On the signed edge domination number of graphs

2011

In this paper, we determine the value of complementary signed domination number for some special class of graphs. We also determine bounds for this parameter and exhibit the sharpness of the bounds. We also characterize graphs attaining the bounds in some special classes.

2017

A b s t r ac t. Let D = (V, A) be a finite simple directed graph (shortly digraph). A function f : V −→ {−1, 1} is called a twin signed dominating function (TSDF) if f (N − [v]) 1 and f (N + [v]) 1 for each vertex v ∈ V. The twin signed domination number of D is γ * s (D) = min{ω(f) | f is a TSDF of D}. In this paper, we initiate the study of twin signed domination in digraphs and we present sharp lower bounds for γ * s (D) in terms of the order, size and maximum and minimum indegrees and outdegrees. Some of our results are extensions of well-known lower bounds of the classical signed domination numbers of graphs.

Journal of Mathematics

Let G = V , E be a graph. A function f : E ⟶ − 1 , + 1 is said to be a signed clique dominating function (SCDF) of G if ∑ e ∈ E K f e ≥ 1 holds for every nontrivial clique K in G . The signed clique domination number of G is defined as γ scl ′ G = min ∑ e ∈ E G f e | f is an SCDF of G . In this paper, we investigate the signed clique domination numbers of join of graphs. We correct two wrong results reported by Ao et al. (2014) and Ao et al. (2015) and determine the exact values of the signed clique domination numbers of P m ∨ K n ¯ and C m ∨ K n .

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.

Electronic Notes in Discrete Mathematics, 2002

Let G = (V , E) be a simple graph on vertex set V and define a function f : V → {−1, 1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γ s (G), is the minimum weight of a signed dominating function on G. We give a sharp lower bound on the signed domination number of a general graph with a given minimum and maximum degree, generalizing a number of previously known results. Using similar techniques we give upper and lower bounds for the signed domination number of some simple graph products: the grid P j × P k , C j × P k and C j × C k . For fixed width, these bounds differ by only a constant.

Algebra and discrete mathematics, 2017

Let D = (V, A) be a finite simple directed graph (digraph). A function f : V −→ {−1, 1} is called a twin signed k-dominating function (TSkDF) if f (N − [v]) ≥ k and f (N + [v]) ≥ k for each vertex v ∈ V. The twin signed k-domination number of D is γ * sk (D) = min{ω(f) | f is a TSkDF of D}. In this paper, we initiate the study of twin signed k-domination in digraphs and present some bounds on γ * sk (D) in terms of the order, size and maximum and minimum indegrees and outdegrees, generalising some of the existing bounds for the twin signed domination numbers in digraphs and the signed k-domination numbers in graphs. In addition, we determine the twin signed k-domination numbers of some classes of digraphs.

Discrete Applied Mathematics, 2009

a b s t r a c t Let G = (V, E) be a graph. A function f : V → {−1, +1} defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. A signed total dominating function f is minimal if there does not exist a signed total dominating function

Discrete Mathematics, 2009

A numerical invariant of directed graphs concerning domination which is named signed domination number γ S is studied in this paper. We present some sharp lower bounds for γ S in terms of the order, the maximum degree and the chromatic number of a directed graph.

Discrete Mathematics, 2004

Let G = (V, E) be a simple graph on vertex set V and define a function f : V → {−1, 1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γ s (G), is the minimum weight of a signed dominating function on G.

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).