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 $v\in V(G)$ 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.

翻译：设图$G$的一组等距路径$S$是"$v$-根植的"，其中$v$是$G$的一个顶点，若$v$是$S$中所有等距路径的一个端点。图的等距路径复杂度，记为$ipco(G)$，是最小整数$k$，使得存在顶点$v\in V(G)$满足以下性质：图$G$中任意等距路径$P$的顶点可被$k$条$v$-根植等距路径覆盖。首先，我们提出一个$O(n^2 m)$时间的算法，用于计算具有$n$个顶点和$m$条边的图的等距路径复杂度。然后我们证明，对于三个看似不相关的图类，即双曲图、(theta, prism, pyramid)-free图和outerstring图，等距路径复杂度保持有界。双曲图在度量图论中被广泛研究。(theta, prism, pyramid)-free图在结构图论中被广泛研究，例如在强完美图定理的背景下。outerstring图类在几何图论和计算几何中被研究。我们的结果还表明，这些（结构上）不同的图类的距离函数比以前认为的更为相似。等距路径复杂度较小有直接的算法意义。具体而言，我们证明如果图$G$的等距路径复杂度以常数为界，则存在一个多项式时间常数因子近似算法用于等距路径覆盖问题，其目标是用最少数量的等距路径覆盖图的所有顶点。这适用于上述所有图类。