We consider the problem of determining the cardinality $\psi(H_{2,k})$ of minimal doubly resolving sets of Hamming graphs $H_{2,k}.$ We prove that for $k \geq 6$ every minimal resolving set of $H_{2,k}$ is also a doubly resolving set, and, consequently, $\psi(H_{2,k})$ is equal to the metric dimension of $H_{2,k},$ which is known from the literature. Moreover, we find an explicit expression for the strong metric dimension of all Hamming graphs $H_{n,k}.
As toroid (polyhedral torus) could not be convex, it is questionable if it is possible to 3-triangulate them (i.e. divide into tetrahedra with the original vertices). Here, we will discuss some examples of toroids to show that for each vertex number n ≥ 7, there exists a toroid for which triangulation is possible. Also we will study the necessary number of tetrahedra for the minimal triangulation.
There are investigated supergroups of some hyperbolic space groups with simplicial fundamental domain. If the vertices of these simplices are out of the absolute, we can truncate them by polar planes of the vertices and the new polyhedra are fundamental ones of the richer groups. In papers of E. Molnár, I. Prok and J. Szirmai the simplices, investigated here, are collected in families F3, F4 and F6. We have constructed at least one new hyperbolic space group for each truncated simplex in these families.
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