Contractions of Polygons in Abstract Polytopes, Part I

There are several well-known constructions of new polytopes from old, such as the pyramid construction. We define a new local construction on abstract polytopes, called digonal contraction, which allows digonal sections to be removed by merging their two edges into a single edge. This contraction also extends to the question of contracting an edge of a polytope by merging its two vertices. Digonal contraction cannot be applied arbitrarily; we present necessary and sufficient conditions for its use. Digonal contraction also has interesting interactions with another new construction, the k-bubble construction introduced by Helfand, which we describe.