Colouring contact graphs of squares and rectilinear polygons de Berg, M.T.; Markovic, A.; Woeginger, G.

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1 Colouring ontat graphs of squares and retilinear polygons de Berg, M.T.; Markovi, A.; Woeginger, G. Published in: nd European Workshop on Computational Geometry (EuroCG 06), 0 Marh - April, Lugano, Switzerland Published: 0/0/06 Doument Version Publisher s PDF, also known as Version of Reord (inludes final page, issue and volume numbers) Please hek the doument version of this publiation: A submitted manusript is the author's version of the artile upon submission and before peer-review. There an be important differenes between the submitted version and the offiial published version of reord. People interested in the researh are advised to ontat the author for the final version of the publiation, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publiation after peer review. The final published version features the final layout of the paper inluding the volume, issue and page numbers. Link to publiation Citation for published version (APA): de Berg, M., Markovi, A., & Woeginger, G. (06). Colouring ontat graphs of squares and retilinear polygons. In nd European Workshop on Computational Geometry (EuroCG 06), 0 Marh - April, Lugano, Switzerland (pp. 7-7) General rights Copyright and moral rights for the publiations made aessible in the publi portal are retained by the authors and/or other opyright owners and it is a ondition of aessing publiations that users reognise and abide by the legal requirements assoiated with these rights. Users may download and print one opy of any publiation from the publi portal for the purpose of private study or researh. You may not further distribute the material or use it for any profit-making ativity or ommerial gain You may freely distribute the URL identifying the publiation in the publi portal? Take down poliy If you believe that this doument breahes opyright please ontat us providing details, and we will remove aess to the work immediately and investigate your laim. Download date:. De. 07

2 EuroCG 06, Lugano, Switzerland, Marh 0 April, 06 Colouring Contat Graphs of Squares and Retilinear Polygons Mark de Berg Aleksandar Markovi Gerhard Woeginger Abstrat We study olourings of ontat graphs of squares and retilinear polygons. Our main results are that (i) it is np-hard to deide if a ontat graph of unit squares is -olourable, and (ii) any ontat graph of a set of retilinear polygons is 6-olourable. Introdution In graph-olouring problems the goal is to assign a olour to eah node in a graph G = (V, E) suh that the resulting olouring satisfies ertain properties. The standard property is that for any edge (u, v) E the nodes u and v have different olours. From now on, whenever we speak of a olouring of a graph we mean a olouring with this property. The minimum number of olours needed to olour a given graph is alled the hromati number of the graph. Two main questions regarding graph olouring are: (i) Given a graph G from a ertain lass of graphs, how quikly an we ompute its hromati number? (ii) What is the hromati number of a given graph lass, that is, the smallest number of olours suh that any graph from the lass an be oloured with that many olours? We are interested in these questions for graphs indued by geometri objets in the plane and, in partiular, by ontat graphs. Let S = {P,..., P n } be a set of geometri objets in the plane. The intersetion graph indued by S is the graph whose nodes orrespond to the objets in S and where there is an edge (P i, P j ) if and only if P i and P j interset. If the objets in S are losed and have disjoint interiors, then the intersetion graph is alled a ontat graph. It has been shown that the lass of ontat graphs of diss is the same as the lass of planar graphs: any ontat graph of diss is planar and any planar graph an be drawn as a ontat graph of diss [6]. By the Four-Colour Theorem [] this implies that any ontat graph of diss is -olourable. More generally, ontat graphs of ompat objets with smooth boundaries are planar, and so they are -olourable. We are interested in olouring ontat graphs of squares and retilinear polygons. (Unless expliitly stated otherwise, whenever we speak of squares or re- Department of Math and CS, TU Eindhoven. MdB, AM and GW are supported by the Netherlands Organisation for Sientifi Researh (NWO) under projet no tilinear polygons we mean axis-parallel squares and axis-parallel retilinear polygons.) Contat graphs of squares are different from ontat graphs of smooth objets, beause four (interior-disjoint) squares an all meet in a ommon point. Thus the obvious embedding of suh a ontat graph where we put a node at the enter of eah square and we onnet the enters of two touhing squares by a two-link path through a touhing point is not neessarily plane. Eppstein et al. [] studied olourings of ontat graphs of squares for the speial ases where the squares form a quadtree subdivision, that is, the set S of squares is obtained by reursively subdividing an initial square in four equal-sized quadrants. They proved that any suh ontat graph is 6-olourable and they gave an example of a quadtree subdivision that requires five olours. (They also onsidered the variant where two squares that only touh in a single vertex are not onsidered neighbours.) Our results. We start by studying the omputational omplexity of olouring ontat graphs. We show that already for a set of unit squares, it is npomplete to deide if the ontat graph is -olourable. Next we study the hromati number of various lasses of ontat graphs. Reall that the obvious embedding of the ontat graph of squares need not be plane. We first prove ontat graphs of unit squares an have a K m as a minor for an arbitrarily large m and, hene, need not be planar. Nevertheless, ontat graphs of unit squares are -olourable and finding a -olouring is quite easy, so our np-ompleteness result on -olouring ompletely haraterizes the omputational omplexity of olouring unit squares. Contat graphs of arbitrarily-sized squares are not always -olourable the quadtree example of Eppstein et al. [] requiring five olours shows this. We prove that the hromati number of the lass of ontat graphs of arbitrarily-sized squares is at most 6. In fat, we prove that any ontat graph of a set of retilinear polygons is 6-olourable. (Even more generally, ontat graphs of polygons whose interior angles are stritly greater than π/5 are 6-olourable.) Thus we obtain the same bound of Eppstein et al., but for a muh larger lass of objets. Moreover, for this lass the bound is tight. To prove our result, we haraterize ontat graphs of retilinear polygons as a ertain subset of -planar graphs, whih are known to be 6-olourable []. This is an extended abstrat of a presentation given at EuroCG 06. It has been made publi for the benefit of the ommunity and should be onsidered a preprint rather than a formally reviewed paper. Thus, this work is expeted to appear in a onferene with formal proeedings and/or in a journal. 7

3 nd European Workshop on Computational Geometry, 06 NP-Completeness of -Colourability In this setion we establish the hardness of - Colourability on ontat graphs of unit squares. Theorem -Colourability on ontat graphs of unit squares is np-omplete. Proof. -Colourability on ontat graphs is obviously in np. To prove that the problem is np-hard we use a redution from the np-omplete problem of -olouring planar graphs of degree at most [5]. Let G = (V, E) be any planar graph on n verties with degree at most. Rosenstiehl and Tarjan [7] showed that we an ompute in polynomial time a visibility representation of G, in whih every vertex u V is represented by a horizontal vertex segment s u and every edge (u, v) E is represented by a vertial edge segment that onnets s u and s v and does not interset any other vertex segment. a a b 0 b b b a a b 0 b b b The onstrution an be done so that (i) all y- oordinates of the vertex segments are multiples of 0, and (ii) all x-oordinates of the edge segments are multiples of and all x-oordinates of the left and right endpoints of the vertex segments are of the form i and j +, respetively, for some integers i < j. The vertex gadget that replaes a vertex segment is as follows; the example shows the gadget for a segment of length 7. vertex segment The grey squares in the onstrution are alled onnetor squares. In order to -olour a vertex gadget, all onnetor squares must reeive the same olour. This olour represents the olour of the orresponding vertex in G. Note that eah edge segment passes through the enter of a onnetor square on both vertex gadgets it onnets. The edge gadget that replaes an edge segment onsists of a basi edge gadget plus zero or more extension bloks. Note that we an generate edge gadgets of vertial length 7 + j for any integer j 0, by using j extension bloks. This suffies beause the y-oordinates of the vertex segments are multiples of 0, and so the distane in between any two onnetor squares we need to onnet by an edge is of the form 0k, for some integer k. Our edge onnetor squares of vertex gadgets extension blok basi edge gadget gadget fores the onnetor squares of the two vertex gadgets it onnets to have different olours. It is easily heked that this implies that the ontat graph of the generated set of squares is -olourable if and only if the original graph G is. Moreover, the entire onstrution an be done in polynomial time. Using a similar proof we an show that - Colourability is np-omplete for ontat graphs of diss, or of any other fixed onvex and ompat shape. Note that for diss (or other smooth shapes) this settles the omplexity of the problem ompletely: ontat graphs of smooth onvex shapes are planar and so they are -olourable, and heking for -olourability is easy. Unit Squares If we draw the ontat graph of a set of unit squares by putting verties at the enters of the squares and drawing edges as straight segments, then the resulting drawing obviously need not be plane. The following theorem shows a stronger result, namely that ontat graphs of unit squares are not planar and that in fat they an have a K m -minor for arbitrarily large m. Theorem For any m, there are ontat graphs of unit squares with a K m -minor. Proof. The squares we will generate to obtain a ontat graph with a K m as minor will all have integer oordinates. The following piture shows the onstrution for m =. Next we explain the various omponents in the onstrution. Consider K m. We all the nodes of the K m super nodes and the edges super edges. For eah super node u we put a blok of m unit squares 7

4 EuroCG 06, Lugano, Switzerland, Marh 0 April, 06 whose lower edges all lie on the same horizontal line. The distane between two adjaent bloks is one unit. In the figure above, the bloks are the four light grey retangles. For eah super edge (u, v) we reate a path of squares as follows. We put two vertial olumns of an even number of squares one on top of the blok reated for u and one on top of the blok reated for v whih have the same height, and we onnet the topmost squares of these olumns by a row of squares. We an do this suh that we do not reate any adjaenies between squares from different paths, exept where a olumn of one path rosses the row of another path. Note that in this ase the two paths atually share a square. Where this happens we add one more square to the top-right of the shared square see the three dark grey squares in the piture above. These extra square allow us to obtain a minor in whih all super edges are represented by disjoint paths, as the next figure shows By ontrating the (nodes orresponding to the) square in eah blok to a super node and ontrating the paths onneting pairs of nodes into super edges we an now obtain our K m as a minor. Note that the onstrution an be done with O(m ) squares (and we an show that at least Ω(m ) are needed). Despite the fat that ontat graphs of unit squares are not planar, they are -olourable. Theorem Any ontat graph of set of unit squares is -olourable, and this number is tight in the worst ase. Proof. The lower bound onstrution is easy just take four squares touhing in a ommon point. For the upper bound, we divide the plane into horizontal strips of the form (, + ) [i, i + ) and assign eah square to the strip ontaining its bottom edge. The squares assigned to a single strip an be oloured with only two olours, and by using the olour pair, for the strips with even i and, for the strips with odd i we obtain a -olouring. Arbitrarily-Sized Squares We now turn our attention to arbitrarily-sized squares. Theorem Any ontat graph of a set of squares is 6-olourable, and there are ontat graphs of squares that need at least five olours. Proof. The upper bound follows from the result in the next setion, where we show that even ontat graphs of retilinear polygons are 6-olourable. It remains to give an example of a set of squares that indues a ontat graph that needs five olours. Eppstein et al. [] already gave suh an example (where the squares form a quadtree subdivision). For ompleteness we provide a different (and slightly smaller) example. We laim that the following graph (whih is also the subgraph of a quadtree) needs at least 5 olours. Suppose for a ontradition that the graph is - olourable. Then, without loss of generality, we an olour the four squares of the middle lique (onsisting of four squares of different sizes) as depited in the following piture (left). We laim that then the top left inner square has olour. Indeed, if it has olour, none of the four squares on its right ould use olour and so one of these squares would need a fifth olour. Similarly, if it has olour, none of the four squares below it ould use olour and of those squares would need a fifth olour. Hene, it has to use olour sine it touhes a square oloured with. Using similar arguments and simple dedution, we arrive to the following partial olouring: Now we observe that the four gray squares form a yle that surrounds a -lique. Moreover, we an easily dedue that none of the squares in the yle an be oloured or. Hene they have to use olour and. But then the surrounded lique annot use or, a ontradition. We onlude that the graph is not -olourable. 7

5 nd European Workshop on Computational Geometry, 06 5 Retilinear Polygons We now turn our attention to ontat squares of retilinear polygons, where we allow the polygons to have holes. We will prove that suh ontat graphs are 6-olourable by showing that they are -planar graphs [], that is, graphs that an be drawn in the plane suh that eah edge has at most one rossing (that is, it rosses at most one other edge and this rossing then onsists of a single point). The following theorem establishes the exat relation between ontat graphs of retilinear polygons and -planar graphs. (We reently learned that a similar result, on the relation between -planar graphs and so-alled -map graphs was already known []. Our proof onerns retilinear maps and is more diret.) Theorem 5 The lass of ontat graphs of retilinear polygons is exatly the lass of -plane graphs in whih every pair of rossing edges is part of a K. Proof. Let S := {P,..., P n } be a set of interiordisjoint retilinear polygons. To prove that the ontat graph of S is -planar, we proeed as follows. First we add a point v i in the interior of every polygon P i, whih is the embedding of the node orresponding to P i. Next, for eah pair of touhing polygons P i, P j we pik a onnetion point q ij P i P j. If P i P j has non-zero length, we pik q ij in the relative interior of P i P j. We then embed the edge (v i, v j ) by the union of two paths from q ij : a path π(q ij, v i ) P i to v i and a path π(q ij, v j ) P j to v j. We do this in suh a way that, for eah P i, the paths from the onnetion points on P i to v i are pairwise disjoint (exept at their shared endpoint v i ). This onnetion point v v v 6 v v an always be done, for example by taking a shortestpath tree rooted at v i whose leaves are the onnetion points on P i. Thus an edge (v i, v j ) an only interset an edge (v k, v l ) when q ij = q kl. Sine any point an be a onnetion point for at most two pairs of polygons, this means that in our embedding every edge intersets at most one other edge. Moreover, sine all four polygons meet on the rossing point, they are part of a -lique. Next we show that every -planar graph G = (V, E) with every pair of rossing edges forming a K is the ontat graph of a set of retilinear polygons. Suh a set an be obtained from a pixelised image of a -planar drawing of G. v 5 Eah polygon is obtained by the vertex it represents and half of eah of its edges, as shown in the piture above. If the edge is not rossing any other, we an deide arbitrarily where to divide it into the two polygons. If it rosses another edge, we ut it at the rossing point, as depited above. We an atually obtain a suitable set of polygons whose total number of verties is linear in O( V ), but the proof is more omplex. This bound is tight sine we an onstrut an instane where one of the polygons has linear omplexity: sine for eah rossing we need a orner, it suffies to have a vertex with a linear number of edges rossing other edges. Note that this proof works for non-retilinear polygons as long as no five of them touh on a single point, whih is always satisfied when the interior angles are stritly bigger than π/5. Sine -planar graphs are 6-olourable and the figure below shows a retilinear representation of K 6, Theorem 5 immediately implies the following. Corollary 6 Any ontat graph of a set of retilinear polygons is 6-olourable, and this number is tight in the worst ase. Aknowledgment We thank one reviewer for pointing out referene []. Referenes [] K. Appel and W. Haken. Every map is four olourable. Bulletin of the Amerian Mathematial Soiety, 8:7 7, 976. [] O. V. Borodin. A new proof of the 6 olor theorem. Journal of Graph Theory, 9():507 5, 995. [] F. J. Brandenburg. On -map graphs and - planar graphs and their reognition problem. CoRR, abs/509.07, 05. [] D. Eppstein, M. W. Bern, and B. L. Huthings. Algorithms for oloring quadtrees. Algorithmia, ():87 9, 00. [5] M. Garey, D. Johnson, and L. Stokmeyer. Some simplified NP-omplete graph problems. Theoretial Computer Siene, ():7 67, 976. [6] P. Koebe. Kontaktprobleme der konformen abbildung. Ber. Verh. Sähs. Akademier der Wissenshaften Leipzig, Math.-Phys. Klasse 88, pages 6, 96. [7] P. Rosenstiehl and R. E. Tarjan. Retilinear planar layouts and bipolar orientations of planar graphs. Disrete & Computational Geometry, : 5,

Directed Rectangle-Visibility Graphs have. Abstract. Visibility representations of graphs map vertices to sets in Euclidean space and

Directed Rectangle-Visibility Graphs have. Abstract. Visibility representations of graphs map vertices to sets in Euclidean space and Direted Retangle-Visibility Graphs have Unbounded Dimension Kathleen Romanik DIMACS Center for Disrete Mathematis and Theoretial Computer Siene Rutgers, The State University of New Jersey P.O. Box 1179,