Minimum eccentricity shortest paths of $K_{2,3}$-minor-free graphs

Abstract: Given a simple, undirected, and unweighted graph $G$, and an integer $R$, the objective of the \textsc{Minimum Eccentricity Shortest Path (MESP)} is to decide whether there exists an \emph{isometric path} $P$ in $G$ such that the distance from every vertex in the graph to its nearest vertex in $P$ is at most $R$. In this paper, we prove that MESP admits an $O(n<sup>4)$-time algorithm on $K_{2,3}$-minor-free graphs. Our algorithm has a cubic running time when the inputs are restricted to a cactus.

• The paper develops an O(n⁴)-time algorithm for MESP on K₂,₃-minor-free graphs by ordering interval holes and encoding possible isometric-path traces in a weighted directed acyclic graph.
• The paper proves that interval holes have tightly bounded interactions: paths meet each hole in at most five maximal subpaths, while outside-vertex projections touch at most four vertices.
• The paper improves the bound to O(n³) on cactus graphs through smaller projections and disjoint confined BFS traversals, while leaving series-parallel and general bounded-treewidth graphs open.

Problem and context

The Minimum Eccentricity Shortest Path (MESP) problem asks, given a simple undirected unweighted graph $G$ and an integer $R$, whether $G$ contains an isometric path $P$ (a shortest path between its endpoints) such that every vertex of $G$ is within distance $R$ of $P$. The problem was introduced by Dragan and Leitert (2608.13158), who established an $O(n^{2k+2})$-time algorithm for deciding whether an isometric path of eccentricity at most $k$ exists, and showed that MESP is W[2]-hard with respect to the desired eccentricity. MESP is NP-complete even on subcubic partial grids — a subclass of planar bipartite graphs of maximum degree 3 — and its complexity on outerplanar graphs, series-parallel graphs, solid grids, and graphs of constant treewidth remains open. Notably, it is unknown whether MESP admits a polynomial-time algorithm even on treewidth-2 graphs.

The paper under review proves two results: MESP admits an $O(n^4)$-time algorithm on $R$0-minor-free graphs, and an $R$1-time algorithm on cactus graphs. Since $R$2-minor-free graphs are planar, have treewidth at most 3, and form a superclass of outerplanar graphs, the first result extends polynomial-time solvability to a class for which no prior technique applied. In particular, the bounded-hyperbolicity and bounded-projection-gap algorithms of Dragan and Leitert do not cover these classes: cycles have unbounded projection gap, and the existing FPT results parameterize by quantities (modular width, distance to cluster graph, maximum leaf number) that are unbounded on outerplanar graphs. The existing constant-factor approximation algorithms likewise give no guarantee better than unbounded factors on outerplanar inputs.

Structural analysis of intervals

The core contribution is a structural characterization of the $R$3-interval $R$4 in a $R$5-minor-free graph: the subgraph whose edges lie on some $R$6-isometric path. Although the number of distinct $R$7-isometric paths can be exponential, the authors establish three facts:

1. Linear ordering of holes: the induced cycles (holes) of $R$8 admit a linear ordering $R$9 such that for any $G$0-isometric path $G$1, the intersections $G$2 appear along $G$3 in this order.
2. Restricted projections: the projection $G$4 of any vertex outside the interval intersects at most four vertices, contained either in one hole or in two consecutive holes of the ordering.
3. Bounded interaction per hole: any $G$5-isometric path intersects each hole in at most five maximal isometric subpaths.

These properties follow from a sequence of minor-exclusion arguments. The key lemmas show that at most two vertices of $G$6 share a distance level from $G$7 (otherwise a $G$8 minor arises), that at most three edges join consecutive distance levels, that every hole forms an $G$9-cutset of $P$0, and that the intersection of any isometric path with a hole induces a path. The bound of five subpaths per hole is shown to be tight via an explicit construction.

Auxiliary graph and weighting scheme

For fixed $P$1, the algorithm builds a directed acyclic auxiliary graph $P$2. Its vertices represent the possible traces of isometric paths through each hole (at most five per hole by the tight bound), plus dummy source $P$3 and sink $P$4. Directed edges connect vertices representing subpaths of consecutive holes whenever their union lies on a common isometric path. Each edge $P$5 carries a set $P$6 — the vertex set of the minimal subpath of some isometric path containing both endpoint subpaths — and a weight

$P$7

where $P$8 captures vertices equidistant from $P$9 and from $G$0, and $G$1 captures vertices all of whose shortest paths to $G$2 pass through the complement $G$3 of $G$4 within the adjacent holes. The central correctness lemma states that the maximum edge weight along a directed $G$5-path equals the eccentricity of the corresponding $G$6-isometric path, proved via the projection lemmas above. A bijection between directed $G$7-paths and $G$8-isometric paths then reduces the problem, for each pair $G$9, to a bottleneck path query solvable by dynamic programming in linear time.

Iterating over all $R$0 pairs yields the overall $R$1 bound: preprocessing (all-pairs BFS distances) costs $R$2, and each pair requires $R$3 time dominated by projection computation via BFS from interval vertices. On cactus graphs, where every block of $R$4 is a hole and projections have size at most 2 (with cut-vertex endpoints when of size 2), the components of $R$5 attach through single holes or unique connector subpaths, so disjoint confined BFS traversals compute all projections in $R$6 total time, reducing the per-pair cost to $R$7 and the total to $R$8.

A methodological point worth noting: the popular approach of taking a diametral pair and any isometric path between them fails here — the paper exhibits a cactus in which no diametral path achieves minimum eccentricity. The interval-decomposition technique is therefore essential rather than incidental.

Limitations and open questions

The result does not resolve the complexity of MESP on general bounded-treewidth graphs, nor on series-parallel graphs (treewidth 2), both explicitly left open. The structural lemmas rely heavily on the exclusion of the $R$9 minor; whether analogous decompositions exist for larger planar subclasses is not established. The running times, while polynomial, are not optimized below $P$0 (or $P$1 on cacti), and no lower bounds or hardness results for these classes are given.

Conclusion

This work establishes polynomial-time solvability of MESP on $P$2-minor-free graphs via a tight structural decomposition of intervals into linearly ordered holes with constant-bounded path interactions, encoded as a linear-size weighted DAG. The technique cleanly separates the combinatorial explosion of isometric paths from the global eccentricity objective, and the open status of MESP on series-parallel and other bounded-treewidth classes marks the natural next target for this approach.