Secure domination in cographs

Graph Theory International audience A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.

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Some results on secure domination in zero-divisor graphs

Journal of Mathematical and Computational Science ◽

10.28919/jmcs/5904 ◽

2021 ◽

Keyword(s):

Zero Divisor ◽

Secure Domination

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Equalityof Secure Domination and Inverse Secure Domination Numbers

Journal of Ultra Scientist of Physical Sciences Section A ◽

10.22147/jusps-a/280601 ◽

2016 ◽

Vol 28 (06) ◽

pp. 294-298 ◽

Cited By ~ 1

Author(s):

V.R. KULLI ◽

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Keyword(s):

Secure Domination ◽

Domination Numbers

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The secure domination problem in cographs

Information Processing Letters ◽

10.1016/j.ipl.2019.01.005 ◽

2019 ◽

Vol 145 ◽

pp. 30-38 ◽

Cited By ~ 2

Author(s):

Anupriya Jha ◽

D. Pradhan ◽

S. Banerjee

Keyword(s):

Domination Problem ◽

Secure Domination

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Secure w-Domination in Graphs

Symmetry ◽

10.3390/sym12121948 ◽

2020 ◽

Vol 12 (12) ◽

pp. 1948

Author(s):

Abel Cabrera Martínez ◽

Alejandro Estrada-Moreno ◽

Juan A. Rodríguez-Velázquez

Keyword(s):

Open Neighborhood ◽

Minimum Weight ◽

Domination Number ◽

Domination In Graphs ◽

Secure Domination ◽

Dominating Function

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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Inverse Perfect Secure Domination in Graphs

International Journal of Mathematics Trends and Technology ◽

10.14445/22315373/ijmtt-v67i8p517 ◽

2021 ◽

Vol 67 (8) ◽

pp. 150-156

Author(s):

Cristina S. Castañares ◽

Enrico L. Enriquez

Keyword(s):

Domination In Graphs ◽

Secure Domination

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