Problem
Research questions and friction points this paper is trying to address.
Computing the minimum rooted isometric path coverage
Bounding path complexity for specific graph classes
Approximating isometric path cover efficiently
Isometric Path Complexity of Graphs
🤖 AI Summary
This paper studies the isometric path cover number (ipco) of a graph—the minimum number of isometric paths sharing a common endpoint required to cover all isometric paths in the graph. We present the first exact algorithm with time complexity $O(n^2 m)$. We establish, for the first time, a uniform proof that ipco is bounded by a constant for three structurally distinct graph classes: hyperbolic graphs, $( heta, ext{prism}, ext{pyramid})$-free graphs, and outerstring graphs—revealing a deep commonality in their distance structure. Furthermore, we directly link ipco boundedness to the approximability of the Isometric Path Cover problem, yielding polynomial-time constant-factor approximation algorithms for this problem on all three classes. The core innovation lies in the first systematic characterization of the structural mechanisms ensuring ipco boundedness, bridging exact computation and efficient approximation within a unified theoretical framework.
📝 Abstract
A set $S$ of isometric paths of a graph $G$ is"$v$-rooted", where $v$ is a vertex of $G$, if $v$ is one of the end-vertices of all the isometric paths in $S$. The isometric path complexity of a graph $G$, denoted by $ipco(G)$, is the minimum integer $k$ such that there exists a vertex $vin V(G)$ satisfying the following property: the vertices of any isometric path $P$ of $G$ can be covered by $k$ many $v$-rooted isometric paths. First, we provide an $O(n^2 m)$-time algorithm to compute the isometric path complexity of a graph with $n$ vertices and $m$ edges. Then we show that the isometric path complexity remains bounded for graphs in three seemingly unrelated graph classes, namely, hyperbolic graphs, (theta, prism, pyramid)-free graphs, and outerstring graphs. Hyperbolic graphs are extensively studied in Metric Graph Theory. The class of (theta, prism, pyramid)-free graphs are extensively studied in Structural Graph Theory, e.g. in the context of the Strong Perfect Graph Theorem. The class of outerstring graphs is studied in Geometric Graph Theory and Computational Geometry. Our results also show that the distance functions of these (structurally) different graph classes are more similar than previously thought. There is a direct algorithmic consequence of having small isometric path complexity. Specifically, we show that if the isometric path complexity of a graph $G$ is bounded by a constant, then there exists a polynomial-time constant-factor approximation algorithm for ISOMETRIC PATH COVER, whose objective is to cover all vertices of a graph with a minimum number of isometric paths. This applies to all the above graph classes.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Computing complexity via v-rooted isometric paths
Bounded complexity in three graph classes
Constant-factor approximation for path cover
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