A graph $H$ is an \emph{induced minor} of a graph $G$ if $H$ can be obtained from $G$ by a sequence of edge contractions and vertex deletions. Otherwise, $G$ is \emph{$H$-induced minor-free}. In this paper, we provide a different proof of the fact that $K_{2,3}$-induced minor-free graphs admit a quasi-isometry with additive distortion to graphs with tree-width at most two. Our proof yields a $O(nm)$-time algorithm which takes as input a $K_{2,3}$-induced minor-free graph with $n$ vertices and $m$ edges, and outputs a tree-width two graph $H$ with the desired additive distortion. For \emph{universally signable} graphs, a subclass of $K_{2,3}$-induced minor-free graphs, the time complexity of our algorithm is linear. As a consequence, we obtain a truly sub-quadratic time additive constant factor approximation algorithm to compute the \emph{diameter} of a universally signable graph. In contrast, assuming the \emph{Strong Exponential Time Hypothesis} (\textsc{SETH}), the diameter of split graphs (a very restricted class of universally signable graphs), cannot be computed in truly sub-quadratic time [Borassi et al. (ENTCS, 2016)].

翻译：若图$H$可通过图$G$经过一系列边收缩与顶点删除操作得到，则称$H$是$G$的一个\emph{诱导子式}。否则，称$G$是\emph{$H$-诱导子式自由的}。本文为“$K_{2,3}$-诱导子式自由图允许与树宽至多为二的图拟等距且具有加性失真”这一事实提供了一个不同的证明。我们的证明产生了一个$O(nm)$时间复杂度的算法，该算法以具有$n$个顶点和$m$条边的$K_{2,3}$-诱导子式自由图作为输入，并输出一个具有所需加性失真的树宽为二的图$H$。对于\emph{普遍可标记图}——$K_{2,3}$-诱导子式自由图的一个子类，我们算法的时间复杂度是线性的。因此，我们获得了一个真正的亚二次时间加性常数因子近似算法，用于计算\emph{普遍可标记图}的\emph{直径}。相比之下，假设\emph{强指数时间假设}（\textsc{SETH}），分裂图（普遍可标记图中一个限制性很强的子类）的直径无法在真正的亚二次时间内计算[Borassi等人（ENTCS，2016）]。