Year: 2026
Type: preprint
Abstract: This work extends the study of mobile resolvability in connected graphs by introducing completely mobile resolving sets. As with mobile resolving sets, we consider stationing a collection of robots at the vertices in a resolving set of a graph. Robots may move one at a time to any adjacent vertex as long as set of occupied vertices remains a resolving set. In this new variant, we ask that every robot can traverse every vertex in the graph through a sequence of permitted moves. Graphs possessing at least one completely mobile resolving set are termed mobile-resolvable. We completely characterise the non-mobile-resolvable graphs as paths of order at least three and subdivided stars with three or more leaves and at most two subdivided edges. For trees, we explicitly determine the completely mobile metric dimension, showing that it exceeds the classical metric dimension by one for most mobile-resolvable trees, and by two for certain subdivided stars. Furthermore, we evaluate this parameter across fundamental graph classes-including cycles, complete bipartite graphs, grid graphs, and circulant graphs.
Source: arXiv (Cornell University)
Institutions: Universidad de Cádiz, The Open University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G745419941, G8537714269
OpenAlex ID: W7221547332