Domination parameters with number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml720" display="inline" overflow="scroll" altimg="si482.gif"><mml:mn>2</mml:mn></mml:math>: Interrelations and algorithmic consequences

Bonomo, Flavia; Brešar, Boštjan; Grippo, Luciano N.; Milanič, Martin; Safe, Martín D.

doi:10.1016/j.dam.2017.08.017

Cited by 16 publications

(8 citation statements)

References 63 publications

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“…Next we derive new bounds on the double total domination number. The first result improves the bound γ ×2,t (G) ≥ γ t (G) + 1 (see [4]) whenever diam(G) ≥ 5.…”

Section: Theorem 6 the Following Statements Hold For Any Graph G Withsupporting

confidence: 53%

“…We suggest the books [12,15] in case the reader is not familiar with another some basic concepts, notation and terminology of graphs. The following upper bounds for the double total domination number were established by Bermudo et al in [1] and independently by Bonomo et al in [4].…”

Section: Proposition 1 the Following Inequality Chains Hold For Any Graph G Withmentioning

confidence: 94%

“…The double total domination number has been well studied by the research community. For instance, in [1,4,17,19,21,22], some combinatorial results were presented, and in [2,3,18,23], some studies on certain product graphs were carried out. Our goal is to making some new remarkable contributions to this domination parameter.…”

Section: Introductionmentioning

confidence: 99%

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New Bounds on the Double Total Domination Number of Graphs

Martínez

Hernández-Mira

2021

Bull. Malays. Math. Sci. Soc.

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Let G be a graph of minimum degree at least two. A set $$D\subseteq V(G)$$ D ⊆ V ( G ) is said to be a double total dominating set of G if $$|N(v)\cap D|\ge 2$$ | N ( v ) ∩ D | ≥ 2 for every vertex $$v\in V(G)$$ v ∈ V ( G ) . The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.

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“…Next we derive new bounds on the double total domination number. The first result improves the bound γ ×2,t (G) ≥ γ t (G) + 1 (see [4]) whenever diam(G) ≥ 5.…”

Section: Theorem 6 the Following Statements Hold For Any Graph G Withsupporting

confidence: 53%

Section: Proposition 1 the Following Inequality Chains Hold For Any Graph G Withmentioning

confidence: 94%

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New Bounds on the Double Total Domination Number of Graphs

Martínez

Hernández-Mira

2021

Bull. Malays. Math. Sci. Soc.

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“…Moreover, strong inapproximability results are known for these variants of domination in the class of split graphs: for every > 0, the dominating set, the total dominating set, and the connected dominating set problem cannot be approximated to within a factor on (1 − ) ln n in the class of n-vertex split graphs, unless P = NP. Theorem 6.5 in [8] provides this result for domination and total domination, while for connected domination the same result follows from the fact that all the three problems are equivalent in any class of split graphs (see Lemma 7.2).…”

mentioning

confidence: 66%

Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1‐Sperner hypergraphs

Boros

Gurvich

Milanič

2019

Journal of Graph Theory

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A hypergraph is said to be 1-Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of 1-Sperner hypergraphs and their structure to graphs. In particular, we consider the classical characterizations of threshold and domishold graphs and use them to obtain further characterizations of these classes in terms of 1-Spernerness, thresholdness, and 2-asummability of their vertex cover, clique, dominating set, and closed neighborhood hypergraphs. Furthermore, we apply a decomposition property of 1-Sperner hypergraphs to derive decomposition theorems for two classes of split graphs, a class of bipartite graphs, and a class of cobipartite graphs. These decomposition theorems are based on certain matrix partitions of the corresponding graphs, giving rise to new classes of graphs of bounded clique-width and new polynomially solvable cases of several domination problems. IntroductionHypergraphs are one of the most fundamental and general combinatorial objects, encompassing various important structures such as graphs, matroids, and combinatorial designs. Results for more specific discrete structures (e.g., graphs) can often be proved more generally, in the context of suitable classes of hypergraphs (see, e.g., Schrijver [57]). Of course, applications of hypergraph theory to structural and optimization aspects of other discrete structures are not restricted to the above phenomenon. Since it is impossible to survey here all kinds of applications of hypergraphs, let us mention only a few more. First, recent work of Hujdurović et al. [37] made use of connections between hypergraphs and binary matrices, acyclic digraphs, and partially ordered sets to study two combinatorial optimization problems motivated by computational biology. Second, a common approach of applying hypergraph theory to graphs is to define and study a hypergraph derived in an appropriate way from a given graph, depending on what type of property of a graph or its vertex or edge subsets one is interested in. This includes matching hypergraphs [1,63], various clique [32,41,47,51,63], independent set [32,63], neighborhood [12,27], separator [13,60], and dominating set hypergraphs [12,13], etc. Close interrelations between hypergraphs and monotone Boolean functions can be useful in such studies, 1 arXiv:1805.03405v3 [math.CO] 29 May 2018 allowing for the transfer and applications of results from the theory of Boolean functions (see, e.g., [21]).In this work, we present several new applications of hypergraphs to graphs. Our starting point is [9] where the class of 1-Sperner hypergraphs was studied. It was shown that such hypergraphs can be decomposed in a particular way and that they form a subclass of the class of threshold hypergraphs studied by Golumbic [31], by Reiterman et al. [56] and, in the equivalent context of threshold monotone Boolean functions, also by Muroga [48] and by Peled and Simeone [52]. (Precise definitions of all relevant concepts will be given in Section 2.)While threshold h...

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“…Bonomo et al [9] obtained the following relationship between the double domination number and the 2-domination number.…”

Section: Upper Bounds On the Double Domination Numbermentioning

confidence: 95%