Classification of Reconfiguration Graphs of Shortest Path Graphs With No Induced $4$-cycles

TL;DR

This paper classifies shortest path reconfiguration graphs that do not contain induced 4-cycles, extending previous results about their structure when girth is at least five, and shows they are composed of specific cycle and path components.

Contribution

It provides a complete classification of shortest path graphs with no induced 4-cycles, expanding understanding of their structural properties beyond girth constraints.

Findings

Shortest path graphs with no induced 4-cycles are characterized.

These graphs are disjoint unions of specific cycle and path structures.

The classification extends previous results on girth and cycle structure.

Abstract

For any graph $G$ with $a,b\in V(G)$, a shortest path reconfiguration graph can be formed with respect to $a$ and $b$; we denote such a graph as $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths from $a$ to $b$ in $G$ while two vertices $U,W$ in $V(S(G,a,b))$ are adjacent if and only if the vertex sets of the paths that represent $U$ and $W$ differ in exactly one vertex. In a recent paper [Asplund et al., \textit{Reconfiguration graphs of shortest paths}, Discrete Mathematics \textbf{341} (2018), no. 10, 2938--2948], it was shown that shortest path graphs with girth five or greater are exactly disjoint unions of even cycles and paths. In this paper, we extend this result by classifying all shortest path graphs with no induced $4$-cycles.