Smaller Extended Formulations for the Spanning Tree Polytope of Bounded-genus Graphs

We give an $O(g^{1/2} n^{3/2} + g^{3/2} n^{1/2})$-size extended formulation for the spanning tree polytope of an $n$-vertex graph embedded on a surface of genus $g$, improving on the known $O(n^2 + g n)$-size extended formulations following from Wong and Martin.Comment: v3: fixed some typo

Reconstructibility of matroid polytopes

We specify what is meant for a polytope to be reconstructible from its graph or dual graph. And we introduce the problem of class reconstructibility, i.e., the face lattice of the polytope can be determined from the (dual) graph within a given class. We provide examples of cubical polytopes that are not reconstructible from their dual graphs. Furthermore, we show that matroid (base) polytopes are not reconstructible from their graphs and not class reconstructible from their dual graphs; our counterexamples include hypersimplices. Additionally, we prove that matroid polytopes are class reconstructible from their graphs, and we present a $O(n^3)$ algorithm that computes the vertices of a matroid polytope from its $n$-vertex graph. Moreover, our proof includes a characterisation of all matroids with isomorphic basis exchange graphs.Comment: 22 pages, 5 figure

Reconstructibility of matroid polytopes

We specify what is meant for a polytope to be reconstructible from its graph or dual graph, and we introduce the problem of class reconstructibility; i.e., the face lattice of the polytope can be determined from the (dual) graph within a given class. We provide examples of cubical polytopes that are not reconstructible from their dual graphs. Furthermore, we show that matroid (base) polytopes are not reconstructible from their graphs and not class reconstructible from their dual graphs; our counterexamples include hypersimplices. Additionally, we prove that matroid polytopes are class reconstructible from their graphs, and we present an O(n3) algorithm that computes the vertices of a matroid polytope from its n-vertex graph. Moreover, our proof includes a characterization of all matroids with isomorphic basis exchange graphs. © 2022 Society for Industrial and Applied Mathematic

Simplicial and Cellular Trees

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed first by Bolker, Kalai and Adin, and more recently by numerous authors, the fundamental topological properties of a tree --- namely acyclicity and connectedness --- can be generalized to arbitrary dimension as the vanishing of certain cellular homology groups. This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley's formula and the matrix-tree theorem. A subtlety of the higher-dimensional case is that enumeration must account for the possibility of torsion homology in trees, which is always trivial for graphs. Cellular trees are the starting point for further high-dimensional extensions of concepts from algebraic graph theory including the critical group, cut and flow spaces, and discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for forthcoming IMA volume "Recent Trends in Combinatorics

The rigidity of infinite graphs

A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in R^d with respect to the classical l^p norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and Henneberg combinatorial characterisations of generic infinitesimal rigidity for finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation of the rigidity of generic finite body-bar frameworks in d-dimensional Euclidean space is generalised to the non-Euclidean l^p norms and to countably infinite graphs. For all dimensions and norms it is shown that a generically rigid countable simple graph is the direct limit of an inclusion tower of finite graphs for which the inclusions satisfy a relative rigidity property. For d>2 a countable graph which is rigid for generic placements in R^d may fail the stronger property of sequential rigidity, while for d=2 the equivalence with sequential rigidity is obtained from the generalised Laman characterisations. Applications are given to the flexibility of non-Euclidean convex polyhedra and to the infinitesimal and continuous rigidity of compact infinitely-faceted simplicial polytopes.Comment: 51 page

Matroid Independence Polytopes and Their Ehrhart Theory

A \emph{matroid} is a combinatorial structure that provides an abstract and flexible model for dependence relations between elements of a set. One way of studying matroids is via geometry: one associates a polytope to a matroid, then uses both combinatorics and geometry to understand the polytope and thereby the original matroid. By a \emph{polytope}, we mean a bounded convex set in Euclidean space $\mathbb{R}^n$ defined by a finite list of linear equations and inequalities, or equivalently as the convex hull of a finite set of points. The best-known polytope associated with a matroid $M$ is its \emph{base polytope} $P(M)$, first introduced by Gel'fand, Goresky, Macpherson and Serganova in 1987~\cite{GGMS}. This dissertation focuses on a closely related construction, the \emph{independence polytope} $Q(M)$, whose combinatorics is much less well understood. Both $P(M)$ and $Q(M)$ are defined as convex hulls of points corresponding to the bases or independence sets, respectively; defining equations and inequalities were given for $P(M)$ by Feichtner and Sturmfels~\cite{Feichtner_Sturmfels} in terms of the ``flacets'' of $M$, and for $Q(M)$ by Schrijver~\cite{Schrijver_B}. One significant difference between the two constructions is that matroid basis polytopes are \emph{generalized permutahedra} as introduced by Postnikov \cite{Beyond}, but independence polytopes do not \emph{a priori} share this structure, so that fewer tools are available in their study. One of the fundamental questions about a polytope is to determine its combinatorial structure as a cell complex: what are its faces of each dimension and which faces contain others? In general it is a difficult problem to extract this combinatorial structure from a geometric description. For matroid base polytopes, the edges (one-dimensional faces) have a simple combinatorial descriptions in terms of the defining matroid, but faces of higher dimension are not understood in general. In Chapter~2 we give an exact combinatorial and geometric description of all the one- and two-dimensional faces of a matroid independence polytope (Theorems~\ref{theorem: Edge Theorem} and ~\ref{theorem: 2 Faces}). One consequence (Proposition~\ref{prop: is a gen perm}) is that matroid independence polytopes can be transformed into generalized permutahedra with no loss of combinatorial structure (at the cost of making the geometry slightly more complicated), which may be of future use. In Chapter~3 we consider polytopes arising from \emph{shifted matroids}, which were first studied by Klivans~\cite{Klivans_Thesis, Klivans_paper}. We describe additional combinatorial structures in shifted matroids, including their circuits, inseparable flats, and flacets, leading to an extremely concrete description of the defining equations and inequalities for both the base and independence polytopes (Theorem~\ref{theorem: pièce de résistance}). As a side note, we observe that shifted matroids are in fact \emph{positroids} in the sense of Postnikov~\cite{Positroid_Postnikov}, although we do not pursue this point of view further. Chapter~4 considers an even more special class of matroids, the \emph{uniform matroids} $U(r,n)$, whose independence polytopes $TC(r,n)=Q_U(r,n)$ are hypercubes in \Rr^n truncated at ``height''~$r$. These polytopes are strongly enough constrained that we can study them from the point of view of Ehrhart theory. For a polytope $P$ whose vertices have integer coordinates, the function i(P,t) = |tP\cap\Zz^n| (that is, the number of integer points in the $t^{th}$ dilate) is a polynomial in $t$ \cite{OG_Ehrhart}, called the \emph{Ehrhart polynomial}. We give two purely combinatorial formulas for the Ehrhart polynomial of $TC(r,n)$, one a reasonably simple summation formula (Theorem~\ref{thm: Ehrhart_Polynomial_Truncated_Cube}) and one a cruder recursive version (Theorem \ref{theorem: Gross_Formula}) that was nonetheless useful in conjecturing and proving the ``nicer'' Theorem~\ref{thm: Ehrhart_Polynomial_Truncated_Cube}. We observe that another fundamental Ehrhart-theoretic invariant, the \emph{$h^*$-polynomial} of $TC(r,n)$, can easily be obtained from work of Li~\cite{Nan_Li} on closely related polytopes called \emph{hyperslabs}. Having computed these Ehrhart polynomials, we consider the location of their complex roots. The integer roots of $i(Q_M,t)$ can be determined exactly even for arbitrary matroids (Theorem \ref{thm: Integer_Roots}), and extensive experimentation using Sage leads us to the conjecture that for all $r$ and $n$, all roots of $TC(r,n)$ have negative real parts. We prove this conjecture for the case $r=2$ (Theorem \ref{thm: Conj_r=2}), where the algebra is manageable, and present Sage data for other values in the form of plots at the end of Chapter~4