The Equidistant Dimension of Graphs
|Author/s||González Herrera, Antonio
|Department||Universidad de Sevilla. Departamento de Didáctica de las Matemáticas|
|Abstract||Asubset S of vertices of a connected graphG is a distance-equalizer set if for every two
distinct vertices x, y ∈ V(G)\S there is a vertex w ∈ S such that the distances from x
and y to w are the same. The equidistant ...
Asubset S of vertices of a connected graphG is a distance-equalizer set if for every two distinct vertices x, y ∈ V(G)\S there is a vertex w ∈ S such that the distances from x and y to w are the same. The equidistant dimension of G is the minimum cardinality of a distance-equalizer set of G. This paper is devoted to introduce this parameter and explore its properties and applications to other mathematical problems, not necessarily in the context of graph theory. Concretely, we first establish some bounds concerning the order, the maximum degree, the clique number, and the independence number, and characterize all graphs attaining some extremal values. We then study the equidistant dimension of several families of graphs (complete and complete multipartite graphs, bistars, paths, cycles, and Johnson graphs), proving that, in the case of paths and cycles, this parameter is related to 3-AP-free sets. Subsequently, we show the usefulness of distance-equalizer sets for constructing doubly resolving sets.
|Funding agencies||European Union’s Horizon 2020
Ministerio de Ciencia e Innovación
|Project ID.||734922 - CONNECT
|Citation||González Herrera, A., Hernando, C. y Mora, M. (2022). The Equidistant Dimension of Graphs. Bulletin of the malaysian mathematical sciences society, 45 (4), 1757-1775. https://doi.org/10.1007/s40840-022-01295-z.|