Solution to some conjectures on mobile position problems
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908461096239104 |
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| author | Shallcross, Ethan Tuite, James Evans, Aoise Krishnakumar, Aditi Boshar, Sumaiyah |
| author_facet | Shallcross, Ethan Tuite, James Evans, Aoise Krishnakumar, Aditi Boshar, Sumaiyah |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16622 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solution to some conjectures on mobile position problems Shallcross, Ethan Tuite, James Evans, Aoise Krishnakumar, Aditi Boshar, Sumaiyah Combinatorics 05C12, 05C69 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. |
| title | Solution to some conjectures on mobile position problems |
| topic | Combinatorics 05C12, 05C69 |
| url | https://arxiv.org/abs/2507.16622 |