Enumeration With Nice Roman Domination Properties

TL;DR

This paper introduces a polynomial-delay enumeration method for minimal Roman dominating functions with certain properties, extending previous algorithms and applying them to various graph classes.

Contribution

It generalizes enumeration techniques for Roman domination, enabling polynomial-delay enumeration of minimal functions with nice properties on specific graph classes.

Findings

Enumeration of minimal perfect Roman dominating functions is possible with polynomial delay.

All minimal maximal Roman dominating functions can be enumerated in O(1.9332^n) time.

Polynomial-delay algorithms exist for minimal connected/total Roman dominating functions on cobipartite graphs and on interval graphs.

Abstract

Although Extension Perfect Roman Domination is NP-complete, all minimal (with respect to the pointwise order) perfect Roman dominating functions can be enumerated with polynomial delay. This algorithm uses a bijection between minimal perfect Roman dominating functions and Roman dominating functions and the fact that all minimal Roman dominating functions can be enumerated with polynomial delay. This bijection considers the set of vertices with value 2 under the functions. In this paper, we will generalize this idea by defining so called nice Roman Domination properties for which we can employ this method. With this idea, we can show that all minimal maximal Roman Dominating functions can be enumerated with polynomial delay in O(1.9332^n) time. Furthermore, we prove that enumerating all minimal connected/total Roman dominating functions on cobipartite graphs can be achieved with polynomial delay. Additionally, we show the existence of a polynomial-delay algorithm for enumerating all minimal connected Roman dominating function on interval graphs. We show some downsides to this method as well.