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Secure Italian domination in graphs

Abstract

An Italian dominating function (IDF) on a graph G is a function f:V(G)→{0,1,2} such that for every vertex v with f(v)=0, the total weight of f assigned to the neighbours of v is at least two, i.e., ∑u∈NG(v)f(u)≥2. For any function f:V(G)→{0,1,2} and any pair of adjacent vertices with f(v)=0 and u with 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) whenever x∈V(G)∖{u,v}. A secure Italian dominating function on a graph G is defined as an IDF f which satisfies that for every vertex v with f(v)=0, there exists a neighbour u with f(u)>0 such that fu→v is an IDF. The weight of f is ω(f)=∑v∈V(G)f(v). The minimum weight among all secure Italian dominating functions on G is the secure Italian domination number of G. This paper is devoted to initiating the study of the secure Italian domination number of a graph. In particular, we prove that the problem of finding this parameter is NP-hard and we obtain general bounds on it. Moreover, for certain classes of graphs, we obtain closed formulas for this novel parameter.

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Details

Category:
Articles
Type:
artykuły w czasopismach
Published in:
JOURNAL OF COMBINATORIAL OPTIMIZATION no. 41, pages 56 - 72,
ISSN: 1382-6905
Language:
English
Publication year:
2021
Bibliographic description:
Dettlaff M., Lemańska M., Rodríguez-Velázquez J.: Secure Italian domination in graphs// JOURNAL OF COMBINATORIAL OPTIMIZATION -Vol. 41, (2021), s.56-72
DOI:
Digital Object Identifier (open in new tab) 10.1007/s10878-020-00658-1
Verified by:
Gdańsk University of Technology

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