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Topological complexity of unordered configuration spaces of certain graphs
Topology and its Applications ( IF 0.6 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.topol.2020.107382
Steven Scheirer

The unordered configuration space of $n$ points on a graph $\Gamma,$ denoted here by $UC^{n}(\Gamma),$ can be viewed as the space of all configurations of $n$ unlabeled robots on a system of one-dimensional tracks, which is interpreted as a graph $\Gamma.$ The topology of these spaces is related to the number of vertices of degree greater than 2; this number is denoted by $m(\Gamma).$ We discuss a combinatorial approach to compute the topological complexity of a "discretized" version of this space, $UD^{n}(\Gamma),$ and give results for certain classes of graphs. In the first case, we show that for a large class of graphs, as long as the number of robots is at least twice the number of essential vertices, then $TC(UD^{n}(\Gamma))=2m(\Gamma)+1.$ In the second, we show that as long as the number of robots is at most half the number of vertex-disjoint cycles in $\Gamma,$ we have $TC(UD^{n}(\Gamma))=2n+1.$

中文翻译:

某些图的无序配置空间的拓扑复杂度

图$\Gamma上$n$个点的无序配置空间,$在此表示为$UC^{n}(\Gamma),$可以看作是一个系统上$n$个未标记机器人的所有配置空间一维轨迹,被解释为图 $\Gamma.$ 这些空间的拓扑与度数大于 2 的顶点数有关;这个数字用 $m(\Gamma) 表示。$ 我们讨论了一种组合方法来计算这个空间的“离散化”版本的拓扑复杂度,$UD^{n}(\Gamma),$ 并给出某些结果图类。在第一种情况下,我们证明对于一大类图,只要机器人的数量至少是本质顶点数量的两倍,那么 $TC(UD^{n}(\Gamma))=2m(\ Gamma)+1.$ 在第二个,
更新日期:2020-11-01
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