Publication Date

Spring 2026

Degree Type

Thesis

Degree Name

Master of Arts (MA)

Department

Mathematics and Statistics

Advisor

Jesse Geneson; Wasin So; Yan Zhang

Abstract

The metric dimension of a graph G is the smallest number of unique vertices (often called “marked vertices” or “landmarks”) such that each vertex has a unique distance to the marked vertices. If a set of vertices S ⊆ V (G) when marked gives all vertices unique distances, it is called a resolving set. Hence if G is a graph with a resolving set S, then for all v, u ∈ V (G) there exists s ∈ S such that d(v, s) ̸= d(u, s). The metric dimension dim(G) = min(|S|). Two widely studied variants exist, one where edges are resolved referred to as the edge metric dimension, and another where both vertices and edges are resolved, the mixed metric dimension. We introduce a new variant of this graph parameter, where vertices are marked at random, including repeats, and we define the expected number of attempts to obtain a resolving set of vertices to be the probabilistic metric dimension, denoted pdim(G). For this, we prove fundamental results including equalities for paths, cycles, complete graphs, balanced spiders, and stars for both standard and edge metric dimension. Additionally, the probabilistic metric dimension of paths, complete graphs, balanced spiders, and stars is proved under mixed metric dimension. We derive upper bounds for any graph and trees and upper bounds for graph operations under standard, edge, and mixed metric dimension.

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