Two graph parameters are said to be coarsely equivalent if they are within constant factors from each other for every graph $G$. Recently, several graph parameters were shown to be coarsely equivalent to tree-length. Recall that the length of a tree-decomposition ${\cal T}(G)$ of a graph $G$ is the largest diameter of a bag in ${\cal T}(G)$, and the tree-length $tl(G)$ of $G$ is the minimum of the length, over all tree-decompositions of $G$. Similarly, the length of a path-decomposition ${\cal P}(G)$ of a graph $G$ is the largest diameter of a bag in ${\cal P}(G)$, and the path-length $pl(G)$ of $G$ is the minimum of the length, over all path-decompositions of $G$. In this paper, we present several graph parameters that are coarsely equivalent to path-length. Among other results, we show that the path-length of a graph $G$ is small if and only if one of the following equivalent conditions is true: (a) $G$ can be embedded to an unweighted caterpillar tree (equivalently, to a graph of path-width one) with a small additive distortion; (b) there is a constant $r\ge 0$ such that for every triple of vertices $u,v,w$ of $G$, disk of radius $r$ centered at one of them intercepts all paths connecting two others; (c) $G$ has a $k$-dominating shortest path with small $k\ge 0$; (d) $G$ has a $k'$-dominating pair with small $k'\ge 0$; (e) some power $G^\mu$ of $G$ is an AT-free (or even a cocomparability) graph for a small integer $\mu\ge 0$.

翻译：若对于任意图$G$，两个图参数彼此相差常数倍，则称它们是粗等价的。最近，多个图参数被证明与树长度粗等价。回顾图$G$的树分解${\cal T}(G)$的长度是指${\cal T}(G)$中袋的最大直径，而图$G$的树长度$tl(G)$是指所有树分解长度的最小值。类似地，图$G$的路径分解${\cal P}(G)$的长度是指${\cal P}(G)$中袋的最大直径，而图$G$的路径长度$pl(G)$是指所有路径分解长度的最小值。本文提出了多个与路径长度粗等价的图参数。除其他结果外，我们证明图$G$的路径长度较小当且仅当以下等价条件之一成立：(a) $G$可以以较小的加性失真嵌入到无权毛毛虫树（等价于路径宽度为一的图）；(b) 存在常数$r\ge 0$，使得对于$G$的任意三个顶点$u,v,w$，以其中一个顶点为中心、半径为$r$的圆盘会截断连接另外两个顶点的所有路径；(c) $G$存在具有较小$k\ge 0$的$k$支配最短路径；(d) $G$存在具有较小$k'\ge 0$的$k'$支配对；(e) $G$的某个幂图$G^\mu$（其中$\mu\ge 0$为较小整数）是AT-free图（甚至可比补图）。