Diameter Constraints in 2-distance Graphs

TL;DR

This paper investigates the diameter properties of 2-distance graphs, establishing bounds on their diameters relative to the original graph and demonstrating sharpness of these bounds, including cases where the 2-distance graph is disconnected.

Contribution

It proves that 2-distance graphs are either disconnected or have diameter at most k+2, with the bound being sharp for even diameters, extending previous understanding of these graphs.

Findings

If diam(G)=k≥3, then diam(G_2) ≤ k+2.

diam(G_2) is either infinite or at most k+2.

The bound is sharp for even k, confirmed with SAT solver experiments.

Abstract

For any finite, simple graph $G = (V,E)$, its $2$-distance graph $G_2$ is a graph having the same vertex set $V$ where two vertices are adjacent if and only if their distance is $2$ in $G$. Connectivity and diameter properties of these graphs have been well studied. For example, it has been shown that if ${\rm diam}(G) = k \geq 3$ then $\lceil \frac{1}{2} k \rceil \leq {\rm diam}(G_2)$, and that this bound is sharp. In this paper, we prove that ${\rm diam}(G_2) = \infty$ (that is, $G_2$ is disconnected) or otherwise ${\rm diam}(G_2) \leq k + 2$. In addition, we show that this inequality is sharp for any even $k$, a result that we verify for some higher orders through judicious use of a \textsc{sat} solver.