Unique Response Roman Domination: Complexity and Algorithms - Banerjee, Sumanta; Chaudhary, Juhi; Pradhan, Dinabandhu (Algorithmica, 2023)
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 0 , 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 0 , 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.