Some properties of $\{k\}$-packing function problem in graphs

TL;DR

This paper investigates the properties of the $\

Contribution

It introduces a relation between different cases of the $\

Findings

A construction method for determining $k$ values where $L_{\\{k\\}}(G)$ is polynomial-time computable.

A relationship established between $\\{1\\}$-packing functions and the independent set number.

Bounds and optimal values for specific graph classes are provided.

Abstract

The recently introduced $\{k\}$-packing function problem is considered in this paper. Special relation between a case when $k=1$, $k\ge 2$ and linear programming relaxation is introduced with sufficient conditions for optimality. For arbitrary simple connected graph $G$ there is construction procedure for finding values of $k$ for which $L_{\{k\}}(G)$ can be determined in the polynomial time. Additionally, relationship between $\{1\}$-packing function and independent set number is established. Optimal values for some special classes of graphs and general upper and lower bounds are introduced.