Minimal Roman Dominating Functions: Extensions and Enumeration

TL;DR

This paper introduces polynomial-delay algorithms for enumerating minimal Roman domination functions and explores the extension problem, revealing distinct behaviors from classical domination problems and providing runtime bounds.

Contribution

It develops the first polynomial-delay enumeration algorithms for minimal Roman domination functions and analyzes the extension problem's complexity.

Findings

Polynomial-delay enumeration algorithms for minimal Roman domination functions.

A polynomial-time algorithm for the Extension Roman Domination problem.

Runtime bounds of d7( ext{RomanUpperbound})^n and d7( ext{RomanLowerbound})^n for graphs.

Abstract

Roman domination is one of the many variants of domination that keeps most of the complexity features of the classical domination problem. We prove that Roman domination behaves differently in two aspects: enumeration and extension. We develop non-trivial enumeration algorithms for minimal Roman domination functions with polynomial delay and polynomial space. Recall that the existence of a similar enumeration result for minimal dominating sets is open for decades. Our result is based on a polynomial-time algorithm for Extension Roman Domination: Given a graph $G = (V,E)$ and a function $f:V\to\{0,1,2\}$, is there a minimal Roman domination function $\Tilde{f}$ with $f\leq \Tilde{f}$? Here, $\leq$ lifts $0< 1< 2$ pointwise; minimality is understood in this order. Our enumeration algorithm is also analyzed from an input-sensitive viewpoint, leading to a run-time estimate of $\Oh(\RomanUpperbound^n)$ for graphs of order n; this is complemented by a lower bound example of $\Omega(\RomanLowerbound^n)$.