Protection of lexicographic product graphs

Klein, Douglas J.; Rodríguez-Velázquez, Juan A.

doi:10.7151/dmgt.2243

Cited by 5 publications

(3 citation statements)

References 21 publications

(39 reference statements)

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“…For a basic introduction to the lexicographic product of two graphs we suggest the books [11,13]. One of the main problems in the study of G • H consists of finding exact values or tight bounds of specific parameters of these graphs and express them in terms of known invariants of G and H. In particular, we cite the following works on domination theory of lexicographic product graphs: (total) domination [18,21,28], Roman domination [25], weak Roman domination [27], rainbow domination [26], super domination [8], doubly connected domination [1], secure domination [15], double domination [3] and total Roman domination [6,4].…”

Section: The Case Of Lexicographic Product Graphsmentioning

confidence: 99%

On the 2-Packing Differential of a Graph

2021

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Let G be a graph of order $${\text {n}}(G)$$ n ( G ) and vertex set V(G). Given a set $$S\subseteq V(G)$$ S ⊆ V ( G ) , we define the external neighbourhood of S as the set $$N_e(S)$$ N e ( S ) of all vertices in $$V(G){\setminus } S$$ V ( G ) \ S having at least one neighbour in S. The differential of S is defined to be $$\partial (S)=|N_e(S)|-|S|$$ ∂ ( S ) = | N e ( S ) | - | S | . In this paper, we introduce the study of the 2-packing differential of a graph, which we define as $$\partial _{2p}(G)=\max \{\partial (S):\, S\subseteq V(G) \text { is a }2\text {-packing}\}.$$ ∂ 2 p ( G ) = max { ∂ ( S ) : S ⊆ V ( G ) is a 2 -packing } . We show that the 2-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of 2-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions. Among other results, we obtain a Gallai-type theorem, which states that $$\partial _{2p}(G)+\mu _{_R}(G)={\text {n}}(G)$$ ∂ 2 p ( G ) + μ R ( G ) = n ( G ) , where $$\mu _{_R}(G)$$ μ R ( G ) denotes the unique response Roman domination number of G. As a consequence of the study, we derive several combinatorial results on $$\mu _{_R}(G)$$ μ R ( G ) , and we show that the problem of finding this parameter is NP-hard. In addition, the particular case of lexicographic product graphs is discussed.

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Section: The Case Of Lexicographic Product Graphsmentioning

confidence: 99%

On the 2-Packing Differential of a Graph

2021

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“…For a basic introduction to the lexicographic product of two graphs we suggest the books [7,12]. One of the main problems in the study of G • H consists of finding exact values or tight bounds for specific parameters of these graphs and express them in terms of known invariants of G and H. In particular, we cite the following works on domination theory of lexicographic product graphs: (total) domination [16,17,18,21], Roman domination [18], weak Roman domination [20], rainbow domination [19], super domination [6], doubly connected domination [2], secure domination [13], double domination [4] and total Roman domination [5].…”

Section: Introductionmentioning

confidence: 99%

Closed formulas for the total Roman domination number of lexicographic product graphs

Martínez

Rodríguez-Velázquez

2021

Ars Math. Contemp.

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Let G be a graph with no isolated vertex and f : V (G) → {0, 1, 2} a function. Let V i = {x ∈ V (G) : f (x) = i} for every i ∈ {0, 1, 2}. We say that f is a total Roman dominating function on G if every vertex in V 0 is adjacent to at least one vertex in V 2 and the subgraph induced by V 1 ∪ V 2 has no isolated vertex. The weight of f is ω(f) = v∈V (G) f (v). The minimum weight among all total Roman dominating functions on G is the total Roman domination number of G, denoted by γ tR (G). It is known that the general problem of computing γ tR (G) is NP-hard. In this paper, we show that if G is a graph with no isolated vertex and H is a nontrivial graph, then the total Roman domination number of the lexicographic product graph G • H is given by γ tR (G • H) = 2γ t (G) if γ(H) ≥ 2, ξ(G) if γ(H) = 1, where γ(H) is the domination number of H, γ t (G) is the total domination number of G and ξ(G) is a domination parameter defined on G.

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“…In this case, for any secure (1, 0)-dominating function f (V 0 , V 1 ), the set V 1 is known as a secure dominating set. This concept was introduced by Cockayne et al [2] and studied further in several papers (e.g., [3][4][5][6][7][8][9]).…”

Section: Introductionmentioning

confidence: 99%

Secure w-Domination in Graphs

2020

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This paper introduces a general approach to the idea of protection of graphs, which encompasses the known variants of secure domination and introduces new ones. Specifically, we introduce the study of secure w-domination in graphs, where w=(w0,w1,…,wl) is a vector of nonnegative integers such that w0≥1. The secure w-domination number is defined as follows. Let G be a graph and N(v) the open neighborhood of v∈V(G). We say that a function f:V(G)⟶{0,1,…,l} is a w-dominating function if f(N(v))=∑u∈N(v)f(u)≥wi for every vertex v with f(v)=i. The weight of f is defined to be ω(f)=∑v∈V(G)f(v). Given a w-dominating function f and any pair of adjacent vertices v,u∈V(G) with f(v)=0 and f(u)>0, the function fu→v is defined by fu→v(v)=1, fu→v(u)=f(u)−1 and fu→v(x)=f(x) for every x∈V(G)\{u,v}. We say that a w-dominating function f is a secure w-dominating function if for every v with f(v)=0, there exists u∈N(v) such that f(u)>0 and fu→v is a w-dominating function as well. The secure w-domination number of G, denoted by γws(G), is the minimum weight among all secure w-dominating functions. This paper provides fundamental results on γws(G) and raises the challenge of conducting a detailed study of the topic.

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Protection of lexicographic product graphs

Cited by 5 publications

References 21 publications

On the 2-Packing Differential of a Graph

On the 2-Packing Differential of a Graph

Closed formulas for the total Roman domination number of lexicographic product graphs

Secure w-Domination in Graphs

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