Moving through Cartesian products, coronas and joins in general position

TL;DR

This paper investigates the general position problem in the context of Cartesian products, corona products and joins, giving upper and lower bounds for general graphs and exact values for families including grids, cylinders, Hamming graphs and prisms of trees.

Abstract

The general position problem asks for large sets of vertices such that no three vertices of the set lie on a common shortest path. Recently a dynamic version of this problem was defined, called the \emph{mobile general position problem}, in which a collection of robots must visit all the vertices of the graph whilst remaining in general position. In this paper we investigate this problem in the context of Cartesian products, corona products and joins, giving upper and lower bounds for general graphs and exact values for families including grids, cylinders, Hamming graphs and prisms of trees.

Figures (6)

• Figure 1: A (non-optimal) general position set (left) and a gp-set (right) of the Petersen graph are shown in red.
• Figure 2: A sequence of legal moves (shown by red arrows) for a $\mathop{\mathrm{Mob_{gp}}}\nolimits$-set of the Petersen graph.
• Figure 3: Canonical gp-set in $K_7\mathop{\mathrm{\,\square\,}}\nolimits K_5$
• Figure 4: The legal moves in the infinite grid.
• Figure 5: Moving robots in $C_5\mathop{\mathrm{\,\square\,}}\nolimits K_2$

Theorems & Definitions (33)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Proposition 2.5
• proof