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Beth Novick

Publications and source records attributed to Beth Novick.

5 recordsLinked to original sources
arXiv2026

Infinite graphs with finite metric dimension

We study the metric dimension (strong and weak) of infinite graphs. In particular, our main interest is characterizing infinite graphs with finite dimension. Our main results: (1) graphs with more than one end have infinite strong dimension; (2) for graphs with a finite number of cycles, the weak dimension is finite if and only if the graph has finitely many vertices of degree three, and the strong dimension is finite if and only if the graph has one end and finitely many vertices of degree three.

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arXiv2022

Metric dimension of growing infinite graphs

We investigate how the metric dimension of infinite graphs change when we add edges to the graph. Our two main results: (1) there exists a growing sequence of graphs (under the subgraph relation, but without adding vertices) for which the metric dimension changes between finite and infinite infinitely many times; (2) finite changes in the edge set can not change the metric dimension from finite to infinite or vice versa.

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arXiv2020

The Threshold Strong Dimension of a Graph

Let $G$ be a connected graph and $u,v$ and $w$ vertices of $G$. Then $w$ is said to {\em strongly resolve} $u$ and $v$, if there is either a shortest $u$-$w$ path that contains $v$ or a shortest $v$-$w$ path that contains $u$. A set $W$ of vertices of $G$ is a {\em strong resolving set} if every pair of vertices of $G$ is strongly resolved by some vertex of $W$. A smallest strong resolving set of a graph is called a {\em strong basis} and its cardinality, denoted $\beta_s(G)$, the {\em strong dimension} of $G$. The {\em threshold strong dimension} of a graph $G$, denoted $\tau_s(G)$, is the smallest strong dimension among all graphs having $G$ as spanning subgraph. A graph whose strong dimension equals its threshold strong dimension is called $\beta_s$-{\em irreducible}. In this paper we establish a geometric characterization for the threshold strong dimension of a graph $G$ that is expressed in terms of the smallest number of paths (each of sufficiently large order) whose strong product admits a certain type of embedding of $G$. We demonstrate that the threshold strong dimension of a graph is not equal to the previously studied threshold dimension of a graph. Graphs with strong dimension $1$ and $2$ are necessarily $\beta_s$-irreducible. It is well-known that the only graphs with strong dimension $1$ are the paths. We completely describe graphs with strong dimension $2$ in terms of the strong resolving graphs introduced by Oellermann and Peters-Fransen. We obtain sharp upper bounds for the threshold strong dimension of general graphs and determine exact values for this invariant for certain subclasses of trees.

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arXiv2017

Reconfiguration graphs of shortest paths

For a graph $G$ and $a,b\in V(G)$, the shortest path reconfiguration graph of $G$ with respect to $a$ and $b$ is denoted by $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths between $a$ and $b$ in $G$. Two vertices in $V(S(G,a,b))$ are adjacent, if their corresponding paths in $G$ differ by exactly one vertex. This paper examines the properties of shortest path graphs. Results include establishing classes of graphs that appear as shortest path graphs, decompositions and sums involving shortest path graphs, and the complete classification of shortest path graphs with girth $5$ or greater. We also show that the shortest path graph of a grid graph is an induced subgraph of a lattice.

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