Twin minus total domination numbers in directed graphs

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.

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.

RAIRO - Operations Research, 2017

Let D = (V, A) be a finite simple directed graph (shortly digraph), N − (v) and N + (v) denote the set of in-neighbors and out-neighbors of a vertex v ∈ V , respectively. A function f : V −→ {−1, 1} is called a twin signed total k-dominating function (TSTkDF) if u∈(N − (v)) f (u) ≥ k and u∈(N + (v)) f (u) ≥ k for each vertex v ∈ V. The twin signed total k-domination number of D is γ * stk (D) = min{ω(f) | f is a TSTkDF of D}, where ω(f) = v∈V f (v) is the weight of f. In this paper, we initiate the study of twin signed total k-domination in digraphs and present different bounds on γ * stk (D). In addition, we determine the twin signed total k-domination number of some classes of digraphs. Our results are mostly extensions of well-known bounds of the twin signed total domination numbers of directed graphs.

Filomat, 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 ?* sk(D) = min{?(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 ?* 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 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.

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.

2010

Let ≥ 1 be an integer, and let D = (V A) be a finite simple digraph, for which −

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.

Miskolc Mathematical Notes, 2016

Let D be a finite and simple digraph with vertex set V .D/. A twin Roman dominating function (TRDF) on D is a labeling f W V .D/ ! f0; 1; 2g such that every vertex with label 0 has an in-neighbor and out-neighbor with label 2. The weight of a TRDF f is the value !.f / D P v2V .D/ f .v/. The twin Roman domination number of a digraph D, denoted by R .D/, equals the minimum weight of a TRDF on D. In this paper we initiate the study of the twin Roman domination number in digraphs. In particular, we present sharp bounds for R .D/ and determine the exact value of the twin Roman domination number for some classes of digraphs.

Discrete Applied Mathematics, 2009

Let D be a digraph. By γ (D) we denote the domintaion number of D and by D − we denote a digraph obtained by reversing all the arcs of D. In this paper we prove that for every δ ≥ 3 and k ≥ 1 there exists a simple strongly connected δ-regular digraph D δ,k such that γ (D − δ,k) − γ (D δ,k) = k. Analogous theorem is obtained for total domination number provided that δ ≥ 4.