# Reconfiguration graphs of shortest paths

## TL;DR

This paper studies the structure and properties of shortest path reconfiguration graphs, revealing their classifications, decompositions, and specific characteristics for certain graph classes like grids and graphs with large girth.

## Contribution

It provides a comprehensive classification of shortest path graphs, including those with girth at least 5, and explores their decompositions and embeddings in lattice structures.

## Key findings

- Classifies graphs that can be shortest path graphs
- Establishes decompositions and sums involving shortest path graphs
- Shows shortest path graph of a grid is an induced subgraph of a lattice

## Abstract

For a graph $G$ and $a,b\in V(G)$, the shortest path reconfiguration graph of $G$ with respect to $a$ and $b$ is denoted by $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths between $a$ and $b$ in $G$. Two vertices in $V(S(G,a,b))$ are adjacent, if their corresponding paths in $G$ differ by exactly one vertex. This paper examines the properties of shortest path graphs. Results include establishing classes of graphs that appear as shortest path graphs, decompositions and sums involving shortest path graphs, and the complete classification of shortest path graphs with girth $5$ or greater. We also show that the shortest path graph of a grid graph is an induced subgraph of a lattice.