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Clemens Huemer

Publications and source records attributed to Clemens Huemer.

At least 19 recordsLinked to original sources
arXiv2026

A Geometric View of Combinatorial Fiedler Theory

Recently, Andrade and Dahl introduced combinatorial Fiedler theory by studying a parameter $b(G)$ defined as the $\ell_1$-analog of the Rayleigh quotient minimization characterization of the algebraic connectivity of a graph $G=(V,E)$. In this work, we study the corresponding maximization problem, which plays the role of the $\ell_1$-analog of the largest Laplacian eigenvalue. We show that the new parameter $B(G)$ associated with this maximization problem admits a simple exact description: it is the average of the two largest vertex degrees of $G$. A unified combinatorial treatment of the minimization and maximization problems is presented first. Later, both optimization problems are reinterpreted in a geometrical setting. The feasible set is identified with a $(n-2)$-dimensional cuboctahedron shell where $n=|V|$. Additional structure is presented for this polyhedron, including the fact that maximizing solutions arise at its vertices and minimizing solutions arise at the centers of its facets. Finally, we analyze the number of optimal vectors for $b(G)$ and $B(G)$ for several graph families. Although the value of $B(G)$ is determined by the two largest degrees, we prove that counting the vectors that attain this value is actually $\#\mathrm{P}$-complete.

cs.CG↗
arXiv2025

On the number of drawings of a combinatorial triangulation

In 1962, Tutte provided a formula for the number of combinatorial triangulations, that is, maximal planar graphs with a fixed triangular face and $n$ additional vertices. In this note, we study how many ways a combinatorial triangulation can be drawn as geometric triangulation, that is, with straight-line segments, on a given point set in the plane. Our central contribution is that there exists a combinatorial triangulation with n vertices that can be drawn in at least $Ω(1,31^n)$ ways on a set of n points as different geometric triangulations. We also show an upper bound on the number of drawings of a combinatorial triangulation on the so-called double chain point set.

math.CO↗
arXiv2024

On polynomials associated to Voronoi diagrams of point sets and crossing numbers

Three polynomials are defined for given sets $S$ of $n$ points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-$k$ Voronoi diagrams of $S$, the circle polynomial with coefficients the numbers of circles through three points of $S$ enclosing $k$ points of $S$, and the $E_{\leq k}$ polynomial with coefficients the numbers of (at most $k$)-edges of $S$. We present several formulas for the rectilinear crossing number of $S$ in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if, and only if, $S$ is in convex position. Further, we present bounds on the location of the roots of these polynomials.

math.CO↗
arXiv2024

Sibson's formula for higher order Voronoi diagrams

Let $S$ be a set of $n$ points in general position in $\mathbb{R}^d$. The order-$k$ Voronoi diagram of $S$, $V_k(S)$, is a subdivision of $\mathbb{R}^d$ into cells whose points have the same $k$ nearest points of $S$. Sibson, in his seminal paper from 1980 (A vector identity for the Dirichlet tessellation), gives a formula to express a point $Q$ of $S$ as a convex combination of other points of $S$ by using ratios of volumes of the intersection of cells of $V_2(S)$ and the cell of $Q$ in $V_1(S)$. The natural neighbour interpolation method is based on Sibson's formula. We generalize his result to express $Q$ as a convex combination of other points of $S$ by using ratios of volumes from Voronoi diagrams of any given order.

cs.CG↗
arXiv2022

Voronoi Diagrams of Arbitrary Order on the Sphere

For a given set of points $U$ on a sphere $S$, the order $k$ spherical Voronoi diagram $SV_k(U)$ decomposes the surface of $S$ into regions whose points have the same $k$ nearest points of $U$. Hyeon-Suk Na, Chung-Nim Lee, and Otfried Cheong (Comput. Geom., 2002) applied inversions to construct $SV_1(U)$. We generalize their construction for spherical Voronoi diagrams from order $1$ to any order $k$. We use that construction to prove formulas for the numbers of vertices, edges, and faces in $SV_k(U)$. These formulas were not known before. We obtain several more properties for $SV_k(U)$, and we also show that $SV_k(U)$ has a small orientable cycle double cover.

cs.CG↗
arXiv2021

The edge labeling of higher order Voronoi diagrams

We present an edge labeling of order-$k$ Voronoi diagrams, $V_k(S)$, of point sets $S$ in the plane, and study properties of the regions defined by them. Among them, we show that $V_k(S)$ has a small orientable cycle and path double cover, and we identify configurations that cannot appear in $V_k(S)$ for small values of $k$. This paper also contains a systematic study of well-known and new properties of $V_k(S)$, all whose proofs only rely on elementary geometric arguments in the plane. The maybe most comprehensive study of structural properties of $V_k(S)$ was done by D.T. Lee (On k-nearest neighbor Voronoi diagrams in the plane) in 1982. Our work reviews and extends the list of properties of higher order Voronoi diagrams.

math.CO↗
arXiv2020

New production matrices for geometric graphs

We use production matrices to count several classes of geometric graphs. We present novel production matrices for non-crossing partitions, connected geometric graphs, and k-angulations, which provide another way of counting the number of such objects. Counting geometric graphs is then equivalent to calculating the powers of a production matrix. Applying the technique of Riordan Arrays to these production matrices, we establish new formulas for the numbers of geometric graphs as well as combinatorial identities derived from the production matrices. Further, we obtain the characteristic polynomial and the eigenvectors of such production matrices.

math.CO↗
arXiv2019

On Maximum-Sum Matchings of Points

Huemer et al. (Discrete Mathematics, 2019) proved that for any two point sets $R$ and $B$ with $|R|=|B|$, the perfect matching that matches points of $R$ with points of $B$, and maximizes the total \emph{squared} Euclidean distance of the matched pairs, verifies that all the disks induced by the matching have a common point. Each pair of matched points $p\in R$ and $q\in B$ induces the disk of smallest diameter that covers $p$ and $q$. Following this research line, in this paper we consider the perfect matching that maximizes the total Euclidean distance. First, we prove that this new matching for $R$ and $B$ does not always ensure the common intersection property of the disks. Second, we extend the study of this new matching for sets of $2n$ uncolored points in the plane, where a matching is just a partition of the points into $n$ pairs. As the main result, we prove that in this case all disks of the matching do have a common point. This implies a big improvement on a conjecture of Andy Fingerhut in 1995, about a maximum matching of $2n$ points in the plane.

cs.CG↗
arXiv2019

On weighted sums of numbers of convex polygons in point sets

Let $S$ be a set of $n$ points in general position in the plane, and let $X_{k,\ell}(S)$ be the number of convex $k$-gons with vertices in $S$ that have exactly $\ell$ points of $S$ in their interior. We prove several equalities for the numbers $X_{k,\ell}(S)$. This problem is related to the Erdős-Szekeres theorem. Some of the obtained equations also extend known equations for the numbers of empty convex polygons to polygons with interior points. Analogous results for higher dimension are shown as well.

math.CO↗
arXiv2019

On circles enclosing many points

We prove that every set of $n$ red and $n$ blue points in the plane contains a red and a blue point such that every circle through them encloses at least $n(1-\frac{1}{\sqrt{2}}) -o(n)$ points of the set. This is a two-colored version of a problem posed by Neumann-Lara and Urrutia. We also show that every set $S$ of $n$ points contains two points such that every circle passing through them encloses at most $\lfloor{\frac{2n-3}{3}}\rfloor$ points of $S$. The proofs make use of properties of higher order Voronoi diagrams, in the spirit of the work of Edelsbrunner, Hasan, Seidel and Shen on this topic. Closely related, we also study the number of collinear edges in higher order Voronoi diagrams and present several constructions.

cs.CG↗
arXiv2019

Coloring decompositions of complete geometric graphs

A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $χ'([G,P])$ of a decomposition $[G,P]$ is the smallest number $k$ for which there exists a $k$-coloring of the elements of $P$ in such a way that: for every element of $P$ all of its edges have the same color, and if two members of $P$ share at least one vertex, then they have different colors. A long standing conjecture of Erdős-Faber-Lovász states that every decomposition $[K_n,P]$ of the complete graph $K_n$ satisfies $χ'([K_n,P])\leq n$. In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic index of several types of decompositions when the vertices of the graph are in general position. We also consider the particular case in which the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions.

math.CO↗
arXiv2019

Matching points with disks with a common intersection

We consider matchings with diametral disks between two sets of points R and B. More precisely, for each pair of matched points p in R and q in B, we consider the disk through p and q with the smallest diameter. We prove that for any R and B such that |R|=|B|, there exists a perfect matching such that the diametral disks of the matched point pairs have a common intersection. In fact, our result is stronger, and shows that a maximum weight perfect matching has this property.

cs.CG↗
arXiv2018

Optimal Grid Drawings of Complete Multipartite Graphs and an Integer Variant of the Algebraic Connectivity

How to draw the vertices of a complete multipartite graph $G$ on different points of a bounded $d$-dimensional integer grid, such that the sum of squared distances between vertices of $G$ is (i) minimized or (ii) maximized? For both problems we provide a characterization of the solutions. For the particular case $d=1$, our solution for (i) also settles the minimum-2-sum problem for complete bipartite graphs; the minimum-2-sum problem was defined by Juvan and Mohar in 1992. Weighted centroidal Voronoi tessellations are the solution for (ii). Such drawings are related with Laplacian eigenvalues of graphs. This motivates us to study which properties of the algebraic connectivity of graphs carry over to the restricted setting of drawings of graphs with integer coordinates.

cs.DM↗
arXiv2017

Caratheodory's Theorem in Depth

Let $X$ be a finite set of points in $\mathbb{R}^d$. The Tukey depth of a point $q$ with respect to $X$ is the minimum number $τ_X(q)$ of points of $X$ in a halfspace containing $q$. In this paper we prove a depth version of Carathéodory's theorem. In particular, we prove that there exists a constant $c$ (that depends only on $d$ and $τ_X(q)$) and pairwise disjoint sets $X_1,\dots, X_{d+1} \subset X$ such that the following holds. Each $X_i$ has at least $c|X|$ points, and for every choice of points $x_i$ in $X_i$, $q$ is a convex combination of $x_1,\dots, x_{d+1}$. We also prove depth versions of Helly's and Kirchberger's theorems.

cs.CG↗
arXiv2016

The intersection graph of the disks with diameters the sides of a convex $n$-gon

Given a convex polygon of $n$ sides, one can draw $n$ disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the undirected graph with vertices the $n$ disks and two disks are adjacent if and only if they have a point in common. We prove that for every convex polygon this graph is planar. Particularly, for $n=5$, this shows that for any convex pentagon there are two disks among the five side disks that do not intersect, which means that $K_5$ is never the intersection graph of such five disks. For $n=6$, we then have that for any convex hexagon the intersection graph of the side disks does not contain $K_{3,3}$ as subgraph.

math.MG↗
arXiv2015

On Cyclic Kautz Digraphs

A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs $MCK(d,\ell)$ and it is derived from the Kautz digraphs $K(d,\ell)$. %It is not common to find non-regular digraphs with minimal diameter given their number of vertices and out-degree. It is well-known that the Kautz digraphs $K(d,\ell)$ have the smallest diameter among all digraphs with their number of vertices and degree. We define the cyclic Kautz digraphs $CK(d,\ell)$, whose vertices are labeled by all possible sequences $a_1\ldots a_\ell$ of length $\ell$, such that each character $a_i$ is chosen from an alphabet containing $d+1$ distinct symbols, where the consecutive characters in the sequence are different (as in Kautz digraphs), and now also requiring that $a_1\neq a_\ell$. The cyclic Kautz digraphs $CK(d,\ell)$ have arcs between vertices $a_1 a_2\ldots a_\ell$ and $a_2 \ldots a_\ell a_{\ell+1}$, with $a_1\neq a_\ell$ and $a_2\neq a_{\ell+1}$. Unlike in Kautz digraphs $K(d,\ell)$, any label of a vertex of $CK(d,\ell)$ can be cyclically shifted to form again a label of a vertex of $CK(d,\ell)$. We give the main parameters of $CK(d,\ell)$: number of vertices, number of arcs, and diameter. Moreover, we construct the modified cyclic Kautz digraphs $MCK(d,\ell)$ to obtain the same diameter as in the Kautz digraphs, and we show that $MCK(d,\ell)$ are $d$-out-regular. Finally, we compute the number of vertices of the iterated line digraphs of $CK(d,\ell)$.

math.CO↗