Independence numbers of polyhedral graphs

TL;DR

This paper determines the minimum size of polyhedral graphs that contain a specified independent set, revealing that for certain parameters, extremal graphs are exactly the vertex-face graphs of maximal planar graphs.

Contribution

It introduces the function p(k,a) for minimal order of polyhedral graphs with given independent set size and characterizes extremal graphs for specific cases.

Findings

Calculated p(k,a) for various k and a.

Identified extremal graphs as vertex-face graphs in specific cases.

Provided structural insights into polyhedral graphs with large independent sets.

Abstract

A polyhedral graph is a $3$-connected planar graph. We find the least possible order $p(k,a)$ of a polyhedral graph containing a $k$-independent set of size $a$ for all positive integers $k$ and $a$. In the case $k = 1$ and $a$ even, we prove that the extremal graphs are exactly the vertex-face (radial) graphs of maximal planar graphs.