A dual view of Roman Domination: The 2-limited packing problem

TL;DR

This paper explores the 2-limited packing problem in graphs, compares it to Roman domination, and demonstrates their duality, especially in trees, enabling solutions to Roman domination via 2-limited packing.

Contribution

It establishes the weak duality between 2-limited packing and Roman domination, and proves strong duality for trees, facilitating new solution methods.

Findings

2-limited packing and Roman domination are weakly dual in general.

In trees, the two problems are strongly dual.

Optimal solutions to one can be used to solve the other in trees.

Abstract

We consider the 2-limited packing problem: for a graph $G=(V,E)$ one seeks to find a maximum cardinality subset $B\subseteq V$, such that, for all $v\in V$, the closed neighbourhood of $v$ contains at most two vertices in $B$. We compare this packing problem to the well-known Roman domination problem by pointing out some similarities and differences in the behaviour of the optimal solutions of both problems and show that these two problems are weakly dual. We show that for trees, the two problems are strongly dual, letting us solve the Roman domination problem by computing an optimal solution to the 2-limited packing problem.