TY - JOUR

T1 - Roman {3}-domination in graphs

T2 - Complexity and algorithms

AU - Chaudhary, Juhi

AU - Pradhan, Dinabandhu

N1 - Funding Information:
Supported by MAThematical Research Impact Centric Support project (MTR/2018/000017), SERB, India.
Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - 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.

AB - 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.

KW - -completeness

KW - Domination

KW - Roman domination

KW - Roman {2}-domination (Italian domination)

KW - Roman {3}-domination (double Italian domination)

UR - http://www.scopus.com/inward/record.url?scp=85139704656&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2022.09.017

DO - 10.1016/j.dam.2022.09.017

M3 - Article

AN - SCOPUS:85139704656

SN - 0166-218X

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

ER -