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We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be realized as a convex polytope.

Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grünbaum-Dress polyhedra, possess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes.

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An obstacle representation of a graph G is a straight-line drawing of G in the plane together with a collection of connected subsets of the plane, called obstacles, that block all non-edges of G while not blocking any of the edges of G. The obstacle number obs(G) is the minimum number of obstacles required to represent G.We study the structure of graphs with obstacle number greater than one. We show that the icosahedron has obstacle number 2, thus answering a question of Alpert, Koch, & Laison asking whether all planar graphs have obstacle number at most 1. We also show that the 1-skeleton of a related polyhedron, the gyroelongated 4-bipyramid, has obstacle number 2. The order of this graph is 10, which is also the order of the smallest known graph with obstacle number 2.Some of our methods involve instances of the Satisfiability problem; we make use of various "SAT solvers" in order to produce computer-assisted proofs.