On maximal Roman domination in graphs: complexity and algorithms

¹ Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, P.R. China
² School of Computer Science of Information Technology, Qiannan Normal University for Nationalities, Tuyun 558000, P.R. China

^* Corresponding author: hqjiang@gzhu.edu.cn

Received: 7 September 2023
Accepted: 7 February 2024

Abstract

For a simple undirected connected graph G = (V, E), a maximal Roman dominating function (MRDF) of G is a function f : V (G) → {0, 1, 2} with the following properties: (i) For every vertex v ∈ {v ∈ V|f(v) = 0}, there exists a vertex u ∈ N(v) such that f(u) = 2. (ii) The set {v ∈ V|f(v) = 0} is not a dominating set of G; In other words, there exists a vertex v ∈ {v ∈ V|f(v) ≠ 0} such that N(v) ∩ {u ∈ V|f(u) = 0} ∅. The weight of an MRDF of G is the sum of its function values over all vertices, denoted as f(G) = ∑_{v∈V (G)} f(v), and the maximal Roman domination number of G, denoted by γ_mR(G), is the minimum weight of an MRDF of G. In this paper, we establish some bounds of the maximal Roman domination number of graphs. Additionally, we develop an integer linear programming formulation to compute the maximal Roman domination number of any graph. Furthermore, we prove that maximal Roman domination problem (MRD) is NP-complete even restricted to star convex bipartite graphs and chordal bipartite graphs. Lastly, we show the maximal Roman domination number of threshold graphs, trees, and block graphs can be computed in linear time.