Unique Response Roman Domination: Complexity and Algorithms

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, V₁, V₂) 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, V₁, V₂) is called a unique response Roman function if for every vertex v with f(v) = 0 , | N_G(v) ∩ V₂| ≤ 1 , and for every vertex v with f(v) = 1 or 2, | N_G(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 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.