Optimal orientations of strong products of paths

Abstract

Let diam min (G) denote the minimum diameter of a strong orientation of G and let G H denote the strong product of graphs G and H. In this paper we prove that diam min (P m P n) = diam(P m P n) for m, n ≥ 5, m = n, and diam min (P m P n) = diam(P m P n) + 1 for m, n ≥ 5, m = n. We also prove that diam min (G H) ≤ max {diam min (G), diam min (H)} for any connected bridgeless graphs G and H.

Minimum strong radius of the strong product of paths

“…From the definition, we can see that in the case of the same factor graph, the cartesian product graph is a subgraph of the strong product graph. At present, the research on product graphs is very rich [11][12][13][14][15][16][17].…”

“…In [9], the upper bound for the strong radius and the strong diameter of cartesian product of graphs are determined. In [17], optimal orientations of strong products of paths are determined.…”

The diameter of strong orientations of strong products of graphs

“…Since E(G✷H) ⊆ E(G ⊠ H) any upper bound for diam min (G✷H) is also an upper bound for diam min (G ⊠ H). To obtain a better bound for diam min (G ⊠ H), we have to show how to orient edges in E(G ⊠ H) \ E(G✷H) so that there will be a short path between any pair of vertices in G⊠ H. This has already been shown for strong products of paths in [11], however here we aim at a general approach which can be applied to any strong product of graphs.…”