Counting the Number of Minimum Roman Dominating Functions of a Graph

TL;DR

This paper presents improved algorithms for counting minimum Roman dominating functions in graphs, achieving faster exponential time complexities with and without polynomial space, by transforming the problem into related combinatorial problems.

Contribution

It introduces two new algorithms with reduced exponential time complexity for counting minimum Roman dominating functions, utilizing problem transformation techniques.

Findings

Algorithms run in O(1.5673^n) time with polynomial space.

Reduced to O(1.5014^n) time using exponential space.

Transformations leverage existing combinatorial algorithms.

Abstract

We provide two algorithms counting the number of minimum Roman dominating functions of a graph on n vertices in O(1.5673^n) time and polynomial space. We also show that the time complexity can be reduced to O(1.5014^n) if exponential space is used. Our result is obtained by transforming the Roman domination problem into other combinatorial problems on graphs for which exact algorithms already exist.