Diameter constraints in 2-distance graphs

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 diam(G) = k ≥ 3 then ⌈(1/2k) ⌉ ≤ diam(G$_2$), and that this bound is sharp. In this paper, we prove that diam(G$_2$) = ∞ (that is, G$_2$ is disconnected) or otherwise diam(G$_2$) ≤ 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 sat solver.