Matroid polytopes with small rank

TL;DR

This paper classifies matroid polytopes with small rank, focusing on independence polytopes $P(M)$ and graphic matroid base polytopes $B(M)$ with rank at most $3$, and investigates their connections to order polytopes, stable set polytopes, and edge polytopes. It provides a complete combinatorial characterization: $P(M)$ has rank either $0$ or at least $3$, with rank $0$ iff $P(M)\cong P(M(\mathcal{A}_s))$ and rank $3$ iff $P(M)\cong P(M(\mathcal{D}_{s_1,s_2,s_3}))$, making low-rank independence polytopes graphic; for graphic matroids of $2$-connected graphs, $B(M(G))$ attains ranks $0$–$3$ exactly in the families listed (via $\mathcal{A}_s$, $\mathcal{B}_{\cdots}$, $\mathcal{C}_{\cdots}$, $\mathcal{D}_{\cdots}$). It also compares five unimodular equivalence classes ${\bf MI}_n$, ${\bf GMB}_n$, ${\bf Order}_n$, ${\bf Stab}_n$, ${\bf Edge}_n$ for $n\le 3$, establishing strict inclusions at $n=1,2$, no cross-inclusions at $n=2$ or $n=3$ between several classes, and that ${\bf GMB}_3\subsetneq {\bf Stab}_3\cup{\bf Edge}_3$. The results connect matroid polytope theory with classical polytopes and inform toric geometry via divisor class groups of the associated toric rings, enriching the understanding of how combinatorial structure controls polyhedral and algebraic properties.

Abstract

For a lattice polytope $P$, the rank of $P$ is defined by $F-(\dim P+1)$, where $F$ is the number of facets of $P$. In this paper, we study matroid polytopes with small rank. More precisely, we characterize matroid independence polytopes and graphic matroid base polytopes with rank at most three. Furthermore, using this characterization, we investigate their relationships with order polytopes, stable set polytopes, and edge polytopes.

Figures (9)

• Figure 1: The graph $\mathcal{A}_s$
• Figure 2: The graph $\mathcal{B}_{s_1,\ldots,s_n,p}$
• Figure 3: The graph $\mathcal{C}_{s,t,p,q}$
• Figure 4: The graph $\mathcal{D}_{s_1,s_2,s_3}$
• Figure 5: The graph $G_1$

Theorems & Definitions (31)

• Theorem 1.1: Theorem \ref{['thm: classifyIM']} and Corollary \ref{['cor:independence']}
• Theorem 1.2: Theorem \ref{['thm:smallmultibasepolytope']}
• Theorem 1.3: Propositions \ref{['prop:n=01']},\ref{['prop:n=2']} and \ref{['prop:n=3']}
• Lemma 2.1: schrijver2003combinatorial
• Lemma 2.2: feichtner2004matroidkim2010flag
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3