Some new results on signed edge domination functions

Czechoslovak Mathematical Journal, 2008

The open neighborhood N G (e) of an edge e in a graph G is the set consisting of all edges having a common end-vertex with e.

The Electronic Journal of Combinatorics, 2011

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce 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 $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum de...

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.

Journal of Combinatorial Theory, 2001

Theory and Applications of Graphs, 2016

For any graph G = (V, E), a subset S ⊆ V dominates G if all vertices are contained in the closed neighborhood of S, that is N [S] = V. The minimum cardinality over all such S is called the domination number, written γ(G). In 1963, V.G. Vizing conjectured that γ(G H) ≥ γ(G)γ(H) where stands for the Cartesian product of graphs. In this note, we define classes of graphs An, for n ≥ 0, so that every graph belongs to some such class, and A0 corresponds to class A of Bartsalkin and German. We prove that for any graph G in class A1, γ(G H) ≥ γ(G) − γ(G) γ(H).

Electronic Journal of Graph Theory and Applications, 2016

A nonnegative signed dominating function (NNSDF) of a graph G is a function f from the vertex set V (G) to the set {−1, 1} such that u∈N [v] f (u) ≥ 0 for every vertex v ∈ V (G). The nonnegative signed domination number of G, denoted by γ N N s (G), is the minimum weight of a nonnegative signed dominating function on G. In this paper, we establish some sharp lower bounds on the nonnegative signed domination number of graphs in terms of their order, size and maximum and minimum degree.

International Journal of Computing Algorithm, 2014

A signed graph based on F is an ordinary graph F with each edge marked as positive or negative. Such a graph is called balanced if each of its cycles includes an even number of negative edges. We find the domination set on the vertices, on bipartite graphs and show that graphs has domination Number on signed graphs, such that a signed graph G may be converted into a balanced graph by changing the signs of d edges. We investigate the number D(F) defined as the largest d(G) such that G is a signed graph based on F. If F is the completebipartite graph with t vertices in each part, then D(f)≤ ½ t²-for some positive constant c.

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.

For any graph $G=(V,E)$, a subset $S\subseteq V$ $dominates$ $G$ if all vertices are contained in the closed neighborhood of $S$, that is $N[S]=V$. The minimum cardinality over all such $S$ is called the domination number, written $\gamma(G)$. In 1963, V.G. Vizing conjectured that $\gamma(G \square H) \geq \gamma(G)\gamma(H)$ where $\square$ stands for the Cartesian product of graphs. In this note, we prove that if $\left|G\right|\geq \gamma(G)\gamma(H)$ and $\left|H\right|\geq \gamma(G)\gamma(H)$, then the conjecture holds. This result quickly implies Vizing's conjecture for almost all pairs of graphs $G,H$ with $\left|G\right|\geq \left|H\right|$, satisfying $\left|G\right|\leq q^{\frac{\left|H\right|}{\log_q\left|H\right|}}$ for $q=\frac{1}{1-p}$ and $p$ the edge probability of the Erd\H{o}s-R\'enyi random graph.

2017

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f(x)geq k$ and (ii) every edge $e$for which $f(e)=-1$ is adjacent to at least one edge $e'$ forwhich $f(e')=2$. The minimum of the values $sum_{ein E}f(e)$,taken over all signed Roman edge $k$-dominating functions $f$ of$G$, is called the signed Roman edge $k$-domination number of $G$and is denoted by $gamma'_{sRk}(G)$. In this paper we establish some new bounds on the signed Roman edge $k$-domination number.

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.

2011

v∈V f(v). The complementary signed domination number of G is defined as γcs(G) = min {w(f) : f is a minimal complementary signed dominating function of G}. 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.

arXiv: Combinatorics, 2019

In a graph $G$, a vertex dominates itself and its neighbors. A subset $D \subseteq V(G)$ is a double dominating set of $G$ if $D$ dominates every vertex of $G$ at least twice. A signed graph $\Sigma = (G,\sigma)$ is a graph $G$ together with an assignment $\sigma$ of positive or negative signs to all its edges. A cycle in a signed graph is positive if the product of its edge signs is positive. A signed graph is balanced if all its cycles are positive. A subset $D \subseteq V(\Sigma)$ is a double dominating set of $\Sigma$ if it satisfies the following conditions: (i) $D$ is a double dominating set of $G$, and (ii) $\Sigma[D:V \setminus D]$ is balanced, where $\Sigma[D:V \setminus D]$ is the subgraph of $\Sigma$ induced by the edges of $\Sigma$ with one end point in $D$ and the other end point in $V \setminus D$. The cardinality of a minimum double dominating set of $\Sigma$ is the double domination number $\gamma_{\times 2}(\Sigma)$. In this paper, we give bounds for the double domi...

Opuscula Mathematica, 2016

Let G = (V, E) be a simple graph. A function f : V → {−1, 1} is called an inverse signed total dominating function if the sum of its function values over any open neighborhood is at most zero. The inverse signed total domination number of G, denoted by γ 0 st (G), equals to the maximum weight of an inverse signed total dominating function of G. In this paper, we establish upper bounds on the inverse signed total domination number of graphs in terms of their order, size and maximum and minimum degrees.

Let G be a graph with vertex set V(G). A function f:V(G)→{-1,1} is a signed dominating function of G if, for each vertex of G, the sum of the values of its neighbors and itself is positive. The signed domination number of a graph G, denoted γ s (G), is the minimum value of ∑ u∈V(G) f(v) over all the signed dominating functions f of G. The signed reinforcement number of G, denoted R s (G), is defined to be the minimum cardinality |S| of a set S of edges such that γ s (G+S)<γ s (G). In this paper, we initialize the study of signed reinforcement number and determine the exact values of R s (G) for several classes of graphs.