Polytopality and Cartesian products of graphs

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal. Even though graphs are perhaps the most prominent feature of polytopes, we are still far from being able to answer several basic questions regarding them. For applications, one of the most important ones is to bound the diameter of the graph in terms of the number of variables and inequalities defining the polytope [San10]. From a theoretical point of view, it is striking that we cannot even efficiently decide whether a given graph occurs as the graph of a polytope or not [RG96]. In this paper, we study how polytopality behaves with respect to some common operations on graphs and polytopes. We start by reviewing in Section 1 some necessary conditions for a graph to be polytopal: Balinski’s Theorem [Bal61], the d-Principal Subdivision Property [Bar67] and the Separation Property [Kle64]. Our guideline is to construct graphs satisfying these properties, but which nonetheless are not graphs of polytopes: we say that these graphs are non-polytopal for “nontrivial reasons”. We present three infinite families of such graphs, to illustrate different methods to prove non-polytopality and to introduce general notions and results useful in the rest of the paper. The second part of this paper is dedicated to the study of the polytopality of Cartesian products of graphs. Cartesian products of polytopal graphs are automatically polytopal, and their polytopality range (i.e. the set of possible dimensions of their realizations) has been the subject of recent research [JZ00, Zie04, SZ10, MPP09]. The main contribution of this paper concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show in Section 2.1 that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide in Section 2.2 a general method to construct (non-simple) polytopal products whose factors are not polytopal. Julian Pfeifle was partially supported by MEC grants MTM2008-03020 and MTM2009-07242, and AGAUR grant 2009 SGR 1040. Vincent Pilaud and Francisco Santos were partially supported by MEC grant MTM2008-04699-C03-02. 1 2 JULIAN PFEIFLE, VINCENT PILAUD, AND FRANCISCO SANTOS 1. Non-polytopal graphs for non-trivial reasons Definition 1.1. A graph G is polytopal if it is isomorphic to the graph of some polytope P . If P is d-dimensional, we say that G is d-polytopal. In small dimension, polytopality is easy to deal with. The first interesting question is 3-polytopality, which is characterized by Steinitz’ “Fundamental Theorem of convex types” (see [Grü03, Zie95] for a discussion on different proofs): Theorem 1.2 (Steinitz [Ste22]). A graph G is the graph of a 3-polytope P if and only if G is planar and 3-connected. Moreover, the combinatorial type of P is uniquely determined by G. The first step to realizing a graph G is to understand the possible face lattice of a polytope whose graph is G. For example, it is often difficult to decide which cycles of G can define 2-faces of a d-polytope realizing G. In dimension 3, graphs of 2-faces are characterized by the following separation condition: Theorem 1.3 (Whitney [Whi32]). Let G be the graph of a 3-polytope P . The graphs of the 2-faces of P are precisely the induced cycles in G that do not separate G. In contrast to the easy 2and 3-dimensional worlds, d-polytopality becomes much more involved as soon as d ≥ 4. For example, neighborly 4-polytopes (whose graph is complete) illustrate the difference between the behavior of 3and 4-dimensional polytopes: (i) Starting from a neighborly 4-polytope, and stacking vertices on undesired edges, Perles observed that every graph is an induced subgraph of the graph of a 4-polytope (while only planar graphs are induced subgraphs of graphs of 3-polytopes). (ii) The existence of combinatorially different neighborly polytopes proves that the 2-faces of a 4-polytope cannot be derived from its graph (compare with Whitney’s Theorem). As a consequence of his work on realization spaces of 4-polytopes, Richter-Gebert underlined several deeper negative results: among others, 4-polytopality is NPhard and cannot be characterized by a finite set of “forbidden minors” (see [RG96, Chapter 9]). 1.1. Necessary conditions for polytopality. The first part of this paper focusses on the following necessary conditions for a graph to be polytopal: Proposition 1.4. A d-polytopal graph G satisfies the following properties: (1) Balinski’s Theorem: G is d-connected [Bal61]. (2) Principal Subdivision Property (d-PSP): Every vertex of G is the principal vertex of a principal subdivision of Kd+1. Here, a subdivision of Kd+1 is obtained by replacing edges by paths, and a principal subdivision of Kd+1 is a subdivision in which all edges incident to a distinguished principal vertex are not subdivided [Bar67]. (3) Separation Property: The maximal number of components into which G may be separated by removing n > d vertices equals fd−1 (