Twin signed k-domination numbers in directed graphs

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.

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.

2010

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

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.

Discussiones Mathematicae Graph Theory, 2017

Let D = (V, A) be a finite simple directed graph (shortly, digraph). A function f : V −→ {−1, 0, 1} is called a twin minus total dominating function (TMTDF) if f (N − (v)) ≥ 1 and f (N + (v)) ≥ 1 for each vertex v ∈ V. The twin minus total domination number of D is γ * mt (D) = min{w(f) | f is a TMTDF of D}. In this paper, we initiate the study of twin minus total domination numbers in digraphs and we present some lower bounds for γ * mt (D) in terms of the order, size and maximum and minimum in-degrees and out-degrees. In addition, we determine the twin minus total domination numbers of some classes of digraphs.

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.

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.

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.