Roman Census: Enumerating and Counting Roman Dominating Functions on Graph Classes

TL;DR

This paper develops efficient algorithms for enumerating and counting minimal Roman dominating functions on various graph classes, improving existing bounds and providing exact formulas for specific graph families.

Contribution

It introduces new enumeration algorithms with improved time bounds for chordal, interval, forest, split, and cobipartite graphs, and offers formulas for paths.

Findings

Enumeration algorithms with specific time bounds for different graph classes.

Matching lower and upper bounds for forests and interval graphs.

Explicit formulas for counting minimal Roman dominating functions on paths.

Abstract

The concept of Roman domination has recently been studied concerning enumerating and counting (WG 2022). It has been shown that minimal Roman dominating functions can be enumerated with polynomial delay, contrasting what is known about minimal dominating sets. The running time of the algorithm could be estimated as $\mathcal{O}(1.9332^n)$ on general graphs of order $n$. In this paper, we focus on special graph classes. More specifically, for chordal graphs, we present an enumeration algorithm running in time $\mathcal{O}(1.8940^n)$. For interval graphs, we can lower this time further to $\mathcal{O}(1.7321^n)$. Interestingly, this also matches (exactly) the known lower bound. We can also provide a matching lower and upper bound for forests, which is (incidentally) the same, namely $\mathcal{O}(\sqrt{3}^n)$. Furthermore, we show an enumeration algorithm running in time $\mathcal{O}(1.4656^n)$ for split graphs and for cobipartite graphs. Our approach also allows to give concrete formulas for counting minimal Roman dominating functions on special graph families like paths.