Extremal Graph Theory for Metric Dimension and Diameter

Abstract

A set of vertices $S$ resolves a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. Let ${\cal G}_{\beta,D}$ be the set of graphs with metric dimension $\beta$ and diameter $D$. It is well-known that the minimum order of a graph in ${\cal G}_{\beta,D}$ is exactly $\beta+D$. The first contribution of this paper is to characterise the graphs in ${\cal G}_{\beta,D}$ with order $\beta+D$ for all values of $\beta$ and $D$. Such a characterisation was previously only known for $D\leq2$ or $\beta\leq1$. The second contribution is to determine the maximum order of a graph in ${\cal G}_{\beta,D}$ for all values of $D$ and $\beta$. Only a weak upper bound was previously known.

Base size, metric dimension and other invariants of groups and graphs

“…A generalization of this bound has been given by Hernando et al [63]. Theorem 3.7 (Hernando et al [63]). Suppose that Γ is a graph with n vertices, diameter d and metric dimension k. Then…”

“…We remark that the proof used by Hernando et al [63] does not carry over to coherent configurations directly, as it uses the fact that graph distance satisfies the triangle inequality, which we do not have for general coherent configurations. (In fact, it holds precisely for distanceregular graphs, where we already have the bound for metric dimension.…”

“…In graph theory, it is a parameter that has appeared in various applications, as diverse as combinatorial optimization [85], pharmaceutical chemistry [39], robot navigation [71] and sonar [90], to name but a few. The paper by Hernando et al [63] contains a considerable bibliography.…”

Metric dimension and pattern avoidance in graphs

“…The next proof is very similar to the proof in [7] that the maximum possible order of a graph of diameter D and metric dimension k is equal to (…”

“…Besides pattern avoidance, researchers have also investigated the maximum size of graphs with a given diameter and metric dimension. In particular, Hernando et al [7] proved that the maximum possible number of vertices in a graph of diameter D and metric dimension k is at most (⌊ 2D [3] proved that graphs with edge metric dimension k and diameter D have at most (D + 1) k edges. Later, Zubrilina sharpened the bound for edge metric dimension to k 2 + kD k−1 + D k [9].…”