Roman-Type Domination on Convex and Chordal Bipartite Graphs: Algorithms and Hardness
Discrete Mathematics
Summary
The authors studied special ways to assign numbers to nodes in two types of bipartite graphs to ensure certain protection properties, known as Roman domination and its variants. They created efficient algorithms that solve these problems quickly for convex bipartite graphs by using their special interval-like structure. However, they showed that for a broader class called chordal bipartite graphs, some of these problems are still very hard to solve (NP-complete). This work highlights where these Roman domination problems can be solved efficiently and where they remain difficult.
Authors
Gautam K. Das, Kamal Santra
Abstract
Roman domination and its variants form an important family of domination-type graph parameters motivated by protection, fault tolerance, and resource allocation. A Roman dominating function of a graph \(G\) is a function \(f:V(G)\rightarrow\{0,1,2\}\) such that every vertex \(v\) with \(f(v)=0\) has a neighbour \(u\) with \(f(u)=2\). The weight of \(f\) is \(w(f)=\sum_{v\in V(G)}f(v)\), and the minimum weight of a Roman dominating function of \(G\) is the Roman domination number, denoted by \(γ_R(G)\). In this paper, we study four variants of Roman domination on two natural subclasses of bipartite graphs, namely convex bipartite graphs and chordal bipartite graphs. On the positive side, we develop a unified left-to-right dynamic programming framework for Roman-\(\{2\}\) domination, double Roman domination, perfect Roman domination, and unique response Roman domination on convex bipartite graphs. The algorithms exploit the interval structure of one bipartition class and represent all unfinished requirements using a constant number of boundary indices. Consequently, each of the four parameters can be computed in \(O(n^6)\) time, where \(n=|V(G)|\). On the negative side, we prove that Roman-\(\{2\}\) domination, perfect Roman domination, and unique response Roman domination remain NP-complete on chordal bipartite graphs. These results establish a clear algorithmic separation between convex bipartite graphs, where the interval ordering yields polynomial-time solvability, and the broader class of chordal bipartite graphs, where several Roman-type domination problems remain computationally intractable.