转向可避免的道路
Shifting paths to avoidable ones
摘要：
图$G$中的导出路径$P$的扩展是导出路径$P'$，使得删除$P'$的端点导致$P$。一个图中的诱导路称为可避免的，如果它的每一个扩张都包含在一个诱导圈中。在2019年，Beisegel、Chudovsky、Gurvich、Milaniban和Servatius证明了每个包含诱导$k$ -顶点路径的图也包含相同长度的可避免诱导路径，并证明了$k = 2$的结果。$k = 1$的病例在1976年由Ohtsuki、Cheung和Fujisawa的一项研究中发现。这个猜想在2020年由Bonamy、Defrain、Hatzel和Thiebaut对所有$k$证明。在本文中，使用类似的方法，我们加强了他们的结果从重构的角度来看。也就是说，我们表明，在每个图中，每个诱导路径可以转化为一个可避免的一个序列的移位，其中两个诱导$k$ -顶点路径是移位的对方，如果他们的工会是诱导路径与$k+1$顶点。我们也得到类似的结果，不一定诱导路径和散步。相反，该语句不能扩展到轨迹或等距路径。
英文摘要：
An extension of an induced path $P$ in a graph $G$ is an induced path $P'$ such that deleting the endpoints of $P'$ results in $P$. An induced path in a graph is said to be avoidable if each of its extensions is contained in an induced cycle. In 2019, Beisegel, Chudovsky, Gurvich, Milanič, and Servatius conjectured that every graph that contains an induced $k$-vertex path also contains an avoidable induced path of the same length, and proved the result for $k = 2$. The case $k = 1$ was known much earlier, due to a work of Ohtsuki, Cheung, and Fujisawa in 1976. The conjecture was proved for all $k$ in 2020 by Bonamy, Defrain, Hatzel, and Thiebaut. In the present paper, using a similar approach, we strengthen their result from a reconfiguration point of view. Namely, we show that in every graph, each induced path can be transformed to an avoidable one by a sequence of shifts, where two induced $k$-vertex paths are shifts of each other if their union is an induced path with $k+1$ vertices. We also obtain analogous results for not necessarily induced paths and for walks. In contrast, the statement cannot be extended to trails or to isometric paths.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
