Dumas, Foucaud, Perez and Todinca (2024) recently proved that every graph whose edges can be covered by $k$ shortest paths has pathwidth at most $O(3^k)$. In this paper, we improve this upper bound on the pathwidth to a polynomial one; namely, we show that every graph whose edge set can be covered by $k$ shortest paths has pathwidth $O(k^4)$, answering a question from the same paper. Moreover, we prove that when $k\leq 3$, every such graph has pathwidth at most $k$ (and this bound is tight). Finally, we show that even though there exist graphs with arbitrarily large treewidth whose vertex set can be covered by $2$ isometric trees, every graph whose set of edges can be covered by $2$ isometric trees has treewidth at most $2$.

翻译：Dumas、Foucaud、Perez与Todinca（2024）近期证明了边集可被$k$条最短路径覆盖的图其路径宽度至多为$O(3^k)$。本文中，我们将该路径宽度的上界改进为多项式级别；具体而言，我们证明边集可被$k$条最短路径覆盖的图其路径宽度为$O(k^4)$，从而回答了同一论文中提出的问题。此外，我们证明当$k\leq 3$时，所有此类图的路径宽度至多为$k$（且该界是紧的）。最后，我们指出尽管存在顶点集可被$2$棵等距树覆盖而树宽任意大的图，但边集可被$2$棵等距树覆盖的图其树宽至多为$2$。