Our terminology is based on [2]. For a graph G, let n, 6, A,y, and x denote the order, the minimum degree, the maximum degree, the domination number, and the chromatic number of G, respectively. Recently Gernert [S] has obtained the following two inequalities for the product yx of the domination number and the chromatic number of a graph. Theorem A. If G is a connected graph with n > 5, then yx <a n2. Theorem B. If G is a regular graph with n 3 5, then yx <ff n2 , In this note we present some improvements of the above inequalities. The following two lemmas will be useful in our proofs.

Discrete Applied Mathematics, 2013

A subset S of vertices in a graph G = (V , E) is a connected dominating set of G if every vertex of V \S is adjacent to a vertex in S and the subgraph induced by S is connected. The minimum cardinality of a connected dominating set of G is the connected domination number γ c (G). The girth g(G) is the length of a shortest cycle in G. We show that if G is a connected graph that contains at least one cycle, then γ c (G) ≥ g(G) − 2, and we characterize the graphs obtaining equality in this bound. We also establish various upper bounds on the connected domination number of a graph, as well as Nordhaus-Gaddum type results.

2011

The k-domination number of a graph is the cardinality of a smallest set of vertices such that every vertex not in the set is adjacent to at least k vertices of the set. We prove two bounds on the k-domination number of a graph, inspired by two conjectures of the computer program Graffiti.pc. In particular, we show that for any graph with minimum degree at least 2k − 1, the k-domination number is at most the matching number.

Discrete Applied Mathematics, 2017

In a graph G, a set D ⊆ V (G) is called 2-dominating set if each vertex not in D has at least two neighbors in D. The 2-domination number γ 2 (G) is the minimum cardinality of such a set D. We give a method for the construction of 2-dominating sets, which also yields upper bounds on the 2-domination number in terms of the number of vertices, if the minimum degree δ(G) is fixed. These improve the best earlier bounds for any 6 ≤ δ(G) ≤ 21. In particular, we prove that γ 2 (G) is strictly smaller than n/2, if δ(G) ≥ 6. Our proof technique uses a weight-assignment to the vertices where the weights are changed during the procedure.