Roman Domination in Convex Bipartite Graphs

TL;DR

This paper introduces a polynomial-time dynamic programming algorithm for solving the Roman domination problem specifically in convex bipartite graphs, where the problem is otherwise NP-complete in general cases.

Contribution

The paper presents the first polynomial-time algorithm for Roman domination in convex bipartite graphs, a special class where the problem was previously NP-complete.

Findings

The algorithm efficiently computes the Roman domination number for convex bipartite graphs.

The approach demonstrates that certain NP-complete problems become tractable within specific graph classes.

Abstract

In the Roman domination problem, an undirected simple graph $G(V,E)$ is given. The objective of Roman domination problem is to find a function $f:V\rightarrow {\{0,1,2\}}$ such that for any vertex $v\in V$ with $f(v)=0$ must be adjacent to at least one vertex $u\in V$ with $f(u)=2$ and $\sum_{u\in V} f(u)$, called Roman domination number, is minimized. It is already proven that the Roman domination problem (RDP) is NP-complete for general graphs and it remains NP-complete for bipartite graphs. In this paper, we propose a dynamic programming based polynomial time algorithm for RDP in convex bipartite graph.