Traversing a graph in general position

TL;DR

This paper introduces the concept of mobile general position sets in graphs, analyzing their properties, providing bounds, and determining exact values for various classes of graphs, with applications in robot movement and graph traversal.

Contribution

It defines the mobile general position number and establishes bounds and exact values for specific graph classes, advancing understanding of graph traversal strategies.

Findings

Bounds on mobile general position number established

Exact values determined for several graph classes

Insights into robot movement strategies in graphs

Abstract

Let $G$ be a graph. Assume that to each vertex of a set of vertices $S\subseteq V(G)$ a robot is assigned. At each stage one robot can move to a neighbouring vertex. Then $S$ is a mobile general position set of $G$ if there exists a sequence of moves of the robots such that all the vertices of $G$ are visited whilst maintaining the general position property at all times. The mobile general position number of $G$ is the cardinality of a largest mobile general position set of $G$. In this paper, bounds on the mobile general position number are given and exact values determined for certain common classes of graphs including block graphs, rooted products, unicyclic graphs, Cartesian products, joins of graphs, Kneser graphs $K(n,2)$, and line graphs of complete graphs.