# Roman {3}-domination in graphs: Complexity and algorithms

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## Abstract

A Roman {3}-dominating function on a graph G is a function f:V(G)→{0,1,2,3} having the property that for any vertex u∈V(G), if f(u)=0, then ∑v∈NG(u)f(v)≥3, and if f(u)=1, then ∑v∈NG(u)f(v)≥2. The weight of a Roman {3}-dominating function f is the sum f(V(G))=∑v∈V(G)f(v) and the minimum weight of a Roman {3}-dominating function on G is called the Roman {3}-domination number of G and is denoted by γ{R3}(G). Given a graph G, ROMAN{3}-DOMINATION asks to find the minimum weight of a Roman {3}-dominating function on G. In this paper, we study the algorithmic aspects of ROMAN{3}-DOMINATION on various graph classes. We show that the decision version of ROMAN{3}-DOMINATION remains NP-complete for chordal bipartite graphs, planar graphs, star-convex bipartite graphs, and chordal graphs. We show that ROMAN{3}-DOMINATION cannot be approximated within a ratio of [Formula presented] for any ɛ>0 unless P=NP for bipartite graphs as well as chordal graphs, whereas ROMAN{3}-DOMINATION can be approximated within a factor of O(lnΔ) for graphs having maximum degree Δ. We also show that ROMAN{3}-DOMINATION is APX-complete for graphs with maximum degree 4. On the positive side, we show that ROMAN{3}-DOMINATION can be solved in linear time for chain graphs and cographs.

Original language English Discrete Applied Mathematics https://doi.org/10.1016/j.dam.2022.09.017 Accepted/In press - 1 Jan 2022

## Keywords

• -completeness
• Domination
• Roman domination
• Roman {2}-domination (Italian domination)
• Roman {3}-domination (double Italian domination)

## ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics
• Applied Mathematics

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