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Register a new userDrawing polytopal graphs with polymake
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This note wants to explain how to obtain meaningful pictures of (possibly high-dimensional) convex polytopes, triangulated manifolds, and other objects from the realm of geometric combinatorics such as tight spans of finite metric spaces and tropical polytopes. In all our cases we arrive at specific, geometrically motivated, graph drawing problems. The methods displayed are implemented in the software system polymake.
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Let $ngeq 3$ and $r_n$ be a $3$-polytopal graph such that for every $3leq ileq n$, $r_n$ has at least one vertex of degree $i$. We find the minimal vertex count for $r_n$. We then describe an algorithm to construct the graphs $r_n$. A dual statement may be formulated for faces of $3$-polytopes. The ideas behind the algorithm generalise readily to solve related problems. Moreover, given a $3$-polytope $t_l$ comprising a vertex of degree $i$ for all $3leq ileq l$, $l$ fixed, we define an algorithm to output for $n>l$ a $3$-polytope $t_n$ comprising a vertex of degree $i$, for all $3leq ileq n$, and such that the initial $t_l$ is a subgraph of $t_n$. The vertex count of $t_n$ is asymptotically optimal, in the sense that it matches the aforementioned minimal vertex count up to order of magnitude, as $n$ gets large. In fact, we only lose a small quantity on the coefficient of the second highest term, and this quantity may be taken as small as we please, with the tradeoff of first constructing an accordingly large auxiliary graph.
We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $Gamma$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$, the ratio between the minimum length of any path from $u$ to $v$ and the Euclidean distance between $u$ and $v$ is small. The maximum such ratio, over all pairs of vertices of $G$, is the spanning ratio of $Gamma$. First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio $1$, a proper straight-line drawing with spanning ratio $1$, and a planar straight-line drawing with spanning ratio $1$ are NP-complete, $exists mathbb R$-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex. Second, we show that moving from spanning ratio $1$ to spanning ratio $1+epsilon$ allows us to draw every graph. Namely, we prove that, for every $epsilon>0$, every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than $1+epsilon$. Third, our drawings with spanning ratio smaller than $1+epsilon$ have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio.
We report on a recent implementation of patchworking and real tropical hypersurfaces in $texttt{polymake}$. As a new mathematical contribution we provide a census of Betti numbers of real tropical surfaces.
Timeslices are often used to draw and visualize dynamic graphs. While timeslices are a natural way to think about dynamic graphs, they are routinely imposed on continuous data. Often, it is unclear how many timeslices to select: too few timeslices can miss temporal features such as causality or even graph structure while too many timeslices slows the drawing computation. We present a model for dynamic graphs which is not based on timeslices, and a dynamic graph drawing algorithm, DynNoSlice, to draw graphs in this model. In our evaluation, we demonstrate the advantages of this approach over timeslicing on continuous data sets.
In this paper we introduce and study the class of d-ball packings arising from edge-scribable polytopes. We are able to generalize Apollonian disk packings and the well-known Descartes theorem in different settings and in higher dimensions. After introducing the notion of Lorentzian curvature of a polytope we present an analogue of the Descartes theorem for all regular polytopes in any dimension. The latter yields to nice curvature relations which we use to construct integral Apollonian packings based on the Platonic solids. We show that there are integral Apollonian packings based on the tetrahedra, cube and dodecahedra containing the sequences of perfect squares. We also study the duality, unicity under Mobius transformations as well as generalizations of the Apollonian groups. We show that these groups are hyperbolic Coxeter groups admitting an explicit matrix representation. An unexpected invariant, that we call Mobius spectra, associated to Mobius unique polytopes is also discussed.
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