Restricted String Representations

A string representation of a graph assigns to every vertex a curve in the plane so that two curves intersect if and only if the represented vertices are adjacent. This work investigates string representations of graphs with an emphasis on the shapes of curves and the way they intersect. We strengthen some previously known results and show that every planar graph has string representations where every curve consists of axis-parallel line segments with at most two bends (those are the so-called $B_2$-VPG representations) and simultaneously two curves intersect each other at most once (those are the so-called 1-string representations). Thus, planar graphs are $B_2$-VPG $1$-string graphs. We further show that with some restrictions on the shapes of the curves, string representations can be used to produce approximation algorithms for several hard problems. The $B_2$-VPG representations of planar graphs satisfy these restrictions. We attempt to further restrict the number of bends in VPG representations for subclasses of planar graphs, and investigate $B_1$-VPG representations. We propose new classes of string representations for planar graphs that we call ``order-preserving.'' Order-preservation is an interesting property which relates the string representation to the planar embedding of the graph, and we believe that it might prove useful when constructing string representations. Finally, we extend our investigation of string representations to string representations that require some curves to intersect multiple times. We show that there are outer-string graphs that require an exponential number of crossings in their outer-string representations. Our construction also proves that 1-planar graphs, i.e., graphs that are no longer planar, yet fairly close to planar graphs, may have string representations, but they are not always 1-string

On the minimum edge size for 2-colorability and realizability of hypergraphs by axis-parallel rectangles

Given a hypergraph H = (V, E) what is the minimum integer λ(H) such that the sub-hypergraph with edges of size at least λ(H) is 2-colorable? We consider the computational problem of finding the smallest such integer for a given hypergraph, and show that it is NP-hard to approximate it to within log m where m = |E|. For most geometric hypergraphs, i.e., those defined on a set of n points by intersecting it with some shapes, it is well known that there is a coloring with 2 colors \u27red\u27 and \u27blue\u27, such that any hyperedge containing c log n points, for some constant c, is bi-chromatic, i.e., contains points of both colors. We observe that indeed, for several such hypergraph families, this is the best possible - i.e., there are some n points where there will always be a hyperedge with Ω(log n) points that is mono-chromatic. These results follow from results on the indecomposability of coverings. We also show that a frequently used hypergraph, used in the literature on indecomposable coverings cannot be realized by axis-parallel rectangles in the plane. This problem was mentioned in a paper of Pach et al. on indecomposable coverings

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum