Twin signed domination numbers in directed graphs

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.

Journal of Combinatorial Optimization, 2019

Let k ≥ 1 be an integer and let D be a digraph with vertex set V (D). A subset S ⊆ V (D) is called a k-dominating set if every vertex not in S has at least k predecessors in S. The k-domination number γ k (D) of D is the minimum cardinality of a k-dominating set in D. We know that for any digraph D of order n, γ k (D) ≤ n. Obviously the upper bound n is sharp for a digraph with maximum in-degree at most k − 1. In this paper we present some lower and upper bounds on γ k (D). Also, we characterize digraphs achieving these bounds. The special case k = 1 mostly leads to well known classical results.

Let D be a finite and simple digraph with vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N − [v] f (x) ≥ k for each v ∈ V (D), where N − [v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set { f 1 , f 2 ,. .. , f d } of distinct signed k-dominating functions of D with the property that ∑ d i=1 f i (v) ≤ 1 for each v ∈ V (D), is called a signed k-dominating family (of functions) of D. The maximum number of functions in a signed k-dominating family of D is the signed k-domatic number of D, denoted by d kS (D). In this note we initiate the study of the signed k-domatic numbers of digraphs and present some sharp upper bounds for this parameter.

Let D be a finite and simple digraph with vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and x∈N − (v) f (x) ≥ k for each v ∈ V (D), where N − (v) consists of all vertices of D from which arcs go into v, then f is a signed total k-dominating function on D. A set {f 1 , f 2 ,. .. , f d } of distinct signed total k-dominating functions of D with the property that d i=1 f i (v) ≤ 1, for each v ∈ V (D), is called a signed total k-dominating family (of functions) of D. The maximum number of functions in a signed total k-dominating family of D is the signed total k-domatic number of D, denoted by d t kS (D). In this note we initiate the study of the signed total k-domatic numbers of digraphs and present some sharp upper bounds for this parameter.

Discrete Mathematics, 1999

An out-domination set of a digraph D is a set S of vertices of D such that every vertex of D-S is adjacent from some vertex of S. The minimum cardinality of an out-domination set of D is thc out-domination number 7+(D). The in-domination number 7 (D) is defined analogously. It is shown that for every digraph D of order n with no isolates, 7-(D)+'/ (D)<~4n/3. Furthermore, the digraphs D for which equality holds are characterized. Other inequalities are also derived.

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.

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.

Opuscula Mathematica

Let D = (V, A) be a finite simple digraph and N (uv) = {u v = uv | u = u or v = v } be the open neighbourhood of uv in D. A function f : A → {−1, +1} is said to be a signed arc total dominating function (SATDF) of D if e ∈N (uv) f (e) ≥ 1 holds for every arc uv ∈ A. The signed arc total domination number γ st (D) is defined as γ st (D) = min{ e∈A f (e) | f is an SATDF of D}. In this paper we initiate the study of the signed arc total domination in digraphs and present some lower bounds for this parameter.

Tamkang Journal of Mathematics, 2016

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arc set $A(D)$. A twin signed Roman dominating function (TSRDF) on the digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the conditions that (i) $\sum_{x\in N^-[v]}f(x)\ge 1$ and $\sum_{x\in N^+[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ (resp. $N^+[v]$) consists of $v$ and all in-neighbors (resp. out-neighbors) of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ has an in-neighbor $v$ and an out-neighbor $w$ for which $f(v)=f(w)=2$. The weight of an TSRDF $f$ is $\omega(f)=\sum_{v\in V(D)}f(v)$. The twin signed Roman domination number $\gamma_{sR}^*(D)$ of $D$ is the minimum weight of an TSRDF on $D$. In this paper, we initiate the study of twin signed Roman domination in digraphs and we present some sharp bounds on $\gamma_{sR}^*(D)$. In addition, we determine the twin signed Roman domination number of some classes of digraphs.

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.

Graphs and Combinatorics, 2008

The closed neighborhood N G [e] of an edge e in a graph G is the set consisting of e and of all edges having an end-vertex in common with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If $\sum_{x\in N[e]}f(x) \geq 1$ for each e ∈ E(G), then f is called a signed edge dominating function of G. The signed edge domination number γ s ′(G) of G is defined as $\gamma_s^\prime(G) = {\text{min}}\{\sum_{e\in E(G)}f(e)\mid f \,\text{is an SEDF of} G\}$ . Recently, Xu proved that γ s ′(G) ≥ |V(G)| − |E(G)| for all graphs G without isolated vertices. In this paper we first characterize all simple connected graphs G for which γ s ′(G) = |V(G)| − |E(G)|. This answers Problem 4.2 of [4]. Then we classify all simple connected graphs G with precisely k cycles and γ s ′(G) = 1 − k, 2 − k.

Discrete Mathematics, 2009

Let γ ′ s (G) be the signed edge domination number of G. In 2006, Xu conjectured that: for any 2-connected graph G of order n(n ≥ 2), γ ′ s (G) ≥ 1. In this article we show that this conjecture is not true. More precisely, we show that for any positive integer m, there exists an m-connected graph G such that γ ′ s (G) ≤ − m 6 |V (G)|. Also for every two natural numbers m and n, we determine γ ′ s (K m,n ), where K m,n is the complete bipartite graph with part sizes m and n. *

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.

Journal of Applied Mathematics and Physics, 2015

This paper is motivated by the concept of the signed k-domination problem and dedicated to the complexity of the problem on graphs. For any fixed nonnegative integer k, we show that the signed k-domination problem is NP-complete for doubly chordal graphs. For strongly chordal graphs and distance-hereditary graphs, we show that the signed k-domination problem can be solved in polynomial time. We also show that the problem is linear-time solvable for trees, interval graphs, and chordal comparability graphs.

Annals of the University of Craiova - Mathematics and Computer Science Series

Let D = (V, A) be a finite simple digraph. A signed double Roman dominating function (SDRD-function) on the digraph D is a function f : V (D) → {−1, 1, 2, 3} satisfying the following conditions: (i) x∈N − [v] f (x) ≥ 1 for each v ∈ V (D), where N − [v] consist of v and all in-neighbors of v, and (ii) if f (v) = −1, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor assigned 3, while if f (v) = 1, then the vertex v must have at least one in-neighbor assigned 2 or 3. The weight of a SDRD-function f is the value x∈V (D) f (x). The signed double Roman domination number (SDRD-number) γ sdR (D) of a digraph D is the minimum weight of a SDRD-function on D. In this paper we study the SDRD-number of digraphs, and we present lower and upper bounds for γ sdR (D) in terms of the order, maximum degree and chromatic number of a digraph. In addition, we determine the SDRD-number of some classes of digraphs.