Minkowski decomposability of polytopes is developed via geometric graphs and decomposing functions; recall, a geometric graph is a realisation in $\R^{d}$ of a graph $G$ where distinct vertices of $G$ correspond to distinct points in $\R^{d}$, edges in $G$ correspond to line segments, and no three vertices are collinear. One advantage of this approach is its versatility. The decomposability of polytopes reduces to the decomposability of geometric graphs, which are not necessarily polytopal. And the decomposability of geometric graphs often revolves around the existence of suitable subgraphs or useful properties in the graphs. Section 6.3 is devoted to the classification of polytopes with at most $2d+1$ vertices into composable and decomposable. The chapter concludes with an interlude on polytopes that admit both a decomposable realisation and an indecomposable realisation. It is one of the few places in the book where we deal with a noncombinatorial property. Whereas the combinatorial type of a polytope can have many distinct realisations, here we will be concerned with concrete realisations of the type and their properties.

The {graph} of a polytope is formed by the vertices and edges of the polytope.We often see this graph as an abstract graph and apply methodology from graph theory. A {polytopal graph} is simply a graph of a polytope. Appendix C reviews the relevant graph-theoretical prerequisites. Graphs of 3-polytopes are planar, and thus we review the topological background to study graphs embedded in a topological space. We then study properties of polytopal graphs. We analyse acyclic orientations of graphs of polytopes in Section 3.5; these orientations are related to the shelling orders of the corresponding dual polytopes. We also examine convex realisations of 3-connected planar graphs and Steinitz’s characterisation of graphs of 3-polytopes. The graph of a 3-polytope contains a subdivision of the graph of the 3-simplex, namely $K^{4}$.Section 3.9 shows that this extends to every dimension. Since $K^{5}$ is the 1-skeleton of a $4$-simplex, the nonplanarity of $K^{5}$ is a special case of a theoremof Flores (1934) and Van Kampen (1932) that states the $d$-skeleton of the $(2d+2)$-simplex cannot be embedded in $\R^{2d}$ (Section 3.10).

A nontrivial graph is \textit {$r$-connected}, for $r\ge 0$, if it has more than $r$ vertices and no two vertices are separated by fewer than $r$ other vertices. Graphs of $d$-polytopes are $d$-connected, according to Balinski (1961). The chapter also discusses a recent result of Pilaud et al (2022) on the edge connectivity of simplicial polytopes. We examine the higher connectivity of strongly connected complexes in Section 4.3. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs.Graphs of $d$-polytopes are $\floor{(d+2)/3}$-linked, but not all graphs of $d$-polytopes are $\floor{d/2}$-linked.Edge linkedness can be defined similarly by replacing vertex-disjoint paths in the definition of linkedness by edge-disjoint paths. The $d$-connectivity of a graph of a $d$-polytope is a particular case of a more general result of Athanasiadis (2009) on the connectivity of $(r,r+1)$-incidence graphs of a $d$-polytope. The chapter ends with a short discussion on the connectivity of incidence graphs.

Convex polytopes, or simply polytopes, are geometric objects in some space $\R^{d}$; in fact, they are bounded intersections of finitely many closed halfspaces in $\R^{d}$.The space $\R^{d}$ can be regarded as a linear space or an affine space, and its linear or affine subspaces can be described by linear or affine equations. We introduce the basic concepts and results from linear algebra that allow the description and analysis of these subspaces. A polytope can alternatively be described as the convex hull of a finite set of points in $\R^{d}$, and so it is a convex set. Convex sets are therefore introduced, as well as their topological properties, with emphasis on relative notions as these are based on a more natural setting, the affine hull of the set. We then review the separation and support of convex sets by hyperplanes. A convex set is formed by fitting together other polytopes of smaller dimensions, its faces; Section 1.7 discusses them.Finally, the chapter studies convex cones and lineality spaces of convex sets; these sets are closely connected to the structure of unbounded convex sets.

Reconstructing a $d$-polytope from its $k$-skeleton ($k\le d-2$) amounts to determining the face lattice of the polytope from the dimension and skeleton. For each $d\ge 4$, there are $d$-polytopes that have isomorphic $(d-3)$-skeleta and yet are not combinatorially isomorphic. But every $d$-polytope is reconstructible from its $(d-2)$-skeleton. Section 5.2 focusses on reconstructions from 2-skeletons and 1-skeletons. It presents an algorithm that reconstructs a $d$-polytope with at most $d-2$ nonsimple vertices from its dimension and 2-skeleton. This result is tight: there are pairs of nonisomorphic $d$-polytopes with $d-1$ nonsimple vertices and isomorphic $(d-3)$-skeleta for each $d\ge 4$.Blind and Mani-Levitska (1987), and later Kalai (1988), showed that a simple polytope can be reconstructed from its dimension and graph. We present a slight generalisation of this result and briefly discuss the theorem of Friedman (2009) stating that the reconstruction can be done in time polynomial in the number of vertices. The chapter ends with variations on the reconstruction problem.

We investigate numbers of faces of polytopes. We begin with the face numbers of 3-polytopes. The characterisation of $f$-vectors of $d$-polytopes ($d\ge 4$) is beyond our current means.In view of this, researchers have considered characterisations of the "projections" of the $f$-vectors, namely the proper subsequences of the $f$-vector; we review the existing results. Section 8.2 gives a proof of a theorem of Xue (2021) on the minimum number of faces of $d$-polytopes with at most $2d$ vertices, answering a conjecture of Grunbaum (2003). This is followed by results on the minimum number of faces of $d$-polytopes with more than $2d$ vertices. We then discuss the lower and upper bound theorems for simplicial polytopes, due to Barnette (1973) and McMullen (1970), respectively, and their extensions such as the $g$-conjecture of McMullen (1971), now the $g$-theorem. The proof of the lower bound theorem connects rigidity theory and the combinatorics polytopes. The chapter ends with a discussion of the flag vector of a polytope. This includes a result of Bayer and Billera (1985) on linear equations for flag vectors like the Dehn--Sommerville’s equations for simplicial polytopes.

The \textit{diameter} of a graph $G$, denoted $\diam G$, is the maximum distance between any two vertices in the graph. The \textit{diameter} of polyhedra is defined as the diameter of their graphs. While the chapter focusses on polytopes, polyhedra also feature in it.There is a connection between diameters of polyhedra and linear programming, and this is partially materialised through the \defn{Hirsch conjectures}, conjectures that relate the diameter of a polyhedra with its dimension and number of facets. We first show that the unbounded and monotonic versions of these conjectures are false (Section 7.2). Early on, Klee and Walkup (1967) realised that problems on the diameter of polyhedra can be reduced to problems on the diameter of simple polyhedra; this and other reductions are the focus of Section 7.3. We also present the counterexample of Santos (2012) for the bounded Hirsch conjecture. We then move to examine lower and upper bounds for the diameter of general polytopes and the diameter of specific polytopes. The final section is devoted to generalisations of polyhedra where diameters may be easier to compute or estimate.

Convex polytopes can be equivalently defined as bounded intersections of finitely many halfspaces in some $\R^{d}$ and as convex hulls of finitely many points in $\R^{d}$.A halfspace is defined by a linear inequality, and each nonempty closed convex set in $\R^{d}$ is the set of solutions of a system of possibly infinite linear inequalities.If we have a finite number of inequalities, the set is a \textit{polyhedron}.Polyhedra are therefore generalisations of polytopes and polyhedral cones. Many assertions in this chapter, such as the facial structure of polytopes, are derived from analogous assertions about polyhedra. We learn how to preprocess objects via projective transformations to simplify the solution of a problem. We then discuss common examples of polytopes. For the visualisation of low-dimensional polytopes, we study Schlegel diagrams, a special type of polytopal complex. We will also examine common results in polytope theory such as the Euler--Poincar\’e--Schl\"afli equation, and a theorem of Bruggesser and Mani (1971) on the existence of shelling orders.The chapter ends with Gale transforms, a useful device to study polytopes with small number of vertices.