## Abstract

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∈ N_{G}(v) such that f(u) = 2 . A function f: V(G) → { 0 , 1 , 2 } induces an ordered partition (V, V_{1}, V_{2}) of V(G), where V_{i}= { v∈ V(G) : f(v) = i} for i∈ { 0 , 1 , 2 } . A function f: V(G) → { 0 , 1 , 2 } with ordered partition (V, V_{1}, V_{2}) is called a unique response Roman function if for every vertex v with f(v) = 0 , | N_{G}(v) ∩ V_{2}| ≤ 1 , and for every vertex v with f(v) = 1 or 2, | N_{G}(v) ∩ V_{2}| = 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 u_{R}(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 n^{1}^{-}^{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.

Original language | English |
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Pages (from-to) | 3889-3927 |

Number of pages | 39 |

Journal | Algorithmica |

Volume | 85 |

Issue number | 12 |

DOIs | |

State | Published - 1 Dec 2023 |

## Keywords

- Domination
- NP-completeness
- Polynomial-time algorithm
- Roman domination
- Unique response Roman domination
- Unique response Roman function

## ASJC Scopus subject areas

- General Computer Science
- Computer Science Applications
- Applied Mathematics