arXiv subjects

Martin Balko

Publications and source records attributed to Martin Balko.

41 records · Page 3Linked to original sources

arXiv2017

Drawing graphs using a small number of obstacles

An obstacle representation of a graph $G$ is a set of points in the plane representing the vertices of $G$, together with a set of polygonal obstacles such that two vertices of $G$ are connected by an edge in $G$ if and only if the line segment between the corresponding points avoids all the obstacles. The obstacle number ${\rm obs}(G)$ of $G$ is the minimum number of obstacles in an obstacle representation of $G$. We provide the first non-trivial general upper bound on the obstacle number of graphs by showing that every $n$-vertex graph $G$ satisfies ${\rm obs}(G) \leq n\lceil\log{n}\rceil-n+1$. This refutes a conjecture of Mukkamala, Pach, and Pálvölgyi. For $n$-vertex graphs with bounded chromatic number, we improve this bound to $O(n)$. Both bounds apply even when the obstacles are required to be convex. We also prove a lower bound $2^{Ω(hn)}$ on the number of $n$-vertex graphs with obstacle number at most $h$ for $h<n$ and a lower bound $Ω(n^{4/3}M^{2/3})$ for the complexity of a collection of $M \geq Ω(n\log^{3/2}{n})$ faces in an arrangement of line segments with $n$ endpoints. The latter bound is tight up to a multiplicative constant.

math.CO↗

arXiv2016

On the Beer index of convexity and its variants

Let $S$ be a subset of $\mathbb{R}^d$ with finite positive Lebesgue measure. The Beer index of convexity $\operatorname{b}(S)$ of $S$ is the probability that two points of $S$ chosen uniformly independently at random see each other in $S$. The convexity ratio $\operatorname{c}(S)$ of $S$ is the Lebesgue measure of the largest convex subset of $S$ divided by the Lebesgue measure of $S$. We investigate the relationship between these two natural measures of convexity. We show that every set $S\subseteq\mathbb{R}^2$ with simply connected components satisfies $\operatorname{b}(S)\leqα\operatorname{c}(S)$ for an absolute constant $α$, provided $\operatorname{b}(S)$ is defined. This implies an affirmative answer to the conjecture of Cabello et al. that this estimate holds for simple polygons. We also consider higher-order generalizations of $\operatorname{b}(S)$. For $1\leq k\leq d$, the $k$-index of convexity $\operatorname{b}_k(S)$ of a set $S\subseteq\mathbb{R}^d$ is the probability that the convex hull of a $(k+1)$-tuple of points chosen uniformly independently at random from $S$ is contained in $S$. We show that for every $d\geq 2$ there is a constant $β(d)>0$ such that every set $S\subseteq\mathbb{R}^d$ satisfies $\operatorname{b}_d(S)\leqβ\operatorname{c}(S)$, provided $\operatorname{b}_d(S)$ exists. We provide an almost matching lower bound by showing that there is a constant $γ(d)>0$ such that for every $\varepsilon\in(0,1)$ there is a set $S\subseteq\mathbb{R}^d$ of Lebesgue measure $1$ satisfying $\operatorname{c}(S)\leq\varepsilon$ and $\operatorname{b}_d(S)\geqγ\frac{\varepsilon}{\log_2{1/\varepsilon}}\geqγ\frac{\operatorname{c}(S)}{\log_2{1/\operatorname{c}(S)}}$.

math.MG↗

arXiv2014

Crossing numbers and combinatorial characterization of monotone drawings of $K_n$

In 1958, Hill conjectured that the minimum number of crossings in a drawing of $K_n$ is exactly $Z(n) = \frac{1}{4} \lfloor\frac{n}{2}\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor\frac{n-3}{2}\right\rfloor$. Generalizing the result by Ábrego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by $x$-monotone curves. In fact, our proof shows that the conjecture remains true for $x$-monotone drawings of $K_n$ in which adjacent edges may cross an even number of times, and instead of the crossing number we count the pairs of edges which cross an odd number of times. We further discuss a generalization of this result to shellable drawings, a notion introduced by Ábrego et al. We also give a combinatorial characterization of several classes of $x$-monotone drawings of complete graphs using a small set of forbidden configurations. For a similar local characterization of shellable drawings, we generalize Carathéodory's theorem to simple drawings of complete graphs.

math.CO↗

arXiv2013

Bounded Representations of Interval and Proper Interval Graphs

Klavik et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals L_v and R_v called bounds. We ask whether there exists a bounded representation in which each interval I_v has its left endpoint in L_v and its right endpoint in R_v. We show that the problem can be solved in linear time for interval graphs and in quadratic time for proper interval graphs. Robert's Theorem states that the classes of proper interval graphs and unit interval graphs are equal. Surprisingly the bounded representation problem is polynomially solvable for proper interval graphs and NP-complete for unit interval graphs [Klav\'ık et al., arxiv:1207.6960]. So unless P = NP, the proper and unit interval representations behave very differently. The bounded representation problem belongs to a wider class of restricted representation problems. These problems are generalizations of the well-understood recognition problem, and they ask whether there exists a representation of G satisfying some additional constraints. The bounded representation problems generalize many of these problems.

cs.DM↗

arXiv2012

Grid Representations and the Chromatic Number

A grid drawing of a graph maps vertices to grid points and edges to line segments that avoid grid points representing other vertices. We show that there is a number of grid points that some line segment of an arbitrary grid drawing must intersect. This number is closely connected to the chromatic number. Second, we study how many columns we need to draw a graph in the grid, introducing some new $\NP$-complete problems. Finally, we show that any planar graph has a planar grid drawing where every line segment contains exactly two grid points. This result proves conjectures asked by David Flores-Peñaloza and Francisco Javier Zaragoza Martinez.

math.CO↗