Metric geodesic covers of graphs

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.

💡 Research Summary

The paper introduces a new invariant for one‑dimensional topological spaces, called the metric geodesic cover number. Given a space X, one may choose any metric that induces the same topology; the metric geodesic cover number is the smallest integer n for which X can be expressed as the union of n geodesics (shortest‑path curves) with respect to that metric. A related notion, the extended metric geodesic cover number, allows X to be embedded in a larger space Y and asks for the smallest n such that X is covered by n geodesics in Y. The extended version enjoys monotonicity under inclusion, but is harder to compute.

The authors first compare their definitions with the well‑studied edge‑geodesic cover, vertex‑geodesic cover, and other related parameters (geodetic number, order dimension, metrizable graphs). They emphasize three key differences: (i) endpoints of geodesics need not be vertices, (ii) edge lengths may be arbitrary positive numbers, and (iii) the underlying space need not be a graph.

The central theoretical contribution is Theorem 1.3, which states that for any finite topological graph X there exists an optimal geodesic cover whose endpoints are vertices of a 2‑subdivision of X (i.e., each original edge is split into two edges). This “retracted” form dramatically reduces the search space: instead of considering arbitrary curves, it suffices to look at geodesics whose endpoints lie among a finite set of subdivision vertices. The proof uses length‑metric arguments and a reduction that contracts any optimal cover onto such a subdivision without increasing its size.

Armed with this reduction, the authors develop a computer‑assisted algorithm. The algorithm enumerates candidate families of geodesics whose endpoints are subdivision vertices, assigns variable lengths to edges, and checks whether a consistent length assignment exists that makes each candidate curve a true geodesic. The feasibility test is reduced to a system of linear inequalities; integer linear programming and branch‑and‑bound techniques prune the search. The implementation, described in Appendix A, successfully computes optimal covers for several small but non‑trivial graphs.

Using the algorithm, the paper determines the metric geodesic cover numbers of several classic graphs: - K₄ (complete graph on four vertices) has cover number 3. - K₅ and K₃,₃ (the non‑planar “utility” graph) each have cover number 4. Moreover, by examining the extended version, the authors prove that any non‑planar graph must have extended cover number at least 4.

The second major result, Theorem 1.4, classifies all spaces with cover number 3. Proposition 5.2 provides an exhaustive catalogue of topological configurations that admit a 3‑geodesic cover. Each configuration can be embedded in the plane, leading to the corollary that any space with cover number 3 or less is planar. Consequently, the extended cover number of any non‑planar graph is ≥ 4.

The paper also presents several illustrative examples that highlight the subtleties of the new parameters:

• Proposition 3.1 gives a simple lower bound ⌈Δ/2⌉ in terms of maximum degree Δ.
• Proposition 3.2 shows an upper bound m + k for a graph with m edges and k isolated self‑loops.
• Proposition 3.3 yields a lower bound ⌈ℓ/2⌉ where ℓ is the number of degree‑1 vertices (leaves), which is tight for trees.
• Example 3.4 (caterpillar graphs) demonstrates that the ordinary cover number can be arbitrarily larger than the extended cover number.
• Example 3.5 (sawtooth graphs) shows that fixing all edge lengths to 1 (the “unweighted” cover number) may be far larger than the weighted cover number, again with an arbitrarily large gap.

In Section 6 the authors discuss connections to recent work on metric polygons. A metric polygon is a space that can be written as a cyclic union of intervals. The authors note that while every metric triangle embeds bi‑Lipschitzly into the plane, the existence of a metric quadrilateral embedding remains open (Question 1.5 in the cited work). Their analysis of K₅ and K₃,₃ shows that these non‑planar graphs, despite having cover number 4, cannot be represented as metric quadrilaterals, providing indirect evidence supporting a positive answer to the open question.

The paper concludes with two open problems. Question 1.5 asks for general techniques to produce lower bounds on (extended) cover numbers, a task that appears much harder than constructing explicit covers. Question 1.6 seeks an analogue of Theorem 1.3 for the extended setting: given a graph X, can one construct a larger graph Y (perhaps the complete graph on the subdivision vertices) such that any optimal extended cover of X appears as a subgraph of Y? Solving this would give a systematic method for computing extended cover numbers.

Overall, the work establishes a new metric‑geometric invariant for graphs, provides both theoretical foundations (bounds, planarity characterization) and practical algorithms (optimal cover computation for small graphs), and opens several avenues for further research in metric graph theory, embedding problems, and combinatorial optimization.