In this paper we generalize the graph associahedron by associating the vertices of graphs with the facets of simple polytopes, and truncating faces of the polytope based on connected subgraphs with restrictions. In the special case where the initial polytope is a hypercube, we examine connected subgraphs of graphs with positive and negative vertices. Certain graphs give us the permutahedron, the associahedron, the type *B*_{n} permutahedron, and polytopes conjectured to be of bi-Catalan combinatorial type.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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