Connectivity of 2-distance graphs

TL;DR

This paper characterizes when the 2-distance graph of a simple graph is connected, providing conditions based on diameter and bipartiteness, and introduces a contraction method for complex cases.

Contribution

It offers a complete characterization of connectivity in 2-distance graphs, including a novel contraction approach for graphs with diameter greater than 2.

Findings

For diameter 2, $D_2(G)$ is connected iff $G$ has no spanning complete bipartite subgraph.

For diameter > 2, connectivity depends on the bipartiteness of a contracted graph $ ilde G$.

Introduces the concept of a maximal Fine set for analyzing connectivity.

Abstract

For a simple graph $G$, the $2$-distance graph, $D_2(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $2$ in the graph $G$. In this paper, we characterize all graphs with connected 2-distance graph. For graphs with diameter 2, we prove that $D_2(G)$ is connected if and only if $G$ has no spanning complete bipartite subgraphs. For graphs with a diameter greater than 2, we define a maximal Fine set and by contracting $G$ on these subsets, we get a new graph $\hat G$ such that $D_2(G)$ is connected if and only if $D_2(\hat G)$ is connected. Especially, $D_2(G)$ is disconnected if and only if $\hat G$ is bipartite.