Solution to some conjectures on mobile position problems

TL;DR

This paper addresses open problems in mobile position and visibility problems in graphs, providing bounds, exact values for specific graph classes, and analyzing the impact of complete mobility constraints.

Contribution

It introduces bounds and exact values for mobile position and visibility numbers, especially for line graphs of complete graphs and grid graphs, advancing understanding of mobile robot placement.

Findings

Bound on mobile numbers in terms of clique number.

Mobile mutual visibility number for line graphs of complete graphs.

Results for grid graphs and the effect of complete mobility.

Abstract

The general position problem for graphs asks for the largest number of vertices in a subset $S \subseteq V(G)$ of a graph $G$ such that for any $u,v \in S$ and any shortest $u,v$-path $P$ we have $S \cap V(P) = \{ u,v\} $, whereas the mutual visibility problem requires only that for any $u,v \in S$ there exists a shortest $u,v$-path with $S \cap V(P) = \{ u,v\} $. In the mobile versions of these problems, robots must move through the network in general position/mutual visibility such that every vertex is visited by a robot. This paper solves some open problems from the literature. We quantify the effect of adding the restriction that every robot can visit every vertex (the so-called \emph{completely mobile} variants), prove a bound on both mobile numbers in terms of the clique number, and find the mobile mutual visibility number of line graphs of complete graphs, strong grids and Cartesian grids.