Geodesic complexity (Q6641610)

The notion of topological complexity was introduced by \textit{M. Farber} [Discrete Comput. Geom. 29, No. 2, 211--221 (2003; Zbl 1038.68130)]. \N\NFrom the introduction: ``The topological complexity of a topological space \(X\), denoted by \(TC(X)\), involves finding the smallest partition of \(X\times X\) into sets on which you can make a continuous choice of a path from \(x_0\) to \(x_1\) for all \((x_0,x_1)\in S\). In [J. Appl. Comput. Topol. 5, No. 1, 141--178 (2021; Zbl 1485.53050)], \textit{D. Recio-Mitter} had the idea of requiring that, if \(X\) is a metric space, the paths be minimal geodesics, i.e., shortest paths between their endpoints. He defined the geodesic complexity, \(GC(X)\), for a metric space \(X\) as follows. If \(X\) is a metric space, \(GC(X)\) is the smallest integer \(k\) such that there is a partition of \(X\times X\) into locally compac sets \(E_i,0\leq i\leq k\), such that for each \(i\), there is a continuous function \(\sigma_i:E_i\rightarrow P(X)\), where \(P(X)\) is the space of all paths in \(X\), and for all \((x_0,x_1)\in E_i,\sigma_i(x_0,x_1)\) is a (minimal) geodesic from \(x_0\) to \(x_1\). Each \(\sigma_i\) is called a geodesic motion planning rule (GMPR).'' \N\NFrom the abstract: ``\dots We survey many of the interesting results obtained by Recio-Mitter. Then we focus on two cases, the cube and the 2-point ordered configuration space of a star graph, with a fairly through sketch of the proof of the geodesic complexity of each''. \N\NThe second section after the Introduction is entitled ``Two robots moving on a star graph'', and at the beginning of it, the author states: ``In this section, we present many of the main ideas in [\textit{D. M. Davis} et al., Algebr. Geom. Topol. 22, No. 2, 785--814 (2022; Zbl 1497.55005)] regarding \( GC(F_\varepsilon(G,2))\), where \(G\) is a star graph with at least four edges''. The third section is entitled ``Geodesic complexity of a cube''. In its opening the author states again: ``In this section, we sketch the proof from [\textit{D. M. Davis}, Tbil. Math. J. 14, No. 1, 149--162 (2021; Zbl 1494.53047)]''. Finally, the article also contains 16 figures, about which the author does not specify whether they are original or have also appeared in the other three articles mentioned.\N\NFor the entire collection see [Zbl 1547.55002].