A cut locus for finite graphs and the farthest point mapping

详细信息    查看全文

• 作者：Alessandro Maddaloni^a ; ^{maddaloni@imada.sdu.dk" class="auth_mail" title="E-mail the corresponding author} ; Carol T. Zamfirescu^b ; ^{czamfirescu@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Graph distance function ; Cut locus ; Farthest point mapping ; Diameter ; Injectivity radius
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 January 2016
• 年：2016
• 卷：339
• 期：1
• 页码：354-364
• 全文大小：442 K

文摘

We reflect upon an analogue of the cut locus, a notion classically studied in Differential Geometry, for finite graphs. The cut locus  C(x)$C\left(x\right)$ of a vertex 915818762ca47bdbcc1192a2c7cac" title="Click to view the MathML source">x$x$ shall be the graph induced by the set of all vertices y$y$ with the property that no shortest path between 915818762ca47bdbcc1192a2c7cac" title="Click to view the MathML source">x$x$ and z$z$, z≠y$z\ne y$, contains y$y$. The cut locus coincides with the graph induced by the vertices realizing the local maxima of the distance function. The function F$F$ mapping a vertex 915818762ca47bdbcc1192a2c7cac" title="Click to view the MathML source">x$x$ to F(x)$F\left(x\right)$, the set of global maxima of the distance function from 915818762ca47bdbcc1192a2c7cac" title="Click to view the MathML source">x$x$, is the farthest point mapping  . Among other things, we observe that if, for a vertex 915818762ca47bdbcc1192a2c7cac" title="Click to view the MathML source">x$x$, C(x)$C\left(x\right)$ is connected, then C(x)$C\left(x\right)$ is the graph induced by F(x)$F\left(x\right)$, and prove that the farthest point mapping has period 2. Elaborating on the analogy with Geometry, we study graphs satisfying Steinhaus’ condition, i.e. graphs for which the farthest point mapping is single-valued and involutive.