Perfect Italian domination in graphs: Complexity and algorithms

An Italian dominating function on a simple undirected graph G is a function f:V(G)⟶{0,1,2} satisfying the condition that for each vertex v with f(v)=0, ∑u∈NG(v)f(u)≥2. An Italian dominating function f on G is called a perfect Italian dominating function on G if for each vertex v with f(v)=0, ∑u∈NG(v...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 319; pp. 271 - 295
Main Authors: Pradhan, D., Banerjee, S., Liu, Jia-Bao
Format: Journal Article
Language:English
Published: Elsevier B.V 15.10.2022
Subjects:
Domination
Italian domination
NP-complete
Perfect Italian domination
Polynomial time algorithm
Domination
NP-complete
Italian domination
Polynomial time algorithm
Perfect Italian domination
ISSN:0166-218X, 1872-6771
Online Access:Get full text
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Summary:An Italian dominating function on a simple undirected graph G is a function f:V(G)⟶{0,1,2} satisfying the condition that for each vertex v with f(v)=0, ∑u∈NG(v)f(u)≥2. An Italian dominating function f on G is called a perfect Italian dominating function on G if for each vertex v with f(v)=0, ∑u∈NG(v)f(u)=2. The weight of a function f on a graph G, denoted by w(f), is the sum ∑v∈V(G)f(v). For a simple undirected graph G, Min-PIDF is the problem of finding the minimum weight of a perfect Italian dominating function on G. First, we discuss the complexity difference between Min-PIDF and the problem of finding the minimum weight of an Italian dominating function. We then establish the NP-completeness of the decision version of Min-PIDF in chordal graphs and investigate the hardness of approximation of Min-PIDF in general graphs. Finally, we present linear time algorithms for computing the minimum weight of a perfect Italian dominating function in block graphs and series-parallel graphs.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2021.08.020