Enumerating independent vertex sets in grid graphs

TL;DR

This paper uses the state matrix recursion algorithm to count independent vertex sets in grid graphs, providing generating functions and analyzing their asymptotic growth rates, including bipartite cases.

Contribution

It applies the state matrix recursion algorithm to enumerate independent vertex sets and bipartite sets in grid graphs, deriving generating functions and growth rate asymptotics.

Findings

Derived generating functions for independent vertex sets in grid graphs

Enumerated bipartite independent vertex sets in grid graphs

Analyzed asymptotic growth rates of these sets

Abstract

A set of vertices in a graph is called independent if no two vertices of the set are connected by an edge. In this paper we use the state matrix recursion algorithm, developed by Oh, to enumerate independent vertex sets in a grid graph and even further to provide the generating function with respect to the number of vertices. We also enumerate bipartite independent vertex sets in a grid graph. The asymptotic behavior of their growth rates is presented.