# Packing functions and graphs with perfect closed neighbourhood matrices

## TL;DR

This paper explores the properties of graphs with perfect closed neighbourhood matrices, providing conditions for optimality in packing functions and characterizing such graphs through structural and clique-node matrix properties.

## Contribution

It introduces a linear programming approach to the k-packing problem, analyzes the perfection of closed neighbourhood matrices, and characterizes graphs with perfect matrices.

## Key findings

- A sufficient condition for optimality based on matrix perfection.
- Characterization of graphs with perfect closed neighbourhood matrices.
- Necessary and sufficient conditions for clique-node closed neighbourhood matrices.

## Abstract

In this work we consider a straightforward linear programming formulation of the recently introduced $\{k\}$-packing function problem in graphs, for each fixed value of the positive integer number $k$. We analyse a special relation between the case $ k = 1$ and $ k \geq 2$ and give a sufficient condition for optimality ---the perfection--- of the closed neighbourhood matrix $N[G]$ of the input graph $G$. We begin a structural study of graphs satisfying this condition. In particular, we look for a characterization of graphs that have perfect closed neighbourhood matrices which involves the property of being a clique-node matrix of a perfect graph. We present a necessary and sufficient condition for a graph to have a clique-node closed neighbourhood matrix. Finally, we study the perfection of the graph of maximal cliques associated to $N[G]$.