Metric geodesic covers of graphs

TL;DR

The paper develops the theory of metric geodesic covers for one-dimensional topological spaces, introducing the metric geodesic cover number and its extended variant. A central result is that any finite graph admits an optimal geodesic cover whose endpoints lie at vertices of the graph's 2-subdivision, enabling a finite, computer-amenable search. The authors classify spaces coverable by three geodesics, showing planarity as a consequence, and demonstrate the approach via computer-assisted calculations for standard graphs: $K_5$, $K_4$, $K_{3,3}$, and $K_{2,3}$. Additional work includes a catalogue of three-geodesic spaces and the development of an algorithm (with appendices) to enumerate candidate covers and verify their realizability through linear programming. The results provide concrete cover numbers for classical graphs and illuminate how metric choices influence geodesic decompositions in graphs.

Abstract

We study the problem of finding, for a given one-dimensional topological space $X$, a cover of $X$ of smallest size by geodesics with respect to some metric. The infimal size of such a set is called the metric geodesic cover number of $X$. We prove reductions enabling us to find, with computer assistance, optimal geodesic covers of a graph and use these to determine the cover number of several standard graphs, including $K_4$, $K_5$ and $K_{3,3}$. We also give a catalogue of topological spaces with cover number $3$, and use it to deduce that any such space must be planar.

Figures (10)

• Figure 1: Geodesic covers of the $n$-star.
• Figure 2: The caterpillar graph and an extended geodesic cover of smallest size.
• Figure 3: Sawtooth graph
• Figure 4: Retracting a geodesic cover.
• Figure 5: Examples of improper geodesics.

Theorems & Definitions (22)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof