The notion of topological complexity was introduced by M. Farber [Discrete Comput. Geom. 29, No. 2, 211–221 (2003; Zbl 1038.68130)]. For a topological space \(X\), the topological complexity TC\((X)\) is the minimal number \(k\), such that there is a partition \(X\times X=E_1\sqcup \dots \sqcup E_k\), with each \(E_i\) locally compact and admitting a continuous function \(\phi_i:E_i\rightarrow P(X)\) such that \(\phi_i(x_0,x_1)\) is a path from \(x_0\) to \(x_1\). The function \(\phi_i\) is called a motion-planning rule. “If \(X\) is the space of configurations of the one or more robots, this models the number of continuous rules required to program the robots or move between any two configurations”.
Later, in 2021, D. Recio-Mitter [J. Appl. Comput. Topol. 5, No. 1, 141–178 (2021; Zbl 1485.53050)] suggested that if \(X\) is a metric space, then one should consider the condition that the paths \(\phi_i(x_0,x_1)\) are minimal geodesics (shortest paths) from \(x_0\) to \(x_1\), and define geodesic complexity, GC\((X)\), to be the smallest \(k\), such that there exists such a decomposition where the \(\phi_i(x_0,x_1)\) are minimal geodesics from \(x_0\) to \(x_1\). The present author specifies that the “Recio-Mitter definition of GC\((X)=k\) involved partitions into sets \(E_0,\dots ,E_k\), which for technical reasons, has become the more common definition of concepts of this sort, but we prefer to stick with Faber’s more intuitive formulation”. Each function \(\phi_i\) is called a geodesic motion-planning rule (GMPR).
The author considers an example from [D. Recio-Mitter, loc. cit.], where \(X\) is the surface of a cube and he mentions that the following result is well known: TC\((X)=\mathbf{TC}(S^2)=3\). Moreover, by a result of Recio-Mitter one has GC(\(X)\geq 4\). In the present paper the author proves:
Theorem 1.1. If \(X\) is the surface of a cube, then GC\((X)=5\).
Remark. For comparison, the author mentions another result by him: For a regular tetrahedron \(T\), one has GC(\(T\))=4 or 5 proved in [D. M. Davis, “Geodesic complexity of a tetrahedron”, Preprint, arXiv:2306.11059], and TC(\(T)=\mathbf{TC}(S^2)=3\). He states that this result relies heavily on the work of the author and M. Guo [Discrete Math. 347, No. 1, Article ID 113709, 10 p. (2024; Zbl 1526.05120)], where they analyzed the isomorphism classes of cut loci on the cube as labeled graphs. In Section 2 of the present paper the author reviews the relevant parts of that work [loc. cit].
In Section 3 it is proved that GC(\(X)\leq 5\), and in Section 4 the inequality GC(\(X)\geq 5\) is proved, “using methods similar to those used by the author in [“Geodesic complexity of a tetrahedron”, Preprint, arXiv:2306.11059] and by M. Sharir and A. Schorr [SIAM J. Comput. 15, 193–215 (1986; Zbl 0612.68090)].”