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The Equidistant Dimension of Graphs

A. González, C. Hernando, M. Mora

Abstract

A subset $S$ of vertices of a connected graph $G$ is a distance-equalizer set if for every two distinct vertices $x, y \in V (G) \setminus S$ there is a vertex $w \in S$ 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 with 3-AP-free sets. Subsequently, we show the usefulness of distance-equalizer sets for constructing doubly resolving sets.

The Equidistant Dimension of Graphs

Abstract

A subset of vertices of a connected graph is a distance-equalizer set if for every two distinct vertices there is a vertex such that the distances from and to are the same. The equidistant dimension of 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 with 3-AP-free sets. Subsequently, we show the usefulness of distance-equalizer sets for constructing doubly resolving sets.

Paper Structure

This paper contains 10 sections, 18 theorems, 19 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Distance-equalizer sets and equidistant dimension
  3. Extremal values
  4. Equidistant dimension of some families of graphs
  5. Complete graphs, complete multipartite graphs and bistars
  6. Paths
  7. Cycles
  8. Johnson graphs
  9. Using distance-equalizer sets for constructing doubly resolving sets
  10. Conclusions and open problems

Key Result

Lemma \oldthetheorem

Let $G$ be a graph. If $S$ is a distance-equalizer set of $G$ and $v$ is a support vertex of $G$, then $S$ contains $v$ or all leaves adjacent to $v$. Consequently,

Figures (3)

  • Figure 1: Black vertices form a distance-equalizer set of minimum size for $P_8$.
  • Figure 3: In this graph $G$, $\psi(G)=dim(G)+eqdim(G)$. Black vertices, gray vertices and the set of leaves are a distance-equalizer set, a resolving set and a doubly resolving set of minimum cardinality, respectively.
  • Figure 4: The graph $G_3$.

Theorems & Definitions (33)

  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • ...and 23 more