Integer domination of Cartesian product graphs

Choudhary, Keerti; Margulies, Susan S.; Hicks, Illya V.

doi:10.1016/j.disc.2015.01.032

Cited by 4 publications

(5 citation statements)

References 8 publications

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“…The concept of {2}-domination was introduced in 1991 [26], and considered later on in several papers. In particular, several recent papers consider the variation of Vizing's conjecture on the domination number of Cartesian products of graphs with respect to this domination invariant, see [8,18,57,58].…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

Domination parameters with number 2: Interrelations and algorithmic consequences

Bonomo

Brešar

Grippo

et al. 2018

Discrete Applied Mathematics

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In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak 2-domination number, γ w2 (G), the 2-domination number, γ 2 (G), the {2}-domination number, γ {2} (G), the double domination number, γ ×2 (G), the total {2}-domination number, γ t{2} (G), and the total double domination number, γ t×2 (G), where G is a graph in which a corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, γ R (G), and two classical parameters, the domination number, γ(G), and the total domination number, γ t (G), we consider 13 domination invariants in graphs. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, a large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain new complexity results regarding the existence of approximation algorithms for the studied invariants, matched with tight or almost tight inapproximability bounds, which hold even in the class of split graphs.

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Section: Definitions and Preliminariesmentioning

confidence: 99%

Domination parameters with number 2: Interrelations and algorithmic consequences

Bonomo

Brešar

Grippo

et al. 2018

Discrete Applied Mathematics

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“…as well as for paired [3,6,17], and fractional domination [9], and the {k}-domination function (integer domination) [2,7,18], and total {k}-domination function [18].…”

Section: G = (V E)mentioning

confidence: 99%

Cartesian product graphs and k-tuple total domination

Kazemi¹,

Pahlavsay²,

Stones³

2018

Filomat

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A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S; the minimum size of a kTDS is denoted γ ×k,t (G). We give a Vizing-like inequality for Cartesian product graphs, namelywhere ρ is the packing number. We also give bounds on γ ×k,t (G H) in terms of (open) packing numbers, and consider the extremal case of γ ×k,t (Kn Km), i.e., the rook's graph, giving a constructive proof of a general formula for γ×2,t(Kn Km).

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“…as well as for paired [4,5,17], and fractional domination [8], and the {k}-domination function (integer domination) [3,6,18], and total {k}-domination function [18]. In 1996, Nowakowski and Rall in [23] made the following Vizing-like conjecture for the upper domination of Cartesian products of graphs.…”

Section: Conjecture 1 (Vizing's Conjecture) For Any Graphs G and H mentioning

confidence: 99%

Upper $k$-tuple total domination in graphs

Kazemi

2017

Pure and Applied Mathematics Quarterly

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Let G = (V, E) be a simple graph. For any integer k ≥ 1, a subset of V is called a k-tuple total dominating set of G if every vertex in V has at least k neighbors in the set. The minimum cardinality of a minimal k-tuple total dominating set of G is called the k-tuple total domination number of G. In this paper, we introduce the concept of upper k-tuple total domination number of G as the maximum cardinality of a minimal k-tuple total dominating set of G, and study the problem of finding a minimal k-tuple total dominating set of maximum cardinality on several classes of graphs, as well as finding general bounds and characterizations. Also, we find some results on the upper k-tuple total domination number of the Cartesian and cross product graphs.Keywords: k-tuple total domination number, upper k-tuple total domination number, Cartesian and cross product graphs, hypergraph, (upper) k-transversal number. MSC(2010): 05C69.

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Integer domination of Cartesian product graphs

Cited by 4 publications

References 8 publications

Domination parameters with number 2: Interrelations and algorithmic consequences

Domination parameters with number 2: Interrelations and algorithmic consequences

Cartesian product graphs and k-tuple total domination

Upper $k$-tuple total domination in graphs

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