Prime vertex-minors of a prime graph

TL;DR

This paper investigates the structure of prime graphs and their vertex-minors, establishing the existence of multiple non-essential vertices unless the graph is locally equivalent to a cycle, with implications for prime graph reductions.

Contribution

It extends previous results by proving the existence of at least two non-essential vertices in prime graphs, and provides new bounds and conditions related to vertex-minors and pivot-minors.

Findings

Every prime graph with at least four vertices has at least two non-essential vertices unless it is locally equivalent to a cycle.

For prime graphs with at least six vertices, there exists a non-adjacent vertex whose removal or pivot operation preserves primality.

Prime graphs have at least three non-essential vertices unless they are locally equivalent to a specific multi-path structure.

Abstract

A graph is prime if it does not admit a partition $(A,B)$ of its vertex set such that $\min\{|A|,|B|\} \geq 2$ and the rank of the $A\times B$ submatrix of its adjacency matrix is at most $1$. A vertex $v$ of a graph is non-essential if at least two of the three kinds of vertex-minor reductions at $v$ result in prime graphs. In 1994, Allys proved that every prime graph with at least four vertices has a non-essential vertex unless it is locally equivalent to a cycle graph. We prove that every prime graph with at least four vertices has at least two non-essential vertices unless it is locally equivalent to a cycle graph. As a corollary, we show that for a prime graph $G$ with at least six vertices and a vertex $x$, there is a vertex $v \ne x$ such that $G \setminus v$ or $G * v \setminus v$ is prime, unless $x$ is adjacent to all other vertices and $G$ is isomorphic to a particular graph on odd number of vertices. Furthermore, we show that a prime graph with at least four vertices has at least three non-essential vertices, unless it is locally equivalent to a graph consisting of at least two internally-disjoint paths between two fixed distinct vertices having no common neighbors. We also prove analogous results for pivot-minors.