Solution to some conjectures on mobile position problems

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.