Recognition of $C_4$-Free and 1/2-Hyperbolic Graphs

Abstract

The shortest-path metric ${\textup{d}}$ of a connected graph $G$ is ${1}/{2}$-hyperbolic if and only if it satisfies ${\textup{d}}(u,v) + {\textup{d}}(x,y) \leq \max \{ {\textup{d}}(u,x) + {\textup{d}}(v,y), {\textup{d}}(u,y) + {\textup{d}}(v,x) \} + 1$, for every $4$-tuple $u$, $x$, $v$, $y$ of $G$. We show that the problem of deciding whether an unweighted graph is ${1}/{2}$-hyperbolic is subcubic equivalent to the problem of determining whether there is a chordless cycle of length $4$ in a graph. An improved algorithm is also given for both problems, taking advantage of fast rectangular matrix multiplication. In the worst case it runs in $O(n^{3.26})$-time.