TY - JOUR

T1 - Unique Response Roman Domination

T2 - Complexity and Algorithms

AU - Banerjee, Sumanta

AU - Chaudhary, Juhi

AU - Pradhan, Dinabandhu

N1 - Funding Information:
Juhi Chaudhary is supported by the European Research Council (ERC) project titled PARAPATH (101039913).
Funding Information:
Dinabandhu Pradhan is supported by MAThematical Research Impact Centric Support project (MTR/2018/000017), SERB, India.
Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2023/1/1

Y1 - 2023/1/1

N2 - A function f: V(G) → { 0 , 1 , 2 } is called a Roman dominating function on G= (V(G) , E(G)) if for every vertex v with f(v) = 0 , there exists a vertex u∈ NG(v) such that f(u) = 2 . A function f: V(G) → { 0 , 1 , 2 } induces an ordered partition (V, V1, V2) of V(G), where Vi= { v∈ V(G) : f(v) = i} for i∈ { 0 , 1 , 2 } . A function f: V(G) → { 0 , 1 , 2 } with ordered partition (V, V1, V2) is called a unique response Roman function if for every vertex v with f(v) = 0 , | NG(v) ∩ V2| ≤ 1 , and for every vertex v with f(v) = 1 or 2, | NG(v) ∩ V2| = 0 . A function f: V(G) → { 0 , 1 , 2 } is called a unique response Roman dominating function (URRDF) on G if it is a unique response Roman function as well as a Roman dominating function on G. The weight of a unique response Roman dominating function f is the sum f(V(G)) = ∑ v∈V(G)f(v) , and the minimum weight of a unique response Roman dominating function on G is called the unique response Roman domination number of G and is denoted by uR(G) . Given a graph G, the Min-URRDF problem asks to find a unique response Roman dominating function of minimum weight on G. In this paper, we study the algorithmic aspects of Min-URRDF. We show that the decision version of Min-URRDF remains NP-complete for chordal graphs and bipartite graphs. We show that for a given graph with n vertices, Min-URRDF cannot be approximated within a ratio of n1-ε for any ε> 0 unless P= NP . We also show that Min-URRDF can be approximated within a factor of Δ+ 1 for graphs having maximum degree Δ . On the positive side, we design a linear-time algorithm to solve Min-URRDF for distance-hereditary graphs. Also, we show that Min-URRDF is polynomial-time solvable for interval graphs, and strengthen the result by showing that Min-URRDF can be solved in linear-time for proper interval graphs, a proper subfamily of interval graphs.

AB - A function f: V(G) → { 0 , 1 , 2 } is called a Roman dominating function on G= (V(G) , E(G)) if for every vertex v with f(v) = 0 , there exists a vertex u∈ NG(v) such that f(u) = 2 . A function f: V(G) → { 0 , 1 , 2 } induces an ordered partition (V, V1, V2) of V(G), where Vi= { v∈ V(G) : f(v) = i} for i∈ { 0 , 1 , 2 } . A function f: V(G) → { 0 , 1 , 2 } with ordered partition (V, V1, V2) is called a unique response Roman function if for every vertex v with f(v) = 0 , | NG(v) ∩ V2| ≤ 1 , and for every vertex v with f(v) = 1 or 2, | NG(v) ∩ V2| = 0 . A function f: V(G) → { 0 , 1 , 2 } is called a unique response Roman dominating function (URRDF) on G if it is a unique response Roman function as well as a Roman dominating function on G. The weight of a unique response Roman dominating function f is the sum f(V(G)) = ∑ v∈V(G)f(v) , and the minimum weight of a unique response Roman dominating function on G is called the unique response Roman domination number of G and is denoted by uR(G) . Given a graph G, the Min-URRDF problem asks to find a unique response Roman dominating function of minimum weight on G. In this paper, we study the algorithmic aspects of Min-URRDF. We show that the decision version of Min-URRDF remains NP-complete for chordal graphs and bipartite graphs. We show that for a given graph with n vertices, Min-URRDF cannot be approximated within a ratio of n1-ε for any ε> 0 unless P= NP . We also show that Min-URRDF can be approximated within a factor of Δ+ 1 for graphs having maximum degree Δ . On the positive side, we design a linear-time algorithm to solve Min-URRDF for distance-hereditary graphs. Also, we show that Min-URRDF is polynomial-time solvable for interval graphs, and strengthen the result by showing that Min-URRDF can be solved in linear-time for proper interval graphs, a proper subfamily of interval graphs.

KW - Domination

KW - NP-completeness

KW - Polynomial-time algorithm

KW - Roman domination

KW - Unique response Roman domination

KW - Unique response Roman function

UR - http://www.scopus.com/inward/record.url?scp=85169819358&partnerID=8YFLogxK

U2 - 10.1007/s00453-023-01171-7

DO - 10.1007/s00453-023-01171-7

M3 - Article

AN - SCOPUS:85169819358

SN - 0178-4617

JO - Algorithmica

JF - Algorithmica

ER -