The graph theory general position problem on some interconnection networks

TL;DR

This paper investigates the general position number in certain interconnection networks, determining it for specific grid subgraphs and Beneš networks using novel labeling techniques and combinatorial theorems.

Contribution

It introduces monotone-geodesic labeling and a Monotone Geodesic Lemma, advancing the understanding of the general position problem in grid-like networks.

Findings

gp-number determined for large classes of grid subgraphs

gp-number bounded for 3D infinite grid

gp-number calculated for Beneš networks

Abstract

Given a graph $G$, the (graph theory) general position problem is to find the maximum number of vertices such that no three vertices lie on a common geodesic. This graph invariant is called the general position number (gp-number for short) of $G$ and denoted by ${\rm gp}(G)$. In this paper, the gp-number is determined for a large class of subgraphs of the infinite grid graph and for the infinite diagonal grid. To derive these results, we introduce monotone-geodesic labeling and prove a Monotone Geodesic Lemma that is in turn developed using the Erd\"os-Szekeres theorem on monotone sequences. The gp-number of the 3-dim infinite grid is bounded. Using isometric path covers, the gp-number is also determined for Bene\v{s} networks.