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Connectivity of $3$-distance graphs

S. R. Musawi, S. H. Jafari

Abstract

For a simple graph $G$, the $3$-distance graph, $D_3(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $3$ in the graph $G$. For a connected graph $G$, we provide some conditions for the connectedness of $D_3(G)$. Also, we characterize all trees and unicyclic graphs with connected $3$-distance graph.

Connectivity of $3$-distance graphs

Abstract

For a simple graph , the -distance graph, , is a graph with the vertex set and two vertices are adjacent if and only if their distance is in the graph . For a connected graph , we provide some conditions for the connectedness of . Also, we characterize all trees and unicyclic graphs with connected -distance graph.
Paper Structure (3 sections, 14 theorems, 1 figure)

This paper contains 3 sections, 14 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Connectivity of $D_3(G)$
  3. Conclusion

Key Result

Lemma 2.1

Let $G$ be a connected graph and $H$ be a 3-induced subgraph of $G$. If $D_3(H)$ is connected and for any $x\in V(G)\setminus V(H)$ there is $y\in V(H)$ such that $d(x,y)\geqslant3$, then $D_3(G)$ is connected.

Figures (1)

  • Figure 1: $H_t$

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • ...and 16 more