Graffiti. pc on the 2-domination number of a graph

Graphs and Combinatorics, 2020

, gave an upper bound for the k-tuple domination number of a graph. Rad (J Combin Math Comb Comput, 2019, in press) presented an improvement of the above bound using the Caro-Wei Theorem. In this paper, using the well-known Brooks' Theorem for vertex coloring and vertex covers, we improve the above bounds on the k-tuple domination number under some certain conditions. In the special case k ¼ 1, we improve the upper bounds for the domination number (

A vertex v in a graph G dominates itself as well as its neighbors. A set S of vertices in G is (1) a dominating set if every vertex of G is dominated by some vertex of S, (2) an open dominating set if every vertex of G is dominated by a vertex of S distinct from itself, and (3) a double dominating set if every vertex of G is dominated by at least two distinct vertices of S. The minimum cardinality of a set S satisfying (1), (2), and (3), respectively, is the domination number, open domination number, and double domination number of G, respectively. We consider the problem of determining the maximum value of each of these domination numbers among all graphs of a given order and size.

Mathematics

A nonempty subset D⊂V of vertices of a graph G=(V,E) is a dominating set if every vertex of this graph is adjacent to at least one vertex from this set except the vertices which belong to this set itself. D⊆V is a total k-dominating set if there are at least k vertices in set D adjacent to every vertex v∈V, and it is a global total k-dominating set if D is a total k-dominating set of both G and G¯. The global total k-domination number of G, denoted by γktg(G), is the minimum cardinality of a global total k-dominating set of G, GTkD-set. Here we derive upper and lower bounds of γktg(G), and develop a method that generates a GTkD-set from a GT(k−1)D-set for the successively increasing values of k. Based on this method, we establish a relationship between γ(k−1)tg(G) and γktg(G), which, in turn, provides another upper bound on γktg(G).

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.

Discrete Mathematics, 2009

A set S of vertices in a graph G is a dominating set of G if every vertex of V (G)\S is adjacent to some vertex in S. The minimum cardinality of a dominating set of G is the domination number of G, denoted as γ (G). Let P n and C n denote a path and a cycle, respectively, on n vertices. Let k 1 (F) and k 2 (F) denote the number of components of a graph F that are isomorphic to a graph in the family {P 3 , P 4 , P 5 , C 5 } and {P 1 , P 2 }, respectively. Let L be the set of vertices of G of degree more than 2, and let G − L be the graph obtained from G by deleting the vertices in L and all edges incident with L. McCuaig and Shepherd [W. McCuaig, B. Shepherd, Domination in graphs with minimum degree two, J. Graph Theory 13 (1989) 749-762] showed that if G is a connected graph of order n ≥ 8 with δ(G) ≥ 2, then γ (G) ≤ 2n/5, while Reed [B.A. Reed, Paths, stars and the number three, Combin. Probab. Comput. 5 (1996) 277-295] showed that if G is a graph of order n with δ(G) ≥ 3, then γ (G) ≤ 3n/8. As an application of Reed's result, we show that if G is a graph of order n ≥ 14 with δ(G) ≥ 2, then γ (G) ≤ 3 8 n + 1 8 k 1 (G − L) + 1 4 k 2 (G − L).

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 δ k (n) = max{δ | γ(n, δ) ≥ k}, k ≥ 1. Our main result is that for any fixed k ≥ 2 there is a constant c k such that for sufficiently large n, n − c k n (k−1)/k ≤ δ k+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, δ k (n)) = k. The case 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 Z n .

Discrete Mathematics

The k-restricted domination number of a graph G is the minimum number d k such that for any subset U of k vertices of G, there is a dominating set in G including U and having at most d k vertices. Some new upper bounds in terms of order and degrees for this number are found.

Mathematica Slovaca, 2007

The k-restricted domination number of a graph G is the minimum number d k such that for any subset U of k vertices of G, there is a dominating set in G including U and having at most d k vertices. Some new upper bounds in terms of order and degrees for this number are found.

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.