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

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.