  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.
