Which nestohedra are removahedra?

TL;DR

This paper characterizes which nestohedra can be realized as removahedra, focusing on those from connected building sets closed under intersection, and establishes a link with chordful graphs.

Contribution

It provides new criteria for realizing nestohedra as removahedra, including two proofs and a characterization involving graphical building sets and chordful graphs.

Findings

Nested complex of certain building sets can be realized as removahedra.

Closure under intersection is sufficient but not necessary for realization.

Graphical building sets correspond to chordful graphs for removahedron realization.

Abstract

A removahedron is a polytope obtained by deleting inequalities from the facet description of the classical permutahedron. Relevant examples range from the associahedra to the permutahedron itself, which raises the natural question to characterize which nestohedra can be realized as removahedra. In this note, we show that the nested complex of any connected building set closed under intersection can be realized as a removahedron. We present two different complementary proofs: one based on the building trees and the nested fan, and the other based on Minkowski sums of dilated faces of the standard simplex. In general, this closure condition is sufficient but not necessary to obtain removahedra. However, we show that it is also necessary to obtain removahedra from graphical building sets, and that it is equivalent to the corresponding graph being chordful (i.e. any cycle induces a clique).