EuroCG 2011, Morschach, Switzerland, March 28{30, 2011 Right Angle Crossing Graphs and 1-planarity∗ Peter Eadesy Giuseppe Liottaz Abstract This paper studies the interplay between two fam- ilies of non-planar drawings that fit into the above A Right Angle Crossing Graph (also called RAC mentioned research directions. graph for short) is a graph that has a straight-line More formally, a drawing of a graph G maps each drawing where any two crossing edges are orthogonal vertex u of G to a distinct point pu in the plane, each to each other. A 1-planar graph is a graph that has edge (u; v) of G a Jordan arc connecting pu and pv and a drawing where every edge is crossed at most once. not passing through any other vertex, and is such that We study the relationship between RAC graphs and any two edges have at most one point in common. A 1-planar graphs in the extremal case that the RAC 1-planar drawing is a drawing of a graph where an graphs have as many edges as possible. It is known edge can be crossed by at most another edge. A 1- that a maximally dense RAC graph with n > 3 ver- planar graph is a graph that has a 1-planar drawing. tices has 4n−10 edges. We show that every maximally A straight-line drawing is a drawing of a graph such dense RAC graph is 1-planar. Also, we show that for that every edge is a straight-line segment. A Right every integer i such that i ≥ 0, there exists a 1-planar Angle Crossing drawing (or RAC drawing, for short) graph with n = 8 + 4i vertices and 4n − 10 edges that is a straight-line drawing where any two crossing edges is not a RAC graph. form right angles at their intersection point. A Right Angle Crossing graph (or RAC graph, for short) is a 1 Introduction graph that has a RAC drawing. Pach and Toth prove that 1-planar graphs with n Recent technological advances have generated tor- vertices have at most 4n−8 edges, which is a tight up- rents of relational data sets that are often represented per bound [13]. Korzhik and Mohar prove that recog- and visually analyzed as graphs drawn in the plane. nizing 1-planar graphs is NP-hard [12]. Suzuki studies The large size of these data sets poses fascinating chal- the combinatorial properties of the so-called optimal lenges to graph drawers: while a considerable portion 1-planar graphs, i.e. those n-vertex 1-planar graph of the existing graph drawing literature showcases ele- having 4n − 8 edges [16]. A limited list of additional gant algorithms and sophisticated data structures un- papers on 1-planar graphs includes [4, 7]. Didimo et der the assumption that the input graph is planar, al. show that a RAC graph with n > 3 vertices has at most graphs are in fact non-planar in practice (see, most 4n − 10 edges and that this bound is tight [5]. e.g., [10, 11]). Argyriou at al. prove that recognizing RAC graphs is In this context, it is worth recalling a few experi- NP-hard [2]. For recent references about RAC graphs mental studies that are motivating some of the current and their variants see also [1, 3, 6, 17]. research directions about drawing non-planar graphs. We focus on the relationship between RAC graphs Huang et al. [8, 9] prove that crossing edges sig- and 1-planar graphs in the extremal case that the nificantly affect human understanding if they form RAC graphs are as dense as possible. A RAC graph acute angle, while crossing edges that form angles is maximally dense if it has n > 3 vertices and 4n−10 π π from about 3 to 2 guarantee good readability prop- edges. We prove the following. erties. Hence it makes sense to study drawings of graphs where such \sharp angle crossings" are forbid- den. Purchase et al. [14, 15, 18]) prove that an edge Theorem 1 Every maximally dense RAC graph is 1- is difficult to read if it is crossed by many other edges. planar. Also, for every integer i such that i ≥ 0, there Another research direction is therefore to study draw- exists a 1-planar graph with n = 8 + 4i vertices and ings of graphs where every edge can only be crossed 4n − 10 edges that is not a RAC graph. a small number of times. ∗Work supported in part by MIUR of Italy under project Section 2 considers a three coloring of the edges of AlgoDEEP prot. 2008TFBWL4 and by an IVFR Grant of the a maximally dense RAC graph and studies the prop- Australian Government. y erties of the subgraphs induced by two of these three School of Information Technologies, University of Sydney colors. Section 3 uses these properties to prove Theo- [email protected] zDip. di Ing. Elettr. e dell'Informaz., Universit`adegli Studi rem 1. For reasons of space, some proofs are sketched di Perugia [email protected] or omitted. This is an extended abstract of a presentation given at EuroCG 2011. 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. 155 27th European Workshop on Computational Geometry, 2011 2 Red-blue-green Coloring of Maximally Dense cannot be a 4-cycle either. By contradiction, assume RAC Graphs that jfextj = 4. Consider first the case that some fence face of Grb has more than 3 edges: Since jfextj = 4 and Let G be a maximally dense RAC graph and let D be a fence face has size at least 4, we have mr +mb ≤ 3n− any RAC drawing of G. Let E be the set of the edges 8. By the inequalities above, we also have mr ≥ frb+1 of D. In [5] the following 3-coloring of the edges of D and mb ≤ n − 3. Since G is maximally dense, we have (and hence of G) is described. Every edge is either a mr + mb + mg = 4n − 10. It follows that mr + mg ≥ red edge or a blue edge, or a green edge. An edge is red 3n−7 > mr+mb, which is however impossible because if and only if it is not crossed by any other edge; a blue we are assuming mb ≥ mg. Lastly, consider the case edge is only crossed by green edges, and a green edge that jfextj = 4 and all fence faces are 3-cycles (which is only crossed by blue edges. We call this 3-coloring implies that there are exactly four fence faces because of the edges of D a red-blue-green coloring of D and jfextj = 4). In every RAC drawing of G, each fence denote it as Πrbg. Let Drb = (V; Er [ Eb) be the sub- face is drawn as a triangle. Hence, for at least one drawing of D consisting of the red and blue edges and of these triangles the angle opposite to the edge that π let Grb be the corresponding subgraph of G. We call belongs to fext must be larger than or equal to 2 . This Grb the red-blue subgraph of G induced by Πrbg and observation, together with Lemma 2, implies that at we call Drb the red-blue sub-drawing of D induced by least one of the fence faces consists of all red edges Πrbg. Note that, by construction, Drb has no crossing in any red-blue-green coloring. We therefore have the j j edges and thus G is a planar graph. We will always fext 3 3 rb following: mr ≥ (frb − 2) + + = frb + . consider G as a planar embedded graph, where the 2 2 2 rb Since mr is an integer, we have mr ≥ frb + 2. By planar embedding is given by D . Analogously we rb mr + mb ≤ n + frb − 2 we obtain mb ≤ n − 4, and by define the red-green subgraph of G induced by Π , rbg mr +mb +mg = 4n−10 we obtain mr +mg ≥ 3n−6. denoted as G , and the red-green sub-drawing of D rg However, Grg is a planar graph and it has the same induced by Π , denoted as D . Also G has the rbg rg rg external face as Grb, that has size 4; so, Grg cannot be planar embedding of Drg, and thus Grg and Grb have a maximal planar graph, a contradiction. It follows the same external face. that fext must be a 3-cycle. The next lemmas will particularly focus on the size and the coloring of some specific faces of the red-blue Lemma 4 Graph Grb is biconnected. graph Grb. We will consider its external face, denoted as fext, and its fence faces, defined as those internal Proof sketch. By Lemmas 2 and 3 the external face faces that share at least one edge with fext. In the of Grb is a 3-cycle consisting of red edges. With a proofs that follow, we denote with mr the number similar reasoning as in the proof of Lemma 3, we ob- of red edges, with mb the number of blue edges, and tain mr + mg ≥ 3n − 7. Since mb ≥ mg we also have with mg the number of green edges. Without loss of mr +mb ≥ 3n−7. Since Grb has at least 3n−7 edges, generality, we will assume from now on that our red- it is biconnected.