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Colouring Contact Graphs of Squares and Rectilinear Polygons Colouring contact graphs of squares and rectilinear polygons Citation for published version (APA): de Berg, M., Markovic, A., & Woeginger, G. (2016). Colouring contact graphs of squares and rectilinear polygons. In 32nd European Workshop on Computational Geometry (EuroCG 2016), 30 March - 1 April, Lugano, Switzerland (pp. 71-74) Document status and date: Published: 01/01/2016 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 02. Oct. 2021 EuroCG 2016, Lugano, Switzerland, March 30–April 1, 2016 Colouring Contact Graphs of Squares and Rectilinear Polygons Mark de Berg∗ Aleksandar Markovic∗ Gerhard Woeginger∗ Abstract tilinear polygons we mean axis-parallel squares and axis-parallel rectilinear polygons.) Contact graphs of We study colourings of contact graphs of squares and squares are different from contact graphs of smooth rectilinear polygons. Our main results are that (i) it objects, because four (interior-disjoint) squares can all is np-hard to decide if a contact graph of unit squares meet in a common point. Thus the obvious embed- is 3-colourable, and (ii) any contact graph of a set of ding of such a contact graph—where we put a node at rectilinear polygons is 6-colourable. the center of each square and we connect the centers of two touching squares by a two-link path through a 1 Introduction touching point—is not necessarily plane. Eppstein et al. [4] studied colourings of contact In graph-colouring problems the goal is to assign a graphs of squares for the special cases where the colour to each node in a graph = (V, E) such that squares form a quadtree subdivision, that is, the set the resulting colouring satisfiesG certain properties. of squares is obtained by recursively subdividing S The standard property is that for any edge (u, v) E an initial square in four equal-sized quadrants. They the nodes u and v have different colours. From now∈ proved that any such contact graph is 6-colourable on, whenever we speak of a colouring of a graph and they gave an example of a quadtree subdivision we mean a colouring with this property. The min- that requires five colours. (They also considered the imum number of colours needed to colour a given variant where two squares that only touch in a single graph is called the chromatic number of the graph. vertex are not considered neighbours.) Two main questions regarding graph colouring are: (i) Given a graph from a certain class of graphs, G how quickly can we compute its chromatic number? Our results. We start by studying the computa- (ii) What is the chromatic number of a given graph tional complexity of colouring contact graphs. We class, that is, the smallest number of colours such that show that already for a set of unit squares, it is np- any graph from the class can be coloured with that complete to decide if the contact graph is 3-colourable. many colours? Next we study the chromatic number of various We are interested in these questions for graphs in- classes of contact graphs. Recall that the obvious em- duced by geometric objects in the plane and, in par- bedding of the contact graph of squares need not be ticular, by contact graphs. Let = P1,...,Pn be plane. We first prove contact graphs of unit squares a set of geometric objects in theS plane.{ The inter-} can have a Km as a minor for an arbitrarily large m section graph induced by is the graph whose nodes and, hence, need not be planar. Nevertheless, contact correspond to the objectsS in and where there is an S graphs of unit squares are 4-colourable and finding a edge (Pi,Pj) if and only if Pi and Pj intersect. If 4-colouring is quite easy, so our np-completeness re- the objects in are closed and have disjoint interiors, S sult on 3-colouring completely characterizes the com- then the intersection graph is called a contact graph. putational complexity of colouring unit squares. Con- It has been shown that the class of contact graphs of tact graphs of arbitrarily-sized squares are not always discs is the same as the class of planar graphs: any 4-colourable—the quadtree example of Eppstein et contact graph of discs is planar and any planar graph al. [4] requiring five colours shows this. We prove that can be drawn as a contact graph of discs [6]. By the the chromatic number of the class of contact graphs of Four-Colour Theorem [1] this implies that any contact arbitrarily-sized squares is at most 6. In fact, we prove graph of discs is 4-colourable. More generally, contact that any contact graph of a set of rectilinear polygons graphs of compact objects with smooth boundaries is 6-colourable. (Even more generally, contact graphs are planar, and so they are 4-colourable. of polygons whose interior angles are strictly greater We are interested in colouring contact graphs of than 2π/5 are 6-colourable.) Thus we obtain the same squares and rectilinear polygons. (Unless explicitly bound of Eppstein et al., but for a much larger class stated otherwise, whenever we speak of squares or rec- of objects. Moreover, for this class the bound is tight. To prove our result, we charaterize contact graphs of ∗Department of Math and CS, TU Eindhoven. MdB, AM and GW are supported by the Netherlands’ Organisation for rectilinear polygons as a certain subset of 1-planar Scientific Research (NWO) under project no. 024.002.003. graphs, which are known to be 6-colourable [2]. This is an extended abstract of a presentation given at EuroCG 2016. It has been made public for the benefit of the community and should be considered a preprint rather than a formally reviewed paper. Thus, this work is expected to appear in a conference with formal proceedings and/or in a journal. 71 32nd European Workshop on Computational Geometry, 2016 c 2 NP-Completeness of 3-Colourability ¬ extension block In this section we establish the hardness of 3- c ¬ Colourability on contact graphs of unit squares. c ¬ Theorem 1 3-Colourability on contact graphs of c unit squares is np-complete. connector squares ¬ basic edge gadget of vertex gadgets c Proof. 3-Colourability on contact graphs is ob- c viously in np. To prove that the problem is np-hard we use a reduction from the np-complete problem of c 3-colouring planar graphs of degree at most 4 [5]. Let = (V, E) be any planar graph on n vertices with degreeG at most 4. Rosenstiehl and Tarjan [7] showed that we can compute in polynomial time a gadget forces the connector squares of the two ver- visibility representation of , in which every vertex tex gadgets it connects to have different colours. It is G easily checked that this implies that the contact graph u V is represented by a horizontal vertex segment su and∈ every edge (u, v) E is represented by a vertical of the generated set of squares is 3-colourable if and ∈ only if the original graph is. Moreover, the entire edge segment that connects su and sv and does not G intersect any other vertex segment. construction can be done in polynomial time. Using a similar proof we can show that 3- a1 a2 a1 a2 Colourability is np-complete for contact graphs of b1 b2 b3 b0 b3 b0 b1 discs, or of any other fixed convex and compact shape. b2 c1 Note that for discs (or other smooth shapes) this set- c1 c2 c2 tles the complexity of the problem completely: con- tact graphs of smooth convex shapes are planar and so The construction can be done so that (i) all y- they are 4-colourable, and checking for 2-colourability coordinates of the vertex segments are multiples of 10, is easy. and (ii) all x-coordinates of the edge segments are multiples of 3 and all x-coordinates of the left and 3 Unit Squares right endpoints of the vertex segments are of the form 3i 1 and 3j + 1 , respectively, for some integers i < j.
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