Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends

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1 Compt Drwings of 1-Plnr Grphs with Right-Angle Crossings nd Few Bends Steven Chplik, Fbin Lipp, Alexnder Wolff, nd Johnnes Zink Lehrstuhl für Informtik I, Universität Würzburg, Germny rxiv: v4 [s.cg] 3 Sep 2018 Abstrt. We study the following lsses of beyond-plnr grphs: 1- plnr, IC-plnr, nd NIC-plnr grphs. These re the grphs tht dmit 1-plnr, IC-plnr, nd NIC-plnr drwing, respetively. A drwing of grph is 1-plnr if every edge is rossed t most one. A 1-plnr drwing is IC-plnr if no two pirs of rossing edges shre vertex. A 1-plnr drwing is NIC-plnr if no two pirs of rossing edges shre two verties. We study the reltions of these beyond-plnr grph lsses to rightngle rossing (RAC ) grphs tht dmit ompt drwings on the grid with few bends. We present four drwing lgorithms tht preserve the given embeddings. First, we show tht every n-vertex NIC-plnr grph dmits NIC-plnr RAC drwing with t most one bend per edge on grid of size O(n) O(n). Then, we show tht every n-vertex 1-plnr grph dmits 1-plnr RAC drwing with t most two bends per edge on grid of size O(n 3 ) O(n 3 ). Finlly, we mke two known lgorithms embedding-preserving; for drwing 1-plnr RAC grphs with t most one bend per edge nd for drwing IC-plnr RAC grphs stright-line. 1 Introdution In grph theory nd grph drwing, beyond-plnr grph lsses hve experiened inresing interest in reent yers. A prominent exmple is the lss of 1-plnr grphs, tht is, grphs tht dmit drwing where eh edge is rossed t most one.the 1-plnr grphs were introdued by Ringel [19] in 1965; Kobourov et l. [16] surveyed them reently. Another exmple tht hs reeived onsiderble ttention re RAC k grphs, tht is, grphs tht dmit poly-line drwing where ll rossings re t right ngles nd eh edge hs t most k bends. The RAC k grphs were introdued by Didimo et l. [8]. Using right-ngle rossings nd few bends is motivted by severl ognitive studies suggesting positive orreltion between lrge rossing ngles or smll urve omplexity nd the redbility of grph drwing [14,15,18]. We investigte the reltionships between (ertin sublsses of) 1-plnr grphs nd RAC k grphs tht dmit drwings on polynomil-size grid. The prior work nd our ontributions re summrized in Fig. 2. A broder overview of beyond-plnr grph lsses is given in reent survey by Didimo et l. [9]. Appers in the Proeedings of the 26th Interntionl Symposium on Grph Drwing nd Network Visuliztion (GD 2018).

2 2 S. Chplik et l. () RAC 0 drwing. (b) IC-plnr drwing. () NIC-plnr drwing. (d) 1-plnr drwing. (e) 1-plnr RAC 1 drwing. Fig. 1: Exmples of different types of drwings. Figs. 1d nd 1e show drwings of the sme grph. Fig. 1e is tken from the Annotted Bibliogrphy on 1-Plnrity [16]. Bsi Terminology. A mpping Γ is lled drwing of the grph G = (V, E) if eh vertex v V is mpped to point in R 2 nd eh edge uv is mpped to simple open Jordn urve in R 2 suh tht the endpoints of this urve re Γ (u) nd Γ (v). For onveniene, we will refer to the points nd simple open Jordn urves of drwing s verties nd edges. The topologilly onneted regions of R 2 \ Γ re the fes of Γ. The unbounded fe of Γ is its outer fe; the other fes re inner fes. Eh fe defines irulr list of bounding edges (resp. edge sides), whih we ll its boundry list. Two drwings of grph G re equivlent when they hve the sme set of boundry lists for their inner fes nd outer fes. Eh equivlene lss of drwings of G is n embedding. A k-bend (poly-line) drwing is drwing in whih every edge is drwn s onneted sequene of t most k + 1 line segments. The (up to) k inner verties of n edge onneting these line segments re lled bend points or bends. A 0-bend drwing is more ommonly referred to s stright-line drwing. A drwing on the grid of size w h is drwing where every vertex, bend point, nd rossing point hs integer oordintes in the rnge [0, w] [0, h]. In ny drwing we require tht verties, bends, nd rossings re pirwise distint points. A drwing is 1-plnr if every edge is rossed t most one. A 1-plnr drwing is independent-rossing plnr (IC-plnr) if no two pirs of rossing edges shre vertex. A 1-plnr drwing is ner-independent-rossing plnr (NIC-plnr) if ny two pirs of rossing edges shre t most one vertex. A drwing is right-ngle-rossing (RAC ) if (i) it is poly-line drwing, (ii) no more thn two edges ross in the sme point, nd (iii) in every rossing point the edges interset t right ngles. We further speilize the notion of RAC drwings. A drwing is RAC k if it is RAC nd k-bend; it is RAC poly if it is RAC nd on grid whose size is polynomil in its number of verties. Exmples for IC-plnr, NIC-plnr, 1-plnr, nd RAC drwings re given in Fig. 1. The plnr, 1-plnr, NIC-plnr, IC-plnr, nd RAC k grphs re the grphs tht dmit rossing-free, 1-plnr, NIC-plnr, IC-plnr, nd RAC k drwing, respetively. More speifilly, RAC poly k is the set of grphs tht dmit RAC poly k drwing. A plne, 1-plne, NIC-plne, nd ICplne grph is grph given with speifi plnr, 1-plnr, NIC-plnr, nd ICplnr embedding, respetively. In 1-plnr embedding the edge rossings re known nd they re stored s if they were verties. We will denote n embedded grph by (G, E) where G is the grph nd E is the embedding of this grph. For

3 Compt Drwings of 1-Plnr Grphs with Right-Angle Crossings 3 E? poly RAC3 = ll grphs RAC 2? RAC 1??? RAC 1-plnr 0 E? App. D1 without B-onfig. App. D2 E? NIC-plnr IC-plnr Se. 3 Se. 2 RAC poly 2 RAC poly 1 RAC poly 0 E??? is ontined in (inl. fixed embeddings) is ontined in (open for fixed embeddings) open if ontined or inomprble open if ontined or ontining or inomprble plnr inomprble Fig. 2: Relting some lsses of (beyond-)plnr grphs nd RAC grphs. Our min results re the ontinment reltionships indited by the thik blue rrows. point p in the plne, let x(p) nd y(p) denote its x- nd y-oordinte, respetively. Given two points p nd q, we denote the stright-line segment onneting them by pq nd its length, the Euliden distne of p nd q, by pq. Previous Work. In the digrm in Fig. 2, we give n overview of the reltionships between lsses of 1-plnr grphs nd RAC k grphs. Clerly, the plnr grphs re subset of the IC-plnr grphs, whih re subset of the NICplnr grphs, whih re subset of the 1-plnr grphs. It is well known tht every plne grph n be drwn with stright-line edges on grid of qudrti size [20,11]. Every IC-plnr grph dmits n IC-plnr RAC 0 drwing but not neessrily in polynomil re [3]. Moreover, there re grphs in RAC poly 0 tht re not 1-plnr [10] nd, therefore, lso not IC-plnr. The lss of RAC 0 grphs is inomprble with the lsses of NIC-plnr grphs [1] nd 1-plnr grphs [10]. Bekos et l. [2] showed tht every 1-plnr grph dmits 1-plnr RAC 1 drwing, but their reursive drwings my need exponentil re. Every grph dmits RAC 3 drwing in polynomil re, but this does not hold if given embedding of the grph must be preserved [8]. Our Contributions. We ontribute four new results; two min results nd two dpttions of prior results. First, we onstrutively show tht every NIC-plne grph dmits RAC 1 drwing in qudrti re; see Setion 2. This improves upon side result by Liott nd Montehini [17], who showed tht every IC-plne grph dmits RAC 2 drwing on grid of qudrti size. Seond, we onstrutively show tht every 1-plne grph dmits RAC 2 drwing in polynomil re; see Setion 3. Beside these two min results, we show how to preserve

4 4 S. Chplik et l. v dummy () rossing s it initilly ppers (b) empty kite nd subdivided originl edge () empty qudrngle (d) divided qudrngle Fig. 3: Modifying the rossings nd omputing the BCO. given embedding when omputing RAC drwings. Preisely, we show Theorem 1 in Appendix D.1 by dpting n lgorithm of Bekos et l. [2] nd we show Theorem 2 in Appendix D.2 by dpting n lgorithm of Brndenburg et l. [3]. Theorem 1. Any n-vertex 1-plne grph dmits n embedding-preserving RAC 1 drwing. It n be omputed in O(n) time. Theorem 2. Any stright-line drwble n-vertex IC-plne grph dmits n embedding-preserving RAC 0 drwing. It n be omputed in O(n 3 ) time. 2 NIC-Plnr 1-Bend RAC Drwings in Qudrti Are In this setion we onstrutively show tht qudrti re is suffiient for RAC 1 drwings of NIC-plnr grphs. We prove the following. Theorem 3. Any n-vertex NIC-plne grph (G, E) dmits NIC-plnr RAC 1 drwing tht respets E nd lies on grid of size O(n) O(n). The drwing n be omputed in O(n) time. Preproessing. Our lgorithm gets n n-vertex NIC-plne grph (G, E) s input. We first im to mke (G, E) bionneted nd plnr so tht we n drw it using the lgorithm by Hrel nd Srds [12]. Around eh rossing in E, we insert up to four dummy edges to obtin empty kites. A kite is K 4 tht is embedded suh tht (i) every vertex lies on the boundry of the outer fe, nd (ii) there is extly one rossing, whih does not lie on the boundry of the outer fe. A kite K s subgrph of grph H is sid to be empty if there is no edge of H\K tht is on n inner fe of K or rosses edges of K. Inserting dummy edge ould rete pir of prllel edges. If this hppens, we subdivide the originl edge prtiipting in this pir by dummy vertex (see the trnsition from Fig. 3 to 3b). Note tht we never rete prllel dummy edges sine G is NIC-plnr. After this, we remove both rossing edges from eh empty kite nd obtin empty qudrngles (see Fig. 3). We store eh suh empty qudrngle in list Q. At the end of the preproessing, we mke the resulting plne grph bionneted vi, e.g., the lgorithm of Hoproft nd Trjn [13]. Sine eh empty qudrngle is ontined in bionneted omponent, no edges re inserted into it. Let (G, E ) be the resulting plne bionneted grph.

5 Compt Drwings of 1-Plnr Grphs with Right-Angle Crossings 5 Drwing Step. Now, we drw grph tht we obtin from (G, E ) by first produing bionneted nonil ordering (BCO) 1. We use the lgorithm by Hrel nd Srds [12], whih is generliztion of the lgorithm of Chrobk nd Pyne [5], whih in turn is bsed on the shift lgorithm of de Frysseix et l. [11]. The lgorithm of Hrel nd Srds onsists of two phses. Given plne bionneted grph H, in the first phse BCO Π of the verties in H is omputed. In the seond phse, H is drwn ording to Π on grid of size (2 V (H) 4) ( V (H) 2). Unlike the lssil shift lgorithm, the lgorithm of Hrel nd Srds omputes the (bionneted) nonil ordering bottom-up, whih we will exploit here. Let Π k = (v 1,..., v k ) be prtil BCO of H fter step k, nd let H k be the plne subgrph of H indued by Π k. We sy tht vertex u is overed by v k if u is on the boundry of the outer fe of H k 1, but not on tht of H k. We perform the following dditionl opertions when we ompute the BCO ˆΠ. Whenever we reh n empty qudrngle q = (, b,, d) of the list Q for the first time, i.e., when the first vertex of q sy is dded to the BCO, we insert n edge inside q from to the vertex opposite in q, tht is, to. We ll the resulting struture divided qudrngle (see Fig. 3d). In two speil ses, we perform further modifitions of the grph. They will help us to gurntee orret reinsertion of the rossing edges in the next step of the lgorithm. Nmely, when we enounter the lst vertex v lst {b,, d} of q, we distinguish three ses. Cse 1: v lst = (see Fig. 4). Here, no opertions re performed. Cse 2: v lst {b, d}, nd the other of {b, d} is overed by (see Fig. 4b). We insert dummy vertex v shift, whih we ll shift vertex, into the urrent BCO diretly before v lst nd mke it djent to nd. Observe tht, if v shift is the k-th vertex in ˆΠ, this still yields vlid BCO sine v shift hs two neighbors in ˆΠ k 1 nd is on the outer fe of the subgrph indued by ˆΠ k 1. Lter, we will remove v shift, but for now it fores the lgorithm of Hrel nd Srds to shift nd wy from eh other before v lst is dded. Cse 3: v lst {b, d}, nd neither b nor d is overed by (see Fig. 4). Let {v lower } = {b, d} \ v lst. We subdivide the edge v lower vi dummy vertex v dummy. If v lower is n originl edge of the input grph, this edge will be bent t v dummy in the finl drwing. We insert v dummy into the urrent BCO diretly before v lower. To obtin divided qudrngle gin, we insert the dummy edge v lower, whih we will remove before we reinsert the rossing edges. This will give us some extr spe inside the tringle (, v dummy, v lower ) for bend point. Inserting v dummy s k-th vertex into ˆΠ keeps ˆΠ vlid sine v dummy uses 1 BCOs re generliztion of nonil orderings tht ssume only bionnetivity (insted of trionnetivity). In BCO of plne grph H, the subgrph H k of H indued by v 1,..., v k is onneted, the edge v 1v 2 lies on the boundry of the outer fe nd ll verties in H H k lie within the outer fe of H k. For k > 2, the vertex v k hs one or more neighbors in H k 1. If v k hs extly one neighbor u in H k 1, then it hs legl support on the outer fe of H k 1, i.e., in the irulr order of djent verties round u, it follows or preedes vertex in H k 1.

6 6 S. Chplik et l. d b () Cse 1; v lst = v shift (b) Cse 2; v lst = d nd b is overed by b d d = v lst b = v lower v dummy () Cse 3; v lst = d nd b is not overed by d y d d e {,} (d) Cse 1 b e {b,d} e {b,d} e {,} (e) Cse 2 b p ross y b v dummy e {b,d} e {,} (f) Cse 3 Fig. 4: Divided qudrngles produed in the three ses of the drwing step () () nd the rossing edges fter the reinsertion step (d) (f) in our lgorithm. For orienttion, lines with slope 1 or 1 re dshed violet. the support edge inident to tht would hve been overed by v lower otherwise. Then, v lower hs t lest two neighbors in ˆΠ k, nmely nd v dummy. We drw the resulting plne bionneted ˆn-vertex grph (Ĝ, Ê) ording to its BCO ˆΠ vi the lgorithm by Hrel nd Srds nd obtin rossing-free drwing ˆΓ. We do not modify the tul drwing phse. Postproessing (Reinserting the Crossing Edges). We refine the underlying grid of ˆΓ by ftor of 2 in both dimensions. Let q = (, b,, d) be qudrngle in Q, where is the first nd v lst the lst vertex in ˆΠ mong the verties in q. From q, we first remove the hord edge nd obtin n empty qudrngle. Then, we distinguish three ses for reinserting the rossing edges tht we removed in the preproessing. These re the sme ses s in the desription of the modified omputtion of the BCO before. In this se distintion we omit some lengthy but stright-forwrd lultions; see Zink s mster s thesis [24] for the detils. Cse 1: v lst = (see Fig. 4). Sine is djent to, b, nd d in Ĝ, it hs the lrgest y-oordinte mong the verties in q. Assume tht y(d) is smller or equl to y(b) sine the other se is symmetri. An exmple of qudrngle in this se before nd fter the reinsertion of the rossing edges is given in Figs. 4 nd 4d, respetively.

7 Compt Drwings of 1-Plnr Grphs with Right-Angle Crossings 7 We will hve rossing point t (x(), y(d)). To this end, we insert the edge with bend t e = (x(), y(d) + 1) nd we insert the edge bd with bend t e bd = (x() + 1, y(d)). Clerly the rossing is t right ngle. Observe tht q is onvex sine is the lst drwn vertex of q nd is djent to b,, nd d in this irulr order in the embedding nd observe tht both bend points lie inside q. Therefore, it follows tht both rossing edges lie ompletely inside q. Cse 2: v lst {b, d}, nd the other of {b, d} is overed by (see Fig. 4b). Assume tht y(d) > y(b); the other se is symmetri. An exmple of qudrngle in this se before nd fter the reinsertion of the rossing edges is given in Figs. 4b nd 4e, respetively. We remove v shift in ddition to removing the edge. We define the rossing point p ross = (x ross, y ross ) s the intersetion point of the lines with slope 1 nd 1 through nd b, respetively. The oordintes of this rossing point re x ross = (x() y() + x(b) + y(b))/2 nd y ross = ( x()+y()+x(b)+y(b))/2. Sine we refined the grid by ftor of 2 in eh dimension, the bove oordintes re both integers. We ple the two bend points onto the sme lines t the losest grid points tht re next to p ross, i.e., we drw the edge with bend point t e = (x ross 1, y ross 1) nd we insert the edge bd with bend point t e bd = (x ross 1, y ross + 1). We do not interset or touh the edge d beuse we shifted fr enough wy from by the extr shift due to v shift. Moreover, the points e nd p ross on the line with slope 1 through re inside the empty qudrngle q sine b is overed by (then b is below the line with slope 1 through ) nd y(b) is t most equl to y(e ). Cse 3: v lst {b, d}, nd neither b nor d is overed by (see Fig. 4). Assume tht y(d) > y(b); gin, the other se is symmetri. An exmple of qudrngle in this se before nd fter the reinsertion of the rossing edges is given in Figs. 4 nd 4f, respetively. Note tht the edge b is dummy edge, whih we inserted during the omputtion of ˆΠ, nd next to this edge, there is the pth v dummy b. This pth is the former edge b. We will reinsert the edges nd bd suh tht they ross in (x(), y(b)). We will bend the edge bd on the line with slope 1 through t y = y(b) beuse from this point we lwys see d inside q. So, we define x bend := x() y with y := y() y(b). First, we remove the dummy edge b. Seond, we insert the edge with bend point t e = (x(), y(b) 1). Third, we insert the edge bd with bend point t e bd = (x bend, y(b)). Note tht e might be below the stright-line segment b sine ould hve been shifted fr wy from. However, e nnot be on or below the pth v dummy b beuse y(v dummy ) < y(e ) nd the slope of the line segment v dummy b is either greter thn 1 or negtive. Therefore, the rossing edges nd bd lie ompletely inside the pentgonl fe (, v dummy, b,, d). Result. After we hve reinserted the rossing edges into eh qudrngle of Q, we remove ll dummy edges nd trnsform the remining dummy verties to bend points. The resulting drwing Γ is RAC 1 drwing tht preserves the embedding of the NIC-plne input grph (G, E). In Appendix A (pge 15), we bound the size of the grid tht our drwings need, s follows. Lemm 4. Every vertex, bend point, nd rossing point of the drwing returned by our lgorithm lies on grid of size t most (16n 32) (8n 16).

8 8 S. Chplik et l () Plnrized rossing where the rossing point beme rossing vertex. d 4 4 d d 2 d 1 2 (b) Enlosing the rossing vertex by subdivided kite. Fig. 5: A rossing point is repled by rossing vertex nd we insert four 2-pths of two dummy edges nd dummy vertex to indue subdivided kite t eh rossing. The verties d 1, d 2, d 3, nd d 4 re the dummy verties of these 2-pths. The shift lgorithm of Hrel nd Srds runs in liner time [12]. Also, our dditionl opertions n be performed in liner time [24]. This proves Theorem 3. We give full exmple of NIC-plne RAC 1 drwing generted by Jv implementtion of our lgorithm in Figs. 9 nd 10 in Appendix B. 3 1-Plnr 2-Bend RAC Drwings in Polynomil Are In this setion we onstrutively prove the following. Theorem 5. Any n-vertex 1-plne grph (G, E) dmits 1-plnr RAC 2 drwing tht respets E nd lies on grid of size O(n 3 ) O(n 3 ). The drwing n be omputed in O(n) time. The ide of our lgorithm is to drw slightly modified, plnrized version of the 1-plne input grph with vrint of the shift lgorithm (by Hrel nd Srds [12]) nd then mnully redrw the rossing edges so tht they ross t right ngles nd hve t most two bends eh. The diffiulty is to find grid points for the bend points nd the rossings so tht the redrwn edges do not touh or ross the surrounding edges drwn by the shift lgorithm. To this end, we refine our grid nd ple the middle prt of eh rossing edge onto horizontl or vertil grid line so tht the edge rossings re t right ngles. Preproessing. Our lgorithm gets n n-vertex 1-plne grph (G, E) s input. First, we plnrize G by repling eh rossing point by vertex (see Fig. 5). We will refer to them s rossing verties. Seond, we enlose eh rossing vertex by subdivided kite, whih is n empty kite where the four boundry edges re subdivided by vertex (see Fig. 5b). We use subdivided kites insted of empty kites to mintin the embedding nd to void dding prllel edges. Third, we mke the grph bionneted using, e.g., the lgorithm of Hoproft nd Trjn [13]. Note tht we do not insert edges into inner fes of subdivided kites beuse ll verties nd edges of subdivided kite re in the sme bionneted omponent. After these three steps, we hve bionneted plne grph (G, E ).

9 Compt Drwings of 1-Plnr Grphs with Right-Angle Crossings 9 We drw (G, E ) using the lgorithm of Hrel nd Srds [12]. This lgorithm returns rossing-free stright-line drwing Γ of (G, E ), whose verties lie on grid of size (2n 4) (n 2), where n is the number of verties of G. Assignment of Edges to Axis-Prllel Hlf-Lines. For eh rossing vertex there re four inident edges in G. They orrespond to two edges of G. Consider the irulr order round in (G, E ). The first nd the third edge inident to orrespond to one edge in (G, E); symmetrilly, the seond nd fourth inident edge orrespond to one edge. To obtin RAC drwing from this, we redrw eh of the four edges round. Consider n edge from vertex of the subdivided kite to the rossing vertex. This edge is then redrwn with bend point b tht lies on n xis-prllel line through. For n exmple how rossing in Γ is repled by RAC rossing, see the trnsition from Fig. 8 to Fig. 8f. In order to obtin right-ngle rossing, we bijetively ssign the four inident edges to the four xis-prllel hlf-lines originting in. We ll suh mpping n ssignment. We do not tke n rbitrry ssignment, but tke re to void extr rossings with edges tht re redrwn or previously drwn. We ll n ssignment A vlid if there is wy to redrw eh edge e with one bend so tht the bend point of e lies on the hlf-line A(e) nd the resulting drwing is plne. To ensure tht our vlid ssignment n be relized on smll grid, we introdue further riteri. We sy tht n edge e 1 depends on nother edge e 2 with respet to n ssignment A if e 2 lies in the ngulr setor between e 1 nd the hlf-line A(e 1 ). In Fig. 6, for exmple, the edge e 3 depends on e 4 nd e 2 depends on e 1, but e 1 nd e 4 do not depend on ny edge. We ll edges (suh s e 1 nd e 4 ) tht do not depend on other edges independent. We define the dependeny depth of n ssignment to be the lrgest integer k with 0 k 3 suh tht there is hin of k +1 edges e 1, e 2,..., e k+1 inident to suh tht e 1 depends on e 2 nd... nd e k depends on e k+1, but there is no suh hin of k+2 edges. For exmple, in Figs. 6, 6b, nd 6, the ssignment hs dependeny depth of 1, wheres in Fig. 6d, the ssignment hs dependeny depth of 0. Showing tht there is vlid ssignment of dependeny depth t most 1 will imply the existene of n pproprite set of grid points for the bend points s formlized in Lemms 7 nd 8. In ft, s we will see in the disussion below, if we ould void dependenies, our drwing would fit on grid of size O(n 2 ) O(n 2 ). Unfortuntely, with our urrent pproh this seems to be unvoidble. We now onstrut n ssignment tht we will show in Lemm 6 to be vlid nd to hve dependeny depth t most 1. The four ses of our ssignment re given in order of priority. Note tht, in Cses 1 nd 2, our ssignment lwys ontins dependenies; see Figs. 6 nd 6b. Note further tht it is enough to speify the ssignment of one edge; the remining ssignment is determined sine the irulr orders of the edges nd the ssigned hlf-lines must be the sme. Cse 1: There is qudrnt q tht ontins ll four inident edges; see Fig. 6. Tke the two inner edges in q nd ssign them to the two hlf-lines tht bound q, while keeping the irulr order. Cse 2: There is qudrnt q tht ontins three inident edges; see Fig. 6b.

10 10 S. Chplik et l. h 2 e 1 e 2e3 e 4 h 1 h 4 h 2 e 1 h 1 e 2 e 3 e 4 h 4 q h 2 e 1 e 2 e 3 e 4 h 1 h 4 h 2 e 2 e 3 h 1 e 1 h 4 e4 q h 3 q h 3 h 3 h 3 () Cse 1: q ontins four edges. h 2 q e 1 h 1 ɛ e 2 e 3 e 4 h 3 h 4 (b) Cse 2: q ontins three edges. h 1 e 1 h 2 h 4 q e 2 e3 e 4 h 3 () Cse 3: q ontins two edges. q e 2 e 3 e 1 h 1 h 2 e 4 h 4 h 3 (d) Cse 4: One edge per qudrnt. h 1 e 1 e 2 h 4 h 2 e 3 e 4 h 3 (e) Cse 1. (f) Cse 2. (g) Cse 3. (h) Cse 4. Fig. 6: The four ses of our ssignment proedure: () (d) indite the ssignment with ornge rrows nd show tht the dependeny depth is lwys t most 1, (e) (f) show tht the ssignment is vlid; the rdius of the light blue disk is ɛ. Consider the edge outside q, sy e 1, nd ssign it to the losest hlf-line h i tht does not bound q. Cse 3: There is qudrnt q tht ontins two inident edges; see Fig. 6. Assign the inident edges in q to their losest hlf-lines. Cse 4: Eh qudrnt ontins extly one inident edge; see Fig. 6d. Assign eh edge to its losest hlf-line in ounter-lokwise diretion. See lso Appendix C, where we prove the following lemm on pge 16. Lemm 6. Our ssignment proedure returns vlid ssignment with dependeny depth t most 1. Note tht Lemm 6 lredy gives us RAC 2 drwing of the input grph, but in order to get (good) bound on the grid size of the drwing, we hve to ple the bend points on grid tht is s orse s possible, but still fine enough to provide us with grid points where we need them: on the hlf-lines emnting from the rossing verties. This is wht the reminder of this setion is bout. Plement of Bend Points on the Grid. In Γ, we hve drwing of subdivided kite for every rossing in the 1-plne input grph. It is n otgon with entrl rossing vertex of degree four in its interior. For n exmple, see Fig. 8. We will redrw the stright-line edges between nd its four djent verties s 1-bend edges ording to the ssignment A omputed in the previous step. The segment of suh 1-bend edge tht ends t will lie on the xis-prllel hlf-line A(). If we pir nd ontente the 1-bend edges tht enter from opposite sides, we obtin two 2-bend edges nd right-ngle rossing in ; see

11 Compt Drwings of 1-Plnr Grphs with Right-Angle Crossings 11 p q () vilble polygon (b) tringle for vlid edge plement given points p nd q Fig. 7: Exmple of n vilble polygon in whih we determine the points p nd q nd with them the tringle for vlid edge plement nd the line segment q. Fig. 8f. It remins to show how the bend points for the edges re pled on the grid. We proeed s follows. First, we determine for eh edge inident to rossing vertex the vilble region into whih we n redrw with bend b on A(). The region between nd the hlf-line A() inside the subdivided kite defines n vilble polygon. Exmples of suh n vilble polygon re given in Figs. 7 nd 8b. Note tht the vilble polygons might overlp (s they do one in Fig. 8b). Observe tht there is only tringle inside eh vilble polygon in whih the new line segment b n be pled. Suh tringle for vlid edge plement is determined by, nd orner point p of the vilble polygon. The point p is the orner point (exluding nd ) for whih the ngle between nd p inside the vilble polygon is the smllest. These tringles for vlid edge plement re depited in Figs. 7b nd 8. Agin, they might overlp. Observe tht in suh tringle, the ngle t nnot beome rbitrrily smll beuse every determining point lies on grid point. Let q be the intersetion point of the line through p nd the hlf-line A(). One n see q s the projetion of p onto A() seen from. Note tht we hve degenerted se if A(). Then, the vilble polygon hs no re nd equls the line segment. In this se let = p = q. Moreover, note tht p n be equl to q beuse the intersetion of A() nd n edge of the subdivided kite is lso orner point of the vilble polygon. This is the only se where p my not be grid point. We will ple the bend point b onto the line segment q, but observe tht the tringles for vlid edge plement of two edges e 1 nd e 2 might overlp if e 1 depends on e 2 in A. To solve this, we first drw the independent edges, then reompute the vilble polygons nd the tringles for vlid edge plement for the other edges, nd finlly drw those edges. Remember tht our ssignment proedure returns only ssignments with dependeny depth t most 1. Let Γ be drwn on grid of size ñ ñ. We refine the grid by ftor of ñ in eh dimension. The next step in our lgorithm relies on the following lemm (whih we prove in Appendix C, pge 19). An importnt tool in our nlysis will be the so-lled Frey sequene [22] of order ñ 1, whih is the sequene of ll redued frtions from 0 to 1 with numertor nd denomintor being positive integers bounded by ñ 1.

12 12 S. Chplik et l () A subdivided kite. The ssignment of edges to hlflines is indited by rrows. (b) Avilble polygons for eh pir of edge nd ssigned hlf-line. () Tringles for vlid edge plement b 3 b 1 b 4 1 b 3 b 1 b 4 1 b 3 b 2 b 4 b 1 1 (d) After the insertion of the bend points of the three independent edges. (e) Avilble polygon nd tringle for vlid edge plement for the edge 2 whih depends on 1. (f) Result fter the insertion of the bend point b 2. Fig. 8: Trnsformtion from plnrized rossing to RAC 2 rossing. Lemm 7. For ny independent edge, the interior of the line segment q ontins t lest one grid point of the refined ñ 2 ñ 2 grid. Using Lemm 7, we pik for eh independent edge ny grid point of q, ple bend point b on it, nd reple the segment by the two segments b nd b. In Fig. 8, the edges 1, 3, nd 4 re independent, but 2 depends on 1. We gin refine the grid by ftor of ñ in eh dimension. The grid size is now ñ 3 ñ 3. For the remining edges inident to rossing vertex, we ompute new vilble polygons nd tringles for vlid edge plement sine we need to tke the 1-bend edges into ount tht were inserted in the previous step. Now the following lemm (proved in Appendix C, pge 22) yields grid points for the bend points of the remining edges. Lemm 8. After hving redrwn the independent edges, the interior of the line segment q of eh edge depending on n independent edge ontins t lest one grid point of the refined ñ 3 ñ 3 grid. For eh remining edge inident to rossing vertex we pik ny grid point of its line segment q nd ple bend point b on it. Agin, we reple by the two line segments b nd b. Result. Finlly, we remove the dummy edges nd dummy verties tht bound the subdivided kites nd interpret the rossing verties s rossing points. We

13 Compt Drwings of 1-Plnr Grphs with Right-Angle Crossings 13 return the resulting RAC 2 drwing Γ. It is drwn on grid of size (8n 3 48n n 64) (4n 3 24n n 32), where n is the number n of verties of G plus 5 times the number of rossings r(e) in E. Note tht r(e) n 2 for 1-plne grphs [6]. If we ignore the bend points, the drwing is on grid of size (2n 4) (n 2), i.e., its size is qudrti. Agin, the lgorithm by Hrel nd Srds [12] nd our modifition run in liner time. Therefore, we onlude the orretness of Theorem 5. 4 Conlusion nd Open Questions We hve shown tht ny n-vertex NIC-plne grph dmits RAC poly 1 drwing in O(n 2 ) re nd tht ny n-vertex 1-plne grph dmits RAC poly 2 drwing in O(n 6 ) re. We hve lso shown how to djust two existing lgorithms for drwing ertin 1-plnr grphs suh tht their embedding is preserved. More preisely, we hve proved tht ny 1-plne grph dmits RAC 1 drwing. This nswers n open question expliitly sked by the uthors of the originl lgorithm [2]. We hve lso proved tht ny stright-line drwble IC-plne grph dmits RAC 0 drwing, where the originl lgorithm did not neessrily preserve the embedding [3].The digrm in Fig. 2 leves some open questions. Does ny 1-plnr grph dmit RAC poly 1 drwing? Cn we drw ny grph in RAC 0 with only right-ngle rossings in polynomil re when we llow one or two bends per edge? Wht is the reltionship between RAC 1 nd RAC poly 2? Cn we ompute RAC poly 2 drwings of 1-plne grphs in o(n 6 ) re? Referenes 1. Bhmier, C., Brndenburg, F.J., Hnuer, K., Neuwirth, D., Reislhuber, J.: NIC-plnr grphs. Disrete Appl. Mth. 232, (2017) Bekos, M.A., Didimo, W., Liott, G., Mehrbi, S., Montehini, F.: On RAC drwings of 1-plnr grphs. Theor. Comput. Si. 689, (2017) Brndenburg, F.J., Didimo, W., Evns, W.S., Kindermnn, P., Liott, G., Montehini, F.: Reognizing nd drwing IC-plnr grphs. Theor. Comput. Si. 636, 1 16 (2016) Chib, N., Ymnouhi, T., Nishizeki, T.: Liner lgorithms for onvex drwings of plnr grphs. In: Bondy, J., Murty, U. (eds.) Progress in Grph Theory, pp Ademi Press, Toronto (1984) 5. Chrobk, M., Pyne, T.H.: A liner-time lgorithm for drwing plnr grph on grid. Inf. Proess. Lett. 54(4), (1995) Czp, J., Hudák, D.: On drwings nd deompositions of 1-plnr grphs. Eletr. J. Comb. 20(2), 54 (2013), rtile/view/v20i2p54 7. Czp, J., Šugerek, P.: Three lsses of 1-plnr grphs. rxiv (2014), org/bs/

14 14 S. Chplik et l. 8. Didimo, W., Edes, P., Liott, G.: Drwing grphs with right ngle rossings. Theor. Comput. Si. 412(39), (2011) Didimo, W., Liott, G., Montehini, F.: A survey on grph drwing beyond plnrity. Arxiv report (2018), Edes, P., Liott, G.: Right ngle rossing grphs nd 1-plnrity. Disrete Appl. Mth. 161(7 8), (2013) de Frysseix, H., Ph, J., Pollk, R.: How to drw plnr grph on grid. Combintori 10(1), (1990) Hrel, D., Srds, M.: An lgorithm for stright-line drwing of plnr grphs. Algorithmi 20(2), (1998) Hoproft, J.E., Trjn, R.E.: Algorithm 447: Effiient lgorithms for grph mnipultion. Commun. ACM 16(6), (1973) Hung, W., Edes, P., Hong, S.: Lrger rossing ngles mke grphs esier to red. J. Vis. Lng. Comput. 25(4), (2014) Hung, W., Hong, S., Edes, P.: Effets of rossing ngles. In: Pro. IEEE VGTC Pifi Visuliztion Symposium (PifiVis 08). pp (2008) Kobourov, S.G., Liott, G., Montehini, F.: An nnotted bibliogrphy on 1-plnrity. Comput. Si. Rev. 25, (2017) Liott, G., Montehini, F.: L-visibility drwings of IC-plnr grphs. Inf. Proess. Lett. 116(3), (2016) Purhse, H.C.: Whih estheti hs the gretest effet on humn understnding? In: Bttist, G.D. (ed.) Pro. 5th Int. Symp. Grph Drwing (GD 97). LNCS, vol. 1353, pp Springer (1997) Ringel, G.: Ein Sehsfrbenproblem uf der Kugel. Abh. Mth. Seminr Univ. Hmburg 29(1 2), (1965) 20. Shnyder, W.: Embedding plnr grphs on the grid. In: Johnson, D.S. (ed.) Pro. 1st ACM-SIAM Symp. Disrete Algorithms (SODA 90). pp (1990), http: //dl.m.org/ittion.fm?id= Thomssen, C.: Retiliner drwings of grphs. J. Grph Theory 12(3), (1988) Wikipedi ontributors: Frey sequene Wikipedi, the free enylopedi (2018), sequene&oldid= , [Online; essed 8-June-2018] 23. Zhng, X.: Drwing omplete multiprtite grphs on the plne with restritions on rossings. At. Mth. Sin. English Ser. 30(12), (2014) Zink, J.: 1-Plnr RAC Drwings with Bends. Mster s thesis, Institut für Informtik, Universität Würzburg (2017), de/pub/theses/2017-zink-mster.pdf

15 Compt Drwings of 1-Plnr Grphs with Right-Angle Crossings 15 A Proofs for Setion 2 To prove Lemm 4 below, we use the following. Lemm 9. Let G be the input grph, G be the grph fter the preproessing, nd Ĝ be the grph fter the omputtion of the BCO. Let n, n, nd ˆn be the number of verties in G, G, nd Ĝ, respetively. It holds tht n 3.4n 4.8 nd ˆn 4n 6. Proof. In the first step of the preproessing, we rete empty kites round every rossing. By reting the empty kites for every rossing, there re edges dded to the grph (but we do not ount edges here) nd there re edges subdivided. When we subdivide n edge, we dd new vertex. There re t most four edges per rossing tht re subdivided. The number r(e) of rossings in NIC-plnr embedding E is bounded by 0.6n 1.2 [7,23]. Using this, we n bound the number n subdivide of verties tht re dded in this step to: n subdivide 4 r(e) 2.4n 4.8 In the seond step of the preproessing, we mke the grph bionneted. To omplish this, we only insert edges nd the number of verties does not inrese. So the number n of verties of the grph G is: n = n + n subdivide 3.4n 4.8 While omputing the BCO ˆΠ, t most one dummy vertex is dded per rossing either s shift vertex in Cse 2 or s dummy vertex in Cse 3. So the number n ˆΠ of verties dded there is: And in totl: n ˆΠ r(e) 0.6n 1.2 ˆn = n + n ˆΠ (3.4n 4.8) + (0.6n 1.2) = 4n 6 Thus, only linerly mny new verties re dded when onstruting G from G nd Ĝ from G. Lemm 4. Every vertex, bend point, nd rossing point of the drwing returned by our lgorithm lies on grid of size t most (16n 32) (8n 16). Proof. The shift lgorithm ples every vertex of the grph Ĝ = ( ˆV, Ê) onto grid point of grid of size (2ˆn 4) (ˆn 2). By the upper bound on ˆn from Lemm 9, we get the following grid size: orser grid size (2(4n 6) 4) ((4n 6) 2) = (8n 16) (4n 8)

16 16 S. Chplik et l. This grid is lter refined by ftor of 2 in both dimensions. This bounds the size of the grid s follow: totl grid size (2(8n 16)) (2(4n 8)) = (16n 32) (8n 16) We ple bend points onto grid points on inner fes only. So the totl size of the drwing nd its underlying grid does not inrese when we dd them. B Full Exmple of Drwing from Setion 2 We hve implemented our lgorithm in Jv. Figure 10 shows drwing of NIC-plne grph produed by this implementtion. The embedded grph in this exmple hs four rossings. For two of these rossings, Cse 2 of our lgorithm pplies (green bkground olor). Cse 1 (yellow bkground olor) nd Cse 3 (red bkground olor) pply to one rossing eh. In prtiulr, in Fig. 10, two pirs of segments with slope +1 nd 1 ross t right ngle nd two pirs of horizontl/vertil segments ross. The drwing in Fig. 9 shows the grph s it is drwn by the shift lgorithm nd before the rossing edges re inserted. Note tht the two divided qudrngles in Cse 2 ontin n dditionl shift vertex nd the one in Cse 3 hs n dditionl 2-pth, whih is lso highlighted nd mkes the qudrngle pentgon with seond hord edge. The drwing in Fig. 10 shows the finl grph drwing fter the rossing edges hve been reinserted in the postproessing step nd fter the dummy edges nd verties hve been removed. The four pirs of rossing edges re highlighted by thik edges. C Proofs for Setion 3 Lemm 6. Our ssignment proedure returns vlid ssignment with dependeny depth t most 1. Proof. Observe tht there is disk with rdius ɛ > 0 entered t suh tht for every point p in this disk, the four line segments 1 p, 2 p, 3 p, 4 p do not ross the boundry of the subdivided kite. In prtiulr, by redrwing edges with bend points in this disk, we need only to worry bout rossings mong the edges inident to, not with edges of the kite. To estblish the lemm, it suffies to onsider the four ses of our ssignment independently. In Figs. 6 6d the dependeny depth is t most 1 in ny of the four ses. Note tht only in Cse 3 other onfigurtions regrding the positions of e 3 nd e 4 re possible, for exmple, when e 3 nd e 4 lie in distint qudrnts or when e 3 nd e 4 lie in the qudrnt opposite q. These lternte onfigurtions result in ll of e 1, e 2, e 3 nd e 4 being independent. Thus, we onlude tht the dependeny depth is lwys t most 1. Now, we ple the bend points. For i = 1,..., 4, we determine the distne ɛ i of b i from, s follows. If edge e i is independent, we simply set ɛ i = ɛ. Otherwise,

17 Compt Drwings of 1-Plnr Grphs with Right-Angle Crossings 17 Fig. 9: Full exmple of n intermedite drwing omputed by our Jv implementtion of the lgorithm from Setion 2. It shows grph drwn by the shift lgorithm before the rossing edges re reinserted. The edges of the divided qudrngles into whih the rossing edges re reinserted in the next step re drwn with thik lines. Note tht two of these qudrngle ontin n dditionl shift vertex nd one n dditionl neighboring 2-pth. Dummy edges nd verties re drwn in gry.

18 18 S. Chplik et l. Fig. 10: Full exmple of finl NIC-plnr RAC 1 drwing omputed by our Jv implementtion of the lgorithm from Setion 2. It is the drwing from Fig. 9 fter the rossing edges hve been reinserted nd the dummy edges of the drwing hve been removed. The four pirs of rossing edges re drwn with thik lines. Note tht two pirs of rossing edges were drwn ording to Cse 2 (green bkground olor), one pir ording to Cse 1 (yellow bkground olor), nd one pir ording to Cse 3 (red bkground olor) of our lgorithm.

19 Compt Drwings of 1-Plnr Grphs with Right-Angle Crossings 19 p q h p h q h q p out () Cse A1. (b) Cse A4. p in () Cse B. Fig. 11: Different ses onerning the nlysis of h = q in the proof of Lemm 7. if e i depends on e j, we first ple b j, ompute the intersetion point x of j b j with A(e i ), nd set ɛ i = x /2. By this simple rule nd the hoie of ɛ it is ler tht no two redrwn edges interset. Hene, the ssignment is vlid. A nie property of neighboring numbers b nd d in the Frey sequene, ssuming b < d, is tht d b = 1 bd. (1) Lemm 7. For ny independent edge, the interior of the line segment q ontins t lest one grid point of the refined ñ 2 ñ 2 grid. Proof. Without loss of generlity, we n ssume tht q is vertil. If x() = x(q) = x(), we hve the degenerted se q =. We do not need to bend the edge in our lgorithm, but, for the ompleteness of the proof, we n esily see tht there re t lest ñ 1 grid points on the refined ñ 2 ñ 2 grid beuse nd q = re grid points of the orser ñ ñ grid. So, without loss of generlity, we n ssume tht x() < x(q) = x(), beuse mirroring the drwing with respet to the line through q does not hnge the struture of the drwing. We n lso ssume tht = (0, 0). Agin, without loss of generlity, we n ssume tht y() 0. If y() = 0, we n furthermore ssume y(q) > 0 (both by the rgument of mirroring ross the x-xis). If y() > 0 nd y(q) < 0, we re fine beuse nd (x(), 0) re both grid points of the orser grid. Between them, there is more thn one grid point of the finer grid. So we ontinue with y() 0 nd y(q) 0. For onveniene, we will work with oordintes on the orser O(ñ) O(ñ) grid in the following se distintion. Moreover, observe tht does not lie on the top- or bottommost row or on the left- or rightmost olumn of the grid sine is enlosed by the dummy edges of divided qudrngle. Therefore, we know tht the differene in the x- nd in the y-oordinte of nd ny other vertex of the drwing is less thn ñ. In prtiulr, we know tht 0 = x() < x(p) x(q) = x() < ñ.

20 20 S. Chplik et l. Now, we distinguish two ses. Cse A: The point p is grid point. The points, p nd q re olliner. For x(p) y(p) nd x() y(), the slopes of p nd re vlues of the Frey sequene of order ñ 1. The slopes re y(p)/x(p) nd y()/x(). One n imgine ll these possible slopes going out from s rys. Without loss of generlity, we n ssume tht the redued frtions of y(p)/x(p) nd y()/x() (or their reiprols) re neighbored frtions in the Frey sequene nd neighbored rys in the piture of the rys going out from. We lso ssume tht y(p)/x(p) nd y()/x() re redued frtions beuse for multiple of one of the Frey numbers, the line segment q ould only be longer nd hve more grid points of the finer grid on it but not fewer. We distinguish the following four subses. Cse A1: y(q) y(), nd y(p) x(p) (see Fig. 11). We hve nd Putting this together, we get nd y() x() re neighbors in the Frey sequene h = q = y(q) y() (2) y(q) = y(p) x(p) x(). h = y(p) ( y(p) x() y() = x() x(p) x(p) y() ). (3) x() Due to y(q) y() nd to Eqution 1, we know tht y(p) leds to ( y(p) h = x() x(p) y() ) = x() x() x(p) > y() x(). Applying this 1 x() x(p) = 1 x(p) > 1 ñ. (4) Cse A2: y(q) y(), nd y(p) x(p) re neighbors in the Frey sequene. This is lmost the sme s Cse A1, only multiplied with 1 beuse now we hve y(p) x(p) < y() x(). Indeed, we hve nd y() x() h = y() y(q) = y() y(p) x(p) x() ( y() = x() x() y(p) ) x(p) 1 = x() x() x(p) = 1 x(p) > 1 ñ.

21 Compt Drwings of 1-Plnr Grphs with Right-Angle Crossings 21 Cse A3: y(q) y(), nd y(p) x(p) y() nd x() re not numbers of the Frey sequene beuse their numertor is greter thn their denomintor, but they n be seen s prt of n extension of the Frey sequene from 1 to +. Their reiprols re neighbors in the Frey sequene. This se is lso similr to A1. Equtions 2 nd 3 still hold, but we need to nd y() x() be reful with Eqution 4 beuse y(p) x(p) sequene. An implition of Eqution 1 is tht re not numbers of the Frey d b d = = 1, whih implies b d = 1. b bd bd Plugging in the Frey numbers x(p) y(p) nd x() y() y(p) x() x(p) y() = 1. x(p) with y(p) < x() y(), we get Using this, we trnsform Eqution 3, whih yields the desired lower bound on h: ( y(p) h = x() x(p) y() ) x() y(p) x() x(p) y() = x() x() x(p) 1 = x() x() x(p) = 1 x(p) > 1 ñ Cse A4: y(q) y(), nd y(p) x(p) y() nd x() re not numbers of the Frey sequene beuse their numertor is greter thn their denomintor, but they n be seen s prt of n extension of the Frey sequene from 1 to +. Their reiprols re neighbors in the Frey sequene. A sketh is given in Fig. 11b. This se is nlogous to Cse A3 in the sme wy s Cse A2 is nlogous to Cse A1. Agin, we n multiply with 1 or lterntively swp ll ourrenes of p nd. Cse B: The point p is not grid point. This sitution my only our if p = q. In this se the point p in the vilble polygon is the intersetion of the ssigned xis-prllel hlf-line nd n edge e of the subdivided kite. We nme the endpoint of e tht is inside the vilble polygon p in nd the endpoint tht is outside p out (see Fig. 11). Clerly, we hve similr sitution s in Cse A. Here, p in is in the position of in Cse A nd p out is in similr position s p in Cse A. The points p in nd p out re verties of G nd, thus, grid points of the ñ ñ grid. The only differene is the order of the points, p, q nd p in, q, p out on eh ommon line. Observe tht the formuls given in Cse A still hold if q lies between p in nd p out insted of lying to the right of both. Therefore, by doing the sme nlysis s in Cse A with exhnged roles of nd p, we get the sme result. To summrize, for both ses nd eh subse, we hve seen tht q > 1/ñ. By refining the ñ ñ grid by ftor of ñ in eh dimension, we get ñ 2 ñ 2 grid

22 22 S. Chplik et l. q ˆb â Fig. 12: The bend point of the edge is pled on grid point of the interior of the line segment q. The point q depends on the plement of ˆb, whih is the bend point of the edge â. where eh grid point of the orser grid is lso grid point of the finer grid. The rossing point is grid point of both grids. On eh of the four xis-prllel hlf-lines emnting from, we reh the next grid point fter distne of 1/ñ. Given tht q > 1/ñ, the interior of the line segment q ontins t lest one grid point. Lemm 8. After hving redrwn the independent edges, the interior of the line segment q of eh edge depending on n independent edge ontins t lest one grid point of the refined ñ 3 ñ 3 grid. Proof. Given Lemm 7, we hve to onsider only edges depending on other edges. All of the following oordintes re reltive to the grid of size ñ 2 ñ 2 tht hs been refined one. We ssume tht the edge depends on the edge â. If the tringle for vlid edge plement of ws not shrunk fter pling ˆb, then the nlysis of Lemm 7 holds here s well. Otherwise we know tht q is the intersetion of âˆb nd the ssigned hlf-line of. We ssume, without loss of generlity, tht x(â) = 0 nd y(â) = 0. Furthermore, we ssume tht x() 0 nd y() 0 beuse mirroring ross some xis-prllel line does not hnge the struture of the drwing. We ssume, without loss of generlity, tht ˆb lies on the hlf-line originting t nd going to positive infinity in the x-dimension beuse âˆb rosses some other xis-prllel hlf-line (here: the one going to negtive infinity in y-dimension) nd, gin, mirroring does not hnge the struture of the drwing. This implies y() > y(â). Our urrent sitution is depited in Fig. 12. Now, we nlyze how short the line segment q n beome in the worst se. The line segment will beome shorter if the x-distne x(ˆb) x() dereses or the y-distne y() y(â) dereses or the x-distne x() x(â) inreses. So q will be shortest if we ssume the most extremes of these vlues, nmely x-distne x(ˆb) x() = 1, nd

23 Compt Drwings of 1-Plnr Grphs with Right-Angle Crossings 23 y-distne y() y(â) = ñ (it nnot beome smller beuse both re points of the orser ñ ñ grid nd y() > y(â)), nd x-distne x() x(â) = (ñ 1) ñ (This is beuse both re grid points on the orser ñ ñ grid. Sine ˆb is on the right side of both, they nnot both be outermost grid points nd, thus, they n only hve distne of ñ 1 on the initil orser grid nd (ñ 1) ñ on the refined grid of Γ.) Hene, for the slope m of âˆb, we get Using this, we determine y(q) by m = ñ ñ 2 ñ + 1. y(q) = ( ñ 2 ñ ) m = ( ñ 2 ñ ) ñ ñ 2 ñ + 1 = ñ3 ñ 2 ñ 2 ñ + 1 Now, we n ompute the length of the line segment q this wy: y() y(q) = ñ ñ3 ñ 2 ñ 2 ñ + 1 = ñ3 ñ 2 + ñ ñ 3 + ñ 2 ñ 2 ñ + 1 ñ = ñ 2 ñ + 1 = 1 ñ > 1 (for ñ > 1) ñ ñ With the sme rgument s in the proof of Lemm 7, we see tht the interior of q ontins lwys t lest one grid point of the refined ñ 3 ñ 3 grid. D Preserving Embeddings In this setion, we show to preserve the embedding when we ompute 1-plnr RAC 1 nd IC-plnr RAC 0 drwings from 1-plne nd stright-line drwble IC-plne grphs, respetively. There re lgorithms known tht ompute suh drwings from 1-plne nd IC-plne grphs, but they hnge the input embedding. We desribe how to modify suh lgorithms so tht the input embedding is preserved in the output. This mens tht the two ontinment reltions shown in the digrm in Fig. 2 with neled E? lso hold for fixed embeddings. D.1 1-Plnr 1-Bend RAC Drwings Bekos et l. [2] desribe n lgorithm for omputing 1-plnr RAC 1 drwings of 1-plnr grphs in liner time. Their lgorithms tke 1-plne grph s input, but the embedding my be hnged during the exeution of the lgorithm, i.e., while the output is indeed drwing of the sme grph, it n indue different 1-plnr embedding. In ft, they expliitly sk if every 1-plnr embedding dmits RAC 1 drwing. We nswer their question in the ffirmtive by desribing how to modify their lgorithm; see Theorem 1. Theorem 1. Any n-vertex 1-plne grph dmits n embedding-preserving RAC 1 drwing. It n be omputed in O(n) time.

24 24 S. Chplik et l. Originl Algorithm. The lgorithm strts with n ugmenttion step. In the 1- plne input grph (G, E), dummy edges re inserted round eh pir of rossing edges to indue empty kites (empty kites re defined in Setion 2). Thereby prllel edges n our. They remove the originl edge from eh set of prllel edges (this hnges the embedding), nd for eh fe of degree two, i.e., fe bounded by two prllel edges, they remove one of the edges. There n still be prllel dummy edges. At the end of the ugmenttion step they tringulte eh fe by inserting dummy edges nd verties to obtin tringulted 1-plne multigrph (G +, E + ). The next step is omputing hierrhil ontrtion of (G +, E + ). For eh set of prllel edges there is n inner grph omponent seprted from the rest of the grph by the two outermost edges of these prllel edges. This inner omponent is ontrted to single thik edge, to whih the informtion bout the ontrted subgrph is sved. This ontrtion opertion is pplied (reursively) to every set of prllel edges. In this wy, they obtin hierrhy of simple 1-plne 3-onneted tringulted grphs. The top-level grph is denoted by (G, E ). The lst step of the lgorithm is drwing the grph. They remove the rossing edges from (G, E ) nd drw it with n lgorithm tht delivers stritly onvex stright-line drwings where the outer fe is presribed onvex polygon. The liner-time lgorithm by Chib et l. [4] fulfills these requirements. They pss, s the presribed polygon, trpezoid if the outer fe hs degree four 2 nd tringle otherwise. Next, they mnully reinsert the rossing edges. For the inner onvex fes, they drw one edge stright-line nd the other edge with bend so tht it rosses the first edge t right ngle. For the outer fes, they bend both edges. This proedure is pplied reursively for eh subgrph ontrted to thik edge. Sine they n presribe the shpe of the outer fe, they n lwys pss shpe tht fits into the free spe next to thik edge to expnd eh subgrph. In the end they remove the dummy edges nd verties tht hve not been prt of the input grph nd obtin 1-plnr RAC 1 drwing of the input grph. Note tht the embedding my hve hnged during the exeution of the ugmenttion step where they hd prllel edges. Our Modifitions. Our modifition in the ugmenttion is to keep the originl edges tht re not prt of rossing. But like them, we remove the prllel originl edges tht ross nother edge. When we remove suh rossing edge e, n empty kite beomes divided qudrngle (see Fig. 13; divided qudrngles re defined in Setion 2). Suppose the edge e rossed e before e s removl. Note tht the edge e nnot hve prllel dummy edges sine these would ross either e or prllel dummy edge of e, but, s stted erlier, no inserted dummy edge results in new rossing nd rossing with e would violte the 1-plnrity. We remember where we removed these edges from beuse we will reinsert them lter. 2 i.e., when rossing on the outer fe ws removed t the beginning of the drwing step