Independence equivalence classes of paths

TL;DR

This paper fully characterizes the independence equivalence classes of paths with an even number of vertices, building on prior work that established uniqueness for odd-length paths.

Contribution

It provides a complete solution to the problem of determining the independence equivalence classes of even-length paths, extending previous results.

Findings

Paths with odd vertices are independence unique.

The independence equivalence class of even-length paths is fully characterized.

The problem posed by Beaton, Brown, and Cameron (2019) is completely solved.

Abstract

The independence equivalence class of a graph $G$ is the set of graphs that have the same independence polynomial as $G$. A graph whose independence equivalence class contains only itself, up to isomorphism, is independence unique. Beaton, Brown and Cameron (2019) showed that paths with an odd number of vertices are independence unique and raised the problem of finding the independence equivalence class of paths with an even number of vertices. The problem is completely solved in this paper.