We introduce and study a new graph parameter, called the \emph{isometric path complexity} of a graph. A path is \emph{isometric} if it is a shortest path between its endpoints. 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 \emph{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, \emph{hyperbolic graphs}, \emph{(theta, prism, pyramid)-free graphs}, and \emph{outerstring graphs}. Hyperbolic graphs are extensively studied in \emph{Metric Graph Theory}. The class of (theta, prism, pyramid)-free graphs are extensively studied in \emph{Structural Graph Theory}, \textit{e.g.} in the context of the \emph{Strong Perfect Graph Theorem}. The class of outerstring graphs is studied in \emph{Geometric Graph Theory} and \emph{Computational Geometry}. Our results also show that the distance functions of these (structurally) different graph classes are more similar than previously thought. Finally, we apply this new concept to the ISOMETRIC PATH COVER problem, whose objective is to cover all vertices of a graph with a minimum number of isometric paths, to all the above graph classes. Indeed, 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.

翻译：我们引入并研究了一个新的图参数，称为图的\emph{等距路径复杂度}。若一条路径是其端点间的最短路径，则称其为\emph{等距路径}。图$G$的一组等距路径$S$是“以$v$为根”的（其中$v$是$G$的一个顶点），如果$v$是$S$中所有等距路径的一个端点。图$G$的\emph{等距路径复杂度}，记作$ipco(G)$，是满足以下性质的最小整数$k$：存在一个顶点$v\in V(G)$，使得$G$中任意等距路径$P$的顶点均可被$k$条以$v$为根的等距路径覆盖。首先，我们给出了一个$O(n^2 m)$时间算法，用于计算具有$n$个顶点和$m$条边的图的等距路径复杂度。随后，我们证明了对于三个看似无关的图类，即\emph{双曲图}、\emph{（theta图、棱柱图、金字塔图）自由图}和\emph{外弦图}，等距路径复杂度保持有界。双曲图在\emph{度量图论}中被广泛研究。（theta图、棱柱图、金字塔图）自由图在\emph{结构图论}中被广泛研究，例如在\emph{强完美图定理}的背景下。外弦图在\emph{几何图论}和\emph{计算几何}中被研究。我们的结果还表明，这些（结构上）不同图类的距离函数比之前认为的更加相似。最后，我们将这一新概念应用于等距路径覆盖问题（该问题的目标是用最少数量的等距路径覆盖图的所有顶点），并涉及上述所有图类。实际上，我们证明：如果图$G$的等距路径复杂度有常数上界，则对于等距路径覆盖问题存在一个多项式时间的常数因子近似算法。