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arXiv · 2306.15301

Connectivity of 2-distance graphs

S. H. Jafari·S. R. Musawi
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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.

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S. H. Jafari, S. R. Musawi. 2023-06-27. Connectivity of 2-distance graphs. https://arxiv.org/abs/2306.15301

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