On the complexity of global Roman domination problem in graphs

2026-01-14

AI Summary

This study investigates the computational complexity of the global Roman domination problem, which seeks a minimum-weight labeling function satisfying the Roman domination condition simultaneously on a graph and its complement. Employing techniques from graph-theoretic modeling, complexity reductions, fine-grained structural analysis, and dynamic programming, the work establishes for the first time that global Roman domination is computationally inequivalent to classical Roman domination. It resolves an open question by proving the problem NP-complete on split graphs and further demonstrates NP-completeness on chordal bipartite graphs, planar bipartite graphs with maximum degree five, and circle graphs. In contrast, the paper presents a linear-time algorithm for cographs, highlighting a tractable case within this otherwise computationally hard landscape.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

Abstract

A Roman dominating function of a graph $G=(V,E)$ is a labeling $f: V \rightarrow{} \{0 ,1, 2\}$ such that for each vertex $u \in V$ with $f(u) = 0$, there exists a vertex $v \in N(u)$ with $f(v) =2$. A Roman dominating function $f$ is a global Roman dominating function if it is a Roman dominating function for both $G$ and its complement $\overline{G}$. The weight of $f$ is the sum of $f(u)$ over all the vertices $u \in V$. The objective of Global Roman Domination problem is to find a global Roman dominating function with minimum weight. The objective of Global Roman Domination is to compute a global Roman dominating function of minimum weight. In this paper, we study the algorithmic aspects of Global Roman Domination problem on various graph classes and obtain the following results. 1. We prove that Roman domination and Global Roman Domination problems are not computationally equivalent by identifying graph classes on which one is linear-time solvable, while the other is NP-complete. 2. We show that Global Roman Domination problem is NP-complete on split graphs, thereby resolving an open question posed by Panda and Goyal [Discrete Applied Mathematics, 2023]. 3. We prove that Global Roman Domination problem is NP-complete on chordal bipartite graphs, planar bipartite graphs with maximum degree five and circle graphs. 4. On the positive side, we present a linear-time algorithm for Global Roman domination problem on cographs.

Problem

• Global Roman Domination
• Roman domination
• graph complement
• minimum weight
• NP-completeness

Innovation

• Global Roman Domination
• NP-completeness
• linear-time algorithm
• graph classes
• computational complexity