Deciding whether there are infinitely many prime graphs with forbidden induced subgraphs

TL;DR

This paper introduces an algorithm to determine whether certain graph classes, defined by forbidden induced subgraphs, contain infinitely many non-isomorphic prime graphs, advancing understanding of graph structure and classification.

Contribution

The paper presents a novel algorithm for deciding the infinitude of prime graphs within classes defined by forbidden induced subgraphs.

Findings

Algorithm successfully determines infinite prime graph classes

Provides a new method for analyzing graph class structures

Enhances understanding of prime graph distribution

Abstract

A homogeneous set of a graph $G$ is a set $X$ of vertices such that $2\le \lvert X\rvert <\lvert V(G)\rvert$ and no vertex in $V(G)-X$ has both a neighbor and a non-neighbor in $X$. A graph is prime if it has no homogeneous set. We present an algorithm to decide whether a class of graphs given by a finite set of forbidden induced subgraphs contains infinitely many non-isomorphic prime graphs.