Structure of Graphs with Locally Restricted Crossings † ‡ § VidaAbstract. Dujmovi´c David Eppstein David R. Wood Wen consider relations between the size, treewidth, andg local crossingk number p (maximum number of crossingsO( (g per+ 1)( edge)k + 1) ofn graphs) embedded on topologicalO((g + 1) surfaces.k) We show that an -vertex graph embedded on a surface of genus withk at most crossings p nper edge has treewidth O( (k + 1)n) and layered treewidthO(k + 1) , and that these bounds are tight up to a constant factor.O((k As+ a 1) special3=4n1=2) case, the -planar graphs with vertices have treewidth and layered treewidth , which are tight bounds thatg < improve m a previouslym known treewidth bound. Analogousg Oresults((m=( areg + proved1)) log2 forg) map graphs defined with respect to any surface. Finally, we show that for , every -edge graph can be embedded on a surface of genus with Keywords. crossings per edge, which is tight to a polylogarithmic factor.k treewidth, pathwidth, layered treewidth, local treewidth, 1-planar, -planar, map graph, graph minor, local crossing number, separator, 1 Introduction This paper studies the structure of graph classes defined by drawings on surfaces in which the crossings are locally restricted in some way.k k-planar k local crossing Thenumber first such example that we considerk are the -planark graphs. A graph is if it can be drawn in the planep × withq × atr most crossings on each edge[p] × [24[q].] × The[r] of the(x; y; graph z)(x + is 1 the; y; zminimum) (x; y; zfor)(x; which y + 1; it z) is (-planarx; y; z)(x; [27 y;, pagesz + 1) 51–53]. An important example is the grid graph, with vertex set and all edges arXiv:1506.04380v2 [math.CO] 20 Feb 2016 of the form or or . A suitable Proc.February 23rd 23, International 2016 Symposium on Graph Drawing and Network Visualization 2015 A preliminary version of this paper entitled “Genus, treewidth, and local crossing number” was published in † , Lecture Notes in [email protected] Science 9411:87–98, Springer, 2015. School of Computer Science and Electrical Engineering, University of Ottawa, Ottawa, Canada ‡ ( ). Supported by NSERC and the Ministry of [email protected] and Innovation, Government of Ontario, Canada. § Department of Computer Science, University of California, Irvine, California, [email protected] ( ). Supported in part by NSF grant CCF-1228639. School of Mathematical Sciences, Monash University, Melbourne, Australia ( ). Supported by the Australian Research Council. 1 p × q × r (r − 1) Figure 1: The grid graph is -planar. (r − 1) linear projection from the natural three-dimensional embedding of this graph to the plane gives a -planar drawing, as illustrated in Figure1. The main way that we describe the structure of a graph is through its treewidth, which is a parameter that measures how similar a graph is to a tree. It is a key measure of the complexity of a graph and is of fundamental importance in algorithmic graph theory and structural graph theory, especially in Robertson and Seymour’s graph minors project. See Section2 for a detailed definition of treewidth. separator Treewidth is closely related to the size of a smallest , a set of vertices whose removal splits the graph into connected components each with at most half the vertices. Graphs of low treewidth necessarily haven small separators, and graphs in which every p p Osubgraph( n) has a small separator have lown treewidth [12, 25]. For example, theO( Lipton-Tarjann) p separator theorem, whichO( n) says that every -vertex planar graph has a separator of order , can be reformulated as every -vertex planar graph has treewidth . Most of our results provide bounds on the treewidth of particular classes of graphs that generalise planarity. In this sense, our results are generalisations of the Lipton-Tarjan separator theorem, and analogous results for other surfaces. k n The starting point for ourO work(k3=4 isn1 the=2) following question: what is the maximum treewidth of -planar graphs on vertices? Grigoriev and Bodlaender [16] studied this question and Theorem 1. The maximum treewidth of -planar -vertex graphs is proved an upper bound of . Wek improven this and give the following tight bound: n p o Θ min n; (k + 1)n : 2 (g; k)-planar g k et al. More generally, a graph is if it can1 be drawn in a surface(2; of k) Euler genus at most with at most crossings on each edge . For instance, Guy [19] investigated the local crossing number of toroidal embeddings—in this notation, the -planar graphs. Theorem 2. The maximum treewidth of (g; k)-planar n-vertex graphs is We again determine an optimal bound on the treewidth of such graphs. n p o Θ min n; (g + 1)(k + 1)n : k = 0 ( ) In both these theorems, the case (withg; k no crossings) is well known [15]. n mOur second contribution is to study theg -planarity of graphs as a functionm of their Ω(minnumberfm of2 edges.=g; m2=n Forg) (global) crossing number, it is known thatO((m a2 graphlog2 g with)=g) vertices and edges drawn on a surface of genus (sufficiently small withΩ( respectm=g) to ) may require crossings, and it can be drawn with crossings [29]. In particular, the lower bound implies that some graphs require crossings per edge on average, and therefore also in the worst case. We prove a nearly-matching upper bound Theorem 3. For every graph G with m edges, for every integer g 1, there is a drawing which implies the above-mentioned upper bound on the total number> of crossings: of G in the orientable surface with at most g handles and with m log2 g O g crossings per edge. G0 g Our third contribution concernsG0 map graphs, which ared defined asG follows. Start with a graph embedded in a surfaceG0 of Euler genus , with each face labelledG a ‘nation’ or a ‘lake’, whereG0 each vertex of is incidentG with at most(g; d)-mapnations. graph Let (0be; d the) graph whose vertices are the nationsd-map graph of , where two vertices are adjacent in if the corresponding faces in share a vertex.(g; 3) Then is called a .A -map graph is gcalled a (plane) ; suchg = 0 graphs have been(g; extensively d) studied [14, 3, 6, 4, 2]. It is easily seen that -map graphs are precisely2 theG graphs of Euler genus at most (which is well known in the case [4]) . So -map graphs provide a natural Euler genus h 2h Euler genus generalisation of graphs embedded in a surface. Note that may contain arbitrarily large 1 c c Euler genus G G The of an orientable surface with handles is . The of a non-orientable surface withG cross-caps is . The of a graph is the minimumg EulerM( genusG) of amedial surface in which G 2 embeds (with no crossings).E(G) M(G) Let be a graph embedded in a surface of Euler genus at most . Let be the graph of . This graph has vertex set where two vertices of are adjacent whenever the corresponding 3 g = 0 H d G Kd cliques even in the case, since if a vertex of is incident with nations then containsG . H G G If is the mapG graph associated with an embedded graph , then considerH the natural vdrawingH of in whichd each vertex of is positioned inside the correspondingv nation, and beachd−2 cd edged−2 e of is drawn as a( curveg; d) through the corresponding(g; b d−2 cd d−2 e) vertex of . If a vertex 2 2 p2 2 of is incident to(g; dnations,) then each edge passingO(d through(g + 1)n)is crossed by at most edges. Thus every -map graph is -planar, and Theorem2 implies that every -map graph has treewidth . We improve on this Theorem 4. The maximum treewidth of ( )-map graphs on vertices is result as follows. g; d n n p o Θ min n; (g + 1)(d + 1)n : layered treewidth (Weg; k prove) our treewidth upper(g; d) bounds by using the concept of [10], which is of independent interest (see Section2). We prove matching lower bounds by finding -planar graphs and -map graphs without small separators and using the known relations between separator size and treewidth. 2 Background and Discussion 2 (0; 1) S G -separator G 1 2 G − S jV (G)j = 2 = 3 For , a set of vertices in a graph is an of if each component of has at most vertices. It is conventional to set or but the precise choice makes no difference to the asymptotic size of a separator. Several results that follow depend on expanders; see [21] for a survey. The following Lemma 5. For every 2 (0 1) there exists 0, such that for all 3 and + 1 folklore result provides a property; of expandersβ > that is the key to ourk applications.> n > k (such that n is even if k is odd), there exists a k-regular n-vertex graph H (called an expander) in which every -separator in H has size at least βn. G G M(G) G M(G) edges in G are consecutive in theM cyclic(G) ordering of edges incident to aG common vertex in the embedding of M.( NoteG) that embeds in theG same surface as , where eachM(G face) of corresponds tovw a vertexG or a face of . Label the faces of thatv correspondw to vertices of as nations, and labelG the faces of that correspondM to(G faces) ofG as lakes.(g; 2) The vertex of corresponding(g; 3) to an edge of is incident to the(g; nations3) corresponding to and (and is incident to no other nations).