Algorithms for graphs on surfaces

Transcription

1 Algorithms for grphs on surfces Éric Colin de Verdière École normle supérieure,

2 ALGORITHMS FOR GRAPHS ON SURFACES Foreword nd introduction Foreword These notes re certinly not in nl shpe, nd comments by e-mil re welcome. The course my deprt from these notes both in content nd presenttion. It is strongly recommended to work on the exercises. Ech exercise is lbeled with one to three strs, supposed to be n indiction of its importnce (in prticulr, depending on whether it is used lter), not of its diculty. Introduction This is n introduction to the computtionl spects of surfces nd grphs drwn on them. This topic hs been subject of ctive reserch, especilly over the lst decde, nd is relted to rther diverse elds nd communities: in computtionl geometry, surfces rise nturlly in vrious pplictions. Opertions in geometric spces such s decomposition, extrction of importnt fetures, nd shortest pth computtion re bsic computtionl geometry tsks tht re relevnt in prticulr for surfces, usully embedded in R 3, or even plnr surfces; in topology, the clssiction of surfces, s discovered in the beginning of the 20th century, is inherently lgorithmic. Surfces ply First version published December 1, Dte of this version: Mrch 19, Ltest version vilble t lgo-grphs-surfces.pdf. Foreword nd introduction prominent role in the deep theories of knots nd three-mnifolds; there re lso mny lgorithmic questions in these res; in grph lgorithms, mny generl grph problems become esier when restricted to plnr grphs (shortest pth, ow nd cut, minimum spnning trees, vertex cover, grph isomorphism, etc.). Grphs on surfces stnd in-between: to which extent do plnr techniques pply? in grph theory, the theory of grph minors founded by Robertson nd Seymour mkes hevy use of grphs embeddble on xed surfce, s well s grphs excluding xed minor. Edge-width nd fcewidth re closely relted to the notion of shortest non-contrctible cycle. In ddition, vrious computtionl tools re needed in pplictions hndling surfces, though we will not describe them in this course: in computer grphics, for texture mpping, morphing, nd visuliztion; in mesh processing nd numericl nlysis, to remesh, simplify topologiclly, pproximte, nd compress surfce; in topologicl nlysis, to build hierrchicl description of shpe, distinguish (un)signicnt fetures, nd compre topologicl fetures of two shpes (lthough this pplies usully to higher-dimensionl geometric dt); in computer-ided geometric design (CAGD) nd geometric modelling. This course is very fr from surveying ll these spects. We im t presenting recent results in computtionl geometry on this subject; for this purpose, mny tools nd concepts need to be introduced, which re of generl interest. The rst chpter introduces plnr grphs from the lgorithmic point of view; it serves s wrm-up to the cse of grphs on surfces. We then introduce surfces, from the topologicl nd computtionl points of view (Chpters 2 nd 3). Algorithms using the cut locus to build short curves nd decompositions of surfces re presented in Chpter 4. Chpter 5 2

3 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs introduces more dvnced notions of topology. All these techniques re combined in Chpter 6 to provide lgorithms to shorten curves up to deformtion. Only prt of the mteril covered in this course ppered in textbooks. For further reding or dierent expositions, mostly on the combintoril spects, recommended books re Armstrong [3], Mohr nd Thomssen [57], nd Stillwell [67]. For the lgorithmic spects nd wider perspective, see the very recent course notes by Erickson [27]. Chpter 1 Plnr grphs Acknowledgments I would like to thnk severl people who suggested some corrections: Je Erickson, Frncis Lzrus, Arthur Milchior, nd Vincent Pilud. 1.1 Topology Preliminries on topology We ssume some fmilirity with bsic topology, but we recll some denitions nonetheless. A topologicl spce is set X with collection of subsets of X, clled open sets, stisfying the three following xioms: the empty set nd X re open; ny union of open sets is open; ny nite intersection of open sets is open. There is, in prticulr, no notion of metric (or distnce, ngle, re) in topologicl spce. The open sets give merely n informtion of neighborhood: neighborhood of x X is set contining n open set contining x. This is lredy lot of informtion, llowing to dene continuity, homeomorphisms, connectivity, boundry, isolted points, dimension.... Speciclly, mp f : X Y is continuous if the inverse imge of ny open set by f is n open set. If X nd Y re two topologicl spces, mp f : X Y is homeomorphism if it is continuous, bijective, nd if its inverse f 1 is lso continuous. A point of detil, ruling out pthologicl spces: the topologicl spces considered in these notes re ssumed to be Husdor, which mens tht two distinct points hve disjoint neighborhoods. 3

4 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs A spce X is connected 1 if it is non-empty nd, for ny points nd b in X, there exists pth in X whose endpoints re nd b. The connected components of topologicl spce X re the clsses of the equivlence reltion on X dened by: is equivlent to b if there exists pth between nd b. A topologicl spce X is disconnected (or seprted) by Y X if nd only if X\Y is not connected; points in dierent connected components of X \ Y re seprted by Y Grph embeddings Figure 1.1. The stereogrphic projection. Exmple 1.1. Most of the topologicl spces here re endowed with nturl metric, which should be forgotten, but dene the topology: R n (n 1); the n-dimensionl sphere S n, i.e., the set of unit vectors of R n+1 ; the n-dimensionl bll B n, i.e., the set of vectors in R n of norm t most 1; in prticulr B 1 = [ 1, 1] nd [0, 1] re homeomorphic; the set of lines in R 2, or more generlly the Grsmnnin, the set of k-dimensionl vector spces in R n. Exercise 1.2 (stereogrphic projection). 99 Prove tht the plne is homeomorphic to S 2 with n rbitrry point removed. (Indiction: see Figure 1.1.) A closed set in X is the complement of n open set. The closure of subset of X is the (unique) smllest closed set contining it. The interior of subset of X is the (unique) lrgest open set contined in it. The boundry of subset of X equls its closure minus its interior. A topologicl spce X is compct if ny set of open sets whose union is X dmits nite subset whose union is still X. A pth in X is continuous mp p : [0, 1] X; its endpoints re p(0) nd p(1). Its reltive interior is the imge by p of the open intervl (0, 1). In this course, unless noted otherwise, ll grphs re undirected nd nite but my hve loops nd multiple edges. A grph yields nturlly topologicl spce: for ech edge e, let X e be topologicl spce homeomorphic to [0, 1]; let X be the disjoint union of the X e ; for e, e, identify (quotient topology), in X, endpoints of X e nd X e if these endpoints correspond to the sme vertex in G. An embedding of G in topologicl spce Y is continuous one-to-one mp from G (viewed s topologicl spce) to Y. Sid dierently, it is crossing-free drwing of G on Y, being the dt of two mps: Γ V, which ssocites to ech vertex of G point of X; Γ E, which ssocites to ech edge e of G pth in X between the imges by Γ V of the endpoints of e, in such wy tht: the mp Γ V is one-to-one (two distinct vertices re sent to distinct points of X); for ech edge e, the reltive interior of Γ E (e) is one-to-one (the imge of n edge is simple pth, except possibly t its endpoints); for ll distinct edges e nd e, the reltive interiors of Γ E (e) nd Γ E (e ) re disjoint (two edges cnnot cross); 1 In this course, the only type of connectivity considered is pth connectivity. 4

5 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs for ech edge e nd for ech vertex v, the reltive interior of Γ E (e) does not meet Γ V (v) (no edge psses through vertex). When we spek of embedded grphs, we sometimes implicitly identify the grph, its embedding, nd the imge of its embedding Plnr grphs nd the Jordn curve theorem In the remining prt of this chpter, we only consider embeddings of grphs into the sphere S 2 or the plne R 2. A grph is plnr if it dmits n embedding into the plne. By Exercise 1.2, this is equivlent to the existence of n embedding into the sphere S 2. The fces of grph embedding re the connected components of the complement of the imge of the vertices nd edges of the grph. Here re the most-often used results in the re. Theorem 1.3 (Jordn curve theorem, reformulted; see [70]). Let G be grph embedded on S 2 (or R 2 ). Then G disconnects S 2 if nd only if it contins cycle. Theorem 1.4 (JordnSchönies theorem; see [70]). Let f : S 1 S 2 be one-to-one continuous mp. Then S 2 \ f(s 1 ) is homeomorphic to two disjoint copies of the open disk. These results re, perhps surprisingly, dicult to prove: the diculty comes from the generlity of the hypotheses (only continuity is required). For exmple, if in the Jordn curve theorem one ssumes tht G is embedded in the plne with polygonl edges, the theorem is not hrd to prove. A grph is cellulrly embedded if its fces re (homeomorphic to) open disks. Henceforth, we only consider cellulr embeddings. It turns out tht grph embedded on the sphere is cellulrly embedded if nd only if it is connected. 2 2 Although this sttement should be intuitively cler, it is not so obvious to prove. It 1.2 Combintorics So fr, we hve considered curves nd grph embeddings in the plne tht re rther generl. In the rest of these notes, we will need to mke some ssumptions: Ech curve, nd ech edge of grph embedding, is piecewise-liner. This llows to ssume tht the complement of n embedding re well-behved. In prticulr, this restriction rules out bizrre objects like spce-lling curves, which dmit no tubulr neighborhood Combintoril representtions of plnr grph embeddings We now focus on the combintoril properties of cellulr grph embeddings in the sphere. Since we re not interested in the geometric properties, it suces to specify how the fces re glued together, or lterntely the cyclic order of the edges round vertex. Embeddings of grphs on the plne re treted similrly: just choose distinguished fce of the embedding into S 2, representing the innite fce of the embedding in the plne. An lgorithmiclly sound wy of representing combintorilly cellulr grph embedding in S 2 is vi mps, which we now describe. The bsic notion is the g, which represents n incidence between vertex, n edge, nd fce of the embedding. Three involutions llow to move to nerby g, nd, by iterting, to visit the whole grph embedding; see Figure 1.2: vi moves to the g with the sme edge-fce incidence, but with dierent vertex incidence; ei moves to the g with the sme vertex-fce incidence, but with dierent edge incidence; moves to the g with the sme vertex-edge incidence, but with dierent fce incidence. my help to use the results of Chpter 2, especilly the fct tht every fce of grph embedding is surfce with boundry. 5

6 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs vi ei Figure 1.2. The gs re represented s line segments prllel to the edges; there re four gs per edge. The involutions vi, ei, nd on the thick g re lso shown. int vertex_degree(flg fl) { int j=0; Flg fl2=fl; do { ++j; fl2=fl2->ei()->fi(); } while (fl2!=fl); return j; } int fce_degree(flg fl) { int j=0; Flg fl2=fl; do { ++j; fl2=fl2->ei()->vi(); } while (fl2!=fl); return j; } Figure 1.3. C++ code for degree computtion. Exmple 1.5. Figure 1.3, left, presents code to compute the degree of vertex, i.e., the number of vertex-edge incidences of this vertex. The function tkes s input g incident with tht vertex. Note tht loop incident with the vertex mkes contribution of two to the degree. Dully, on the right, code to compute the degree of fce (the number of edge-fce incidences of this fce) is shown. Ech g lso hs pointer to the underlying vertex, edge, nd fce (clled respectively vu, eu, fu). Ech such vertex, edge, or fce contins no informtion on the incident elements, only informtion needed in the lgorithms (we will see exmples lter). This lso llows to test whether two gs re incident to the sme vertex, edge, or fce. If needed, one my similrly put some informtion in the vertex-edge, edge-fce, vertex-fce, nd vertex-edge-fce incidences. Note tht g is not necessrily uniquely dened by its triple (vertex, edge, nd fce), s shows the exmple of grph with single vertex nd single (loop) edge. The complexity of grph G = (V, E) is V + E. The complexity of cellulr grph embedding is the totl number of gs involved, which is liner in the number of edges (every edge bers four gs), nd lso in the number of vertices, edges, nd fces. Therefore the complexity of grph cellulrly embedded in the plne nd of one of its embeddings re linerly relted Dulity nd Euler's formul A dul grph of cellulr grph embedding G = (V, E) on S 2 is grph embedding G dened s follows: put one vertex f of G in the interior of ech fce f of G; for ech edge e of G, crete n edge e in G crossing e nd no other edge of G (if e seprtes fces f 1 nd f 2, then e connects f1 nd f 2 ). See Figure 1.4. A dul grph embedding is lso cellulr. The combintoril representtion of the dul grph is unique. Actully, with the mp representtion, dulizing is esy: simply replce with vi nd vice-vers. This in prticulr 6

7 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs we hve v e + f = 2. Hence this formul lso holds for every embedding of connected grph in the plne. Proof. Let T be the edge set of spnning tree of G. The dul edges of its complement, (E \ T ), is lso spnning tree. The number of edges of G is e = T + (E \ T ), which, by Exercise 1.6, equls (v 1) + (f 1). Exercise 1.9 (esy direction of Kurtowski's theorem). 99 Show tht the complete grph with 5 vertices, K 5, is not plnr. Indiction: Use Euler's formul nd double-counting on the number of vertex-edge nd edge-fce incidences. Also show tht the biprtite grph K 3,3 (with 6 vertices v 1, v 2, v 3, w 1, w 2, w 3 nd 9 edges, connecting every possible pir {v i, w j }) is not plnr. Figure 1.4. Dulity. proves tht dulity is n involution: G = G. Exercise 1.6 (esy). 999 Every tree (cyclic connected grph) with v vertices nd e edges stises v e = 1. Lemm 1.7. Let G = (V, E) be cellulr grph embedding in S 2, nd let G = (F, E ) be its dul grph. Furthermore, let E E. Then (V, E ) is cyclic if nd only if (F, (E \ E ) ) is connected. In prticulr, (V, E ) is spnning tree if nd only if (F, (E \ E ) ) is spnning tree. Proof. (V, E ) is cyclic if nd only if S 2 \ E is connected, by the Jordn curve theorem 1.3. Furthermore, S 2 \E is connected if nd only if (F, (E\ E ) ) is connected: Two points x nd x in fces f nd f of G cn be connected by pth voiding E nd not entering ny fce other thn f nd f if nd only if f nd f re djcent by some edge not in E, i.e. if nd only if f nd f re djcent in (F, (E \ E ) ). Corollry 1.8 (Euler's formul for cellulr grph embeddings in S 2 ). For every cellulr grph embedding in S 2 with v vertices, e edges, nd f fces, Side Note: Brycentric Subdivision Let G be cellulr embedding of grph on S 2. By overlying G with its dul grph G, we obtin qudrngultion: cellulr embedding of grph G + where ech fce hs degree four. See Figure 1.4. Every fce of G + is incident with four vertices: one vertex v G of G, one vertex v G of G, nd two vertices tht re the intersection of n edge of G nd n edge of G. If, within ech fce, we connect v G with v G, we obtin tringultion, clled the brycentric subdivision of G (Figure 1.5). It lso provides nother convenient wy to visulize the gs nd their opertions. Ech tringle in the brycentric subdivision corresponds to g. Its three neighbors re the gs rechble vi the opertions vi, ei, nd. 1.3 Algorithms Algorithms for plnr grphs tke s input combintoril representtion of n embedding of the grph. 7

8 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs fces. Then removing e yields cellulr grph embedding, denoted by G\e. The dul opertion is contrction: let e be n edge of G tht is incident with two dierent vertices (i.e., tht is not loop), then we my contrct e by identifying its two incident vertices; the resulting grph embedding is denoted by G/e. Obviously, these two opertions preserve the plnrity. Figure 1.5. Figure 1.4. The brycentric subdivision of the prt of the grph shown in Minimum spnning tree lgorithm Let G = (V, E) be cellulr grph embedding in S 2, with weight function w : E R on its edges. Let n be its complexity. Theorem A minimum spnning tree of G cn be computed in O(n) time. We note tht, by Lemm 1.7, E E is minimum spnning tree of G if nd only if (E \ E ) is mximum spnning tree of G (where the weight of dul edge equls the weight of the corresponding priml edge). Exercise Prove tht connected plnr grph hs either vertex or fce with degree t most three. We introduce two opertions to trnsform cellulr grph embedding in S 2 into nother one. These opertions (together with their reverses) re clled Euler opertions. Let e be n edge of G tht is incident with two dierent Proof of Theorem The two following dul rules llow to build inductively the set of edges T (G) of minimum spnning tree of G: Let v be vertex of G. If ll edges incident with v re loops, then G hs exctly one vertex, so there is unique, empty, spnning tree. Otherwise, let e be minimum-weight edge incident exctly once with v. Necessrily, edge e belongs to minimum spnning tree of G. Hence T (G/e) e is minimum spnning tree of G; let f be fce of G. If ll edges incident with f hve f on both sides, then G hs exctly one fce, so G is tree, nd there is unique spnning tree, G itself. Otherwise, let e be mximum-weight edge incident exctly once with f. Then e does not belong to minimum spnning tree of G (becuse e belongs to mximum spnning tree of G ). It follows tht T (G \ e) is minimum spnning tree of G. The number of itertions of this lgorithm is O(n). Assuming we cn pick vertex v or fce f with degree O(1) (whose existence is gurnteed by Exercise 1.11) in constnt mortized time, we hve liner-time lgorithm. Indeed, without loss of generlity ssume we hve vertex v with degree O(1); the dul cse is similr. Determining which edges incident to v re loops tkes O(1) time. If ll of them re loops, then the recursion stops; otherwise, nding minimum-weight edge e tht is not loop cn clerly be done in O(1) time. Also, contrcting e cn be done in O(1) time, since there re O(1) gs to updte: this uses the fct tht one vertex incident with e hs degree O(1). It remins to explin how to compute in O(1) mortized time vertex or fce with degree t most three. For this purpose, we mintin bucket B ( list) contining ll vertices nd fces of degree t most three (nd possibly other vertices nd fces, possibly some of them being destroyed in the course of the lgorithm fter they re put in the bucket). Initilly, put ll vertices nd fces in B. When contrcting or deleting n edge e, only 8

9 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs the degrees of the vertices nd fces incident with e cn chnge, so we put them in the bucket before contrcting or deleting e. Therefore in totl O(n) vertices nd fces re put into B. To nd vertex or fce of degree t most three in the current grph, pick n element of B, check in O(1) time whether it still belongs to the current grph nd, if so, whether it hs degree t most three. If it is not the cse, remove it from B nd proceed with the next element. Since O(n) elements in totl re put in B, lso O(n) elements re removed from B, so the totl time spent to nd vertices nd fces with degree t most three is O(n) Seprtors Let G be grph. Here we ssume tht G is weighted: every vertex gets non-negtive weight, nd ll the weights sum up to t most one. (As n importnt specil cse, one could choose 1/n for the weight of ech of the n vertices of G. However, we will need the generl cse t some point.) The weight of G is the sum of the weights of its vertices. A seprtor for G is set S of vertices such tht every connected component of G S hs weight t most 1/2. Seprtors of smll size, when they exist, re very useful, nd often llow for ecient divide-nd-conquer strtegies. The purpose of this section is to show how to compute eciently (optimlly) smll seprtors in plnr grphs. Let us rst focus on two specil cses. The rst one studies seprtors for trees nd is the mother of ll exmples for grph seprtors. Proposition Let T be weighted tree with n vertices nd edges. In O(n) time, one cn compute seprtor for T mde of single vertex. Proof. Root T t n rbitrry vertex r. Using trversl of the tree, one cn lbel ech vertex v of T with the sum of the weights of v nd ll its descendents. This llows to compute in O(d) time the weight of ech component of T v, for n rbitrry vertex v of degree d. Now, strting t n rbitrry vertex v of the tree: if ll components of T v hve weight t most 1/2, return v; otherwise, replce v with the neighbor w of v belonging to the lrgest connected component of T v. When the lgorithm returns, its result is correct. Furthermore, it moves from vertex to vertex long edges, but it never visits the sme vertex twice, becuse T is tree nd becuse, in the second cse, the component of T w contining v hs weight t most one minus the weight of the component of T v contining w, nd this is t most 1/2. The running time is thus proportionl to the sum of the degrees of the vertices, which is O(n). The second specil cse considers plnr grphs with smll rdius. Lemm Let G be weighted plnr grph with n vertices nd edges. Let U be spnning tree of G rooted t some vertex r, such tht ny vertex is t distnce t most d from r in U. Then seprtor for G of size t most 3d + 1 cn be computed in O(n) time. Proof. Without loss of generlity, we my ssume tht G is tringulted. Indeed, we cn without hrm itertively remove edges forming fces of degree one or two nd then tringulte every fce of degree t lest four. Dene weights on the vertices of the dul grph G by chrging the weight of ech vertex of G to the dul of exctly one incident fce of G. Now G qulies s weighted grph. T := G U is tree. By Proposition 1.12, we cn compute vertex c of T such tht ech connected component of T c hs weight t most 1/2. Let t be the tringle of G contining c. Let S be the subgrph of G tht is the union of the three edges of t together with the three shortest pths from the vertices of t to the root r. S hs t most 3d + 1 vertices. There remins to prove tht ech component C of G S hs weight t most 1/2. To see this, note tht S splits the plne into severl connected regions (Figure 1.6), nd C belongs entirely to single region. By the Jordn curve theorem, the vertices of G inside tht region belong to single component of T c nd therefore hve totl weight t most 1/2. Furthermore, when we ssigned weights to G, ll the weight of C ws chrged into tht region. In other words, C hs weight t most 1/2. 9

10 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs c Figure 1.6. The sitution in the proof of Lemm Finlly, here is the generl result on plnr grphs. Theorem Let G be plnr grph with n vertices nd edges. Then one cn compute in O(n) time seprtor for G of size O( n). One cn prove tht this result is optiml: Ech seprtor of the n n- grid hs size Ω( n). Proof of Theorem We cn obviously ssume tht G is connected. Let r be n rbitrry vertex of G. Compute bredth-rst serch tree T in G rooted t r. For every vertex v of G, dene its level to be its distnce to r. Let l 1 be weighted medin level of vertex in G: nmely, the totl weight of the vertices with level lower (resp., higher) thn l 1 is t most 1/2. Let l 0 be the lrgest level smller thn or equl to l 1 contining t most n vertices. Similrly, let l 2 be the smllest level lrger thn or equl to l 1 contining t most n vertices. (It my be tht these levels contin no vertices.) These computtions tke O(n) time. Here re the key properties of the levels l 0 nd l 2 : The levels strictly between l 0 nd l 2 contin t lest n+1 vertices, so there re less thn n such levels. Hence l 2 l 0 n. Let S be the set of vertices t levels l 0 nd l 2. Then S hs size t most 2 n. This is crucil since the output of the lgorithm will contin S. r T S Any connected component of G S with weight lrger thn 1/2 is conned between levels l 0 nd l 2. So let G be the subgrph of G induced by the vertices with levels strictly between l 0 nd l 2. To prove the theorem, it suces to build seprtor of G with O( n) vertices. The trick is to notice tht G is contined in plnr grph of rdius O( n), nd to pply Lemm 1.13, which concludes. In more detil, put n dditionl vertex r in the fce of G tht contins the vertices of G of level smller thn l 0. (These vertices induce connected subgrph of G, nd therefore belong to the sme fce of G.) Then we connect r to ll the vertices of level l 0. Alterntively, remove ll vertices of G with level t lest l 2 nd contrct to single vertex r ll edges of T whose incident vertices hve level less thn l 0. Let G be the resulting grph. It is plnr nd contins G. Furthermore, G hs spnning tree with depth O( n). (Tke the edges of the bredth- rst serch tree T of G tht pper in G, together with ll edges incident with r ; ll these edges form spnning tree rooted t r with depth t most l 2 l 0.) Becuse of this, we cn pply Lemm 1.13 to G, which concludes the proof Shortest pths Let G = (V, E) be grph where ech edge hs non-negtive length, nd let s be vertex of G. A shortest pth tree is spnning tree rooted t s tht contins shortest pth from s to ech vertex in G. Dijkstr's lgorithm (with the pproprite dt structure for the priority queue, for exmple Fiboncci heps) llows to compute shortest pth tree in O( E + V log V ) time. The following result improves the result for plnr grphs. Its proof is (omitted nd) extremely complicted; one mjor ingredient is vst generliztion of Theorem Theorem Given weighted grph embedding in S 2, shortest pth tree from given vertex cn be computed in time liner in the complexity of the grph. 10

11 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs Figure 1.7. Illustrtion of the sttement of Tutte's theorem. 1.4 Representtions of plnr grphs To conclude this chpter, we give nd prove two results of independent interest regrding how we cn build stright-line embeddings of grphs in the plne nd in the three-dimensionl spce Tutte's brycentric embedding theorem A grph is 3-connected if it is connected nd if it is still connected fter removing zero, one, or two vertices nd their incident edges. Theorem Let G = (V, E) be 3-connected grph without loops or multiple edges. Assume G is embedded on R 2. Let v 1,..., v k be the vertices of the outer fce. Assign unique positions f(v) in R 2 for ech vertex v, such tht the f(v i ), i = 1,..., k re mpped to the vertices of convex polygon (respecting the order of the vertices); the imge of every vertex v dierent from the v i 's is brycenter with strictly positive coecients of the imges of its neighbors in G. Then drwing stright-line edges between the imge points gives n embedding of G. Given such 3-connected grph, it is lwys possible to chieve the conditions of the theorem. For exmple, choose the brycentric coecients to be ll equl to one. The brycentric condition yields n ne system, which is solvble by n rgument of dominnt digonl. Equivlently, one my view the edges s springs with the sme rigidity, nd the interior vertices s being free to move. The equilibrium of this physicl system is met when the energy is minimized [62, p. 124]. We refer to the vertices v 1,..., v k s exterior vertices, nd to the other ones s interior vertices. Let v be n interior vertex of G. Let h : R 2 R be n ne function vnishing on f(v). If ll the neighbors of v lie on h 1 (0), we sy v is h-inctive. Otherwise, v is h-ctive. In this cse, v hs neighbors in both h 1 ((0, )) nd h 1 ((, 0)). In prticulr one cn nd rising pth from v to n exterior vertex: pth whose vlue of h strictly increses. Similrly one cn nd flling pth. Proposition The imge of every interior vertex of G is in the interior of the convex polygon f(v 1 )... f(v k ). Proof. Let h be n ne form such tht the polygon f(v 1 )... f(v k ) lies in h 1 ((0, )). If there is vertex (whose imge is) in h 1 ((, 0)), then consider the one tht hs minimum vlue of h. Since it is brycenter with positive coecients of its neighbors, ll its neighbors must hve the sme vlue of h. By induction nd connectivity of G, some exterior vertex must hve tht vlue of h, which is not possible. Therefore, ech interior vertex lies in the interior or on the boundry of the polygon f(v 1 )... f(v k ). Let v be n interior vertex; ssume v lies on n edge of the outer polygon, whose supporting line is h 1 (0). Then ll the neighbors of v re h-inctive. Thus, ll interior vertices tht cn be reched from v by pth using only interior vertices lie on h 1 (0). This contrdicts the 3-connectivity of G, becuse removing the two exterior vertices on h 1 (0) destroys the connectivity of G. Proposition For ny ne form h, there is no h-inctive vertex. See Figure 1.8 for n illustrtion of the next lemm. Lemm 1.19 (Ylemm). Let w 1, w 2, w 3 nd v be pirwise distinct vertices of grph H. Assume, for i = 1, 2, 3, tht there is pth P i from w i to v which voids the w j 's (for j i). Then there exist three pths P i, from w i to common vertex v, which re pirwise disjoint (except t v ). 11

12 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs w 1 w 2 w 3 Figure 1.8. The sitution in the Ylemm. v v v w 1 w 2 w 3 v Figure 1.9. A summry of the proof of Proposition Proof. First, using P 1 nd P 2, we esily get (simple) pth R from w 1 to w 2, so tht R nd P 1 hve the sme rst edge w 1 z. Then we consider the pth P 3. If this pth P 3 intersects R, let v be the rst vertex of intersection on P 3. v splits R in two prts, which we cll P 1 (from w 1 to v ) nd P 2 (from w 2 to v ); P 3 is the prt of P 3 going from w 3 to v, with loops removed (if ny). The P i 's stisfy the property stted in the lemm. If P 3 does not intersect R, we cll v the lst vertex on P 1 (when going from w 1 to v) which is lso on R. Such vertex exists nd is dierent from w 1 becuse w 1 z is the rst edge of R nd P 1. Let P 3 be the pth dened by P 3 followed by the prt of the pth P 1 which goes from v to v, with loops removed (if ny). v splits R in two prts, which we cll P 1 nd P 2. The pths P i 's stisfy the desired property. Proof of Proposition For the ske of contrdiction, ssume v is n h-inctive vertex. We prove the existence of subdivision of K 3,3 in G: subgrph of G such tht, fter contrcting edges, we get K 3,3. Thus G cnnot be plnr. See Figure 1.9. Let G(h) be the subgrph of G induced by the vertices on h 1 (0). Since G is 3-connected, there re, in G(h), three distinct h-ctive vertices w 1, w 2, nd w 3 nd three pths P i connecting v with w i, such tht, for ny i, the pth P i contins no vertex w j for j i. Indeed, let w be vertex of G so tht h(f(w)) 0. By connectivity of G, tke pth from v to w nd cll w 1 the rst h-ctive vertex on this pth. Do the sme in G {w 1 }, nd choose w 2, by 2-connectivity. Similrly, use 3-connectivity to select w 3 in G {w 1, w 2 }. Applying then the Ylemm in G(h), we get the existence of vertex v in G(h), together with three distinct pths (except t v ) P i from w i to v in G(h). We cn build rising pths Q i from ech of the w i to vertex x mximizing the vlue of h. Then, the Ylemm llows us to ssume, by chnging x nd the Q i 's if necessry, tht these three pths re disjoint (except t x). Similrly, we cn build flling pths R i from ech of the w i to vertex y nd pply the Ylemm. Using the pths P i, Q i nd R i, which re ll pirwise disjoint except t their endpoints, nd the vertices x, v, y nd w 1, w 2, w 3, we get subdivision of the grph K 3,3. This contrdicts the plnrity of G. By Proposition 1.18, the convex hull of the neighbors of n interior vertex v is non-degenerte polygon, nd v lies in its interior. We now tringulte the fces of G, except the outer fce: fces re dded to split the fces of G into tringles, without dding vertices (this is done in purely combintoril wy). It still holds tht v is in the interior of the convex hull of its neighbors, which redily implies tht v is brycenter with positive coecients of them. To summrize, the hypotheses of the theorem re stised, but we cn now even ssume tht every fce of G, except mybe the outer fce, is tringle. Proposition Let uvy nd uvz be two tringles of G shring the edge uv. Let h be n ne form vnishing on f(u) nd f(v). Then h(f(y))h(f(z)) < 0. 12

13 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs Proof of Proposition Lemm 1.21 shows tht, whenever one tringle is non-degenerte, then its incident tringles re non-degenerte. Necessrily, ny tringle hving one exterior edge is non-degenerte (Proposition 1.17). So every tringle is non-degenerte. The result follows. We cn now conclude the proof of Tutte's theorem. Figure Two cses for the proof of Lemm Vertices u nd v, together with flling pths, re highlighted. The proof relies on the following lemm. Lemm If h(f(y)) > 0, then h(f(z)) < 0. Proof. By ssumption u, v, nd y re h-ctive. Find strictly flling pths going from u nd v to n exterior vertex. The flling pths my shre vertex. In ny cse, we get simple circuit C in G using uv whose set of vertices re on the strictly negtive side of h, except u nd v. See Figure (We my need some exterior vertices if the flling pths do not shre vertex.) On the originl (not necessrily stright-line) embedding of G, the circuit C bounds disk. Let S be the set of vertices in the interior of this disk; S contins no exterior vertices, so every vertex in S is brycenter with positive coecients of its neighbors. Under f, ll the vertices of C re mpped to the hlf-spce h 0 except u nd v, which re mpped to h = 0; therefore, s in the proof of Proposition 1.17, ll the vertices in S belong to the open hlf-spce h < 0. Since h(f(y)) > 0, the vertex y cnnot belong to C or S. In the originl embedding of G, the circuit C uses edge uv, nd y is outside C. Therefore z must be inside C (i.e., in S) or on C. In the former cse, s seen bove, we get h(f(z)) < 0. In the ltter, since u nd v re the only vertices of C on the line h = 0, we lso get h(f(z)) < 0. Proof of Theorem As we discussed erlier, consequence of Proposition 1.18 is tht we my ssume tht G is tringulted (except possibly for the outer fce). Since the tringles re non-degenerte by Proposition 1.20, it suces to prove tht the interiors of two distinct tringles re disjoint. For the ske of contrdiction, let be point of R 2 in the interior of two tringles t nd t. Shoot ry from to the boundry of the polygon f(v 1 )... f(v k ) voiding the imge of every vertex. Whenever the ry leves t, by Proposition 1.20, it enters nother tringle. So we get sequence of tringles t = t 1, t 2,..., t where t is the unique tringle incident to the boundry edge tht is on the end of the ry. Similrly, we get sequence of tringles t = t 1, t 2,..., t. Going bck in both sequences from t, we pss from tringle to n unmbiguously dened preceding tringle. Since we strt with the sme tringle, we get t = t. In prticulr, we hve: Corollry 1.22 (FárySteinWgner's theorem). Every plnr grph cn be drwn in the plne with stright line edges. Furthermore, if the grph is 3-connected, then we cn choose the fces to be convex, since it is the cse in Tutte embedding. This ltter fct is in prticulr importnt in the next ppliction Steinitz' theorem Every convex polytope in R 3 hs set of vertices (extreml points) nd edges. This is clled the 1-skeleton of the polytope. In this section, we will prove the following theorem. 13

14 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs Theorem 1.23 (Steinitz' theorem (1922)). Let G = (V, E) be grph without loops or multiple edges. Then G is plnr 3-connected grph if nd only if it is the 1-skeleton of convex polytope in R 3. The if prt is the esiest direction: Imgine big physicl model of your convex polytope, where the fces nd the interior of the polytope re trnsprent. If you sit outside the polytope, close enough to the center of fce, you will see no crossing between the edges. In other words, the 1-skeleton is plnr. We omit the proof tht the 1-skeleton of 3- dimensionl polytope is 3-connected; this cn be proved directly without too much trouble, nd follows from more generl theorem by Blinski (see the proof for exmple in Ziegler's book [75, Sect. 3.5]). Let ω : E (0, ) be function from the (undirected) edges of G to set of strictly positive coecients. (In the sequel, we could tke ω to be constnt, equl to one.) Let f : V R 2 be the corresponding Tutte equilibrium given by Theorem 1.16, where every vertex v is brycenter with coecients ω uw1,..., ω uwm of its neighbors w 1,..., w m. We ctully ssume tht f : V R 3 mps the vertices into the plne z = 1 of R 3. To every interior fce f of G we ssocite vector q f in R 3. We choose n rbitrry interior fce f 0, for which q f0 = 0. The other q f 's re dened by the following formul: For every interior edge uv with left fce f 1 nd right fce f 2, we dene where denotes the cross-product in R 3. q f1 = ω uv (f(u) f(v)) + q f2 (1.1) Lemm The vectors q f re well-dened. Proof. First note tht exchnging u exchnges f 1 nd f 2 in (1.1), nd thus gives q f2 = ω vu (f(v) f(u)) + q f1, which rewrites q f2 = ω uv ( f(u) f(v)) + q f1 ; this is exctly Eqution (1.1). Let v be vertex of G, nd w 1,..., w m be its neighbors. We get: ( m m ) ω vwi (f(v) f(w i )) = f(v) ω vwi (f(w i ) f(v)) = 0. i=1 i=1 f 0 p p Figure Illustrtion of the proof of Lemm We wish to prove tht the denition of q f is the sme, whichever of the two pths p nd p in G we choose. This is done by choosing n intermedite pth p tht contins less vertices of G on both sides; since there re less vertices of G between p nd p or between p nd p thn between p nd p, we my ssume by induction on the number of enclosed vertices tht choosing p or p, nd similrly p or p, does not ect the computtion of q f. Therefore, Eqution (1.1) gives consistent vectors q f for ll fces round n interior vertex. Now, strting from the initil fce f 0, we my dene the vlue of q f by choosing n rbitrry sequence of fces from f 0. In other words, in the dul grph G, every pth strting t f0 nd ending t some fce f gives vlue of q f. (Incidentlly, this shows tht the function q, if it exists, is unique.) We need to check tht this vlue does not depend on the prticulr pth chosen. For this purpose, consider two such pths p nd p in G. We my ssume tht p nd p use distinct vertex sets of G except t the endpoints f0 nd f. Then by the Jordn curve theorem 1.3, p nd p enclose set of fces of G. The result is proven by induction on the number of fces of G enclosed by p nd p : The cse of one fce is the previous prgrph. For the induction step, build one pth p in G tht is in-between p nd p, nd pply induction. See Figure We dene piecewise liner function g from the union of the interior fces to R by setting, for every point x in fce f, g(x) = x q f. Lemm This mp g is well-dened. p f 14

15 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs Proof. We only need to prove tht, whenever x belongs to n edge uv incident with fces f 1 nd f 2, the vlue of g(x) is the sme, whichever fce f 1 or f 2 we choose for the computtion; in other words, x q f1 = x q f2. By linerity it suces to prove the result for x = f(u) nd x = f(v). f(v) q f1 = f(v) ω uv (f(u) f(v)) + q f2 = f(v) q f2. A similr computtion holds for f(v). Lemm Let uv be n edge with left fce f 1 nd right fce f 2. Assume x is in the fce f 1. Then x q f1 < x q f2. Proof. x q f2 x q f1 = ω uv x f(u) f(v) = ω uv det(x, f(u), f(v)) > 0, by our orienttion convention (recll tht the lst coordinte of the points x, f(u), nd f(v) is one) nd the fct tht ω uv > 0. Sketch of proof of Theorem Recll tht the position of vertex v is f(v) in R 3, ctully in the plne z = 1. We just move verticlly f(v) to height g(f(v)). Let F (v) be the new position. Let P be the convex hull of the F (v). Lemm 1.26 implies tht every interior edge uv is n edge of P, becuse every such edge is vlley; the sme clerly holds for the exterior edges. It is cler tht the lifts of ll vertices on given fce re coplnr, nd therefore ech fce of P is convex polygon. Therefore, P is convex polytope. There is one subtlety, however: the vertices of the outer fce re not necessrily coplnr; though, if the outer fce is incident with three vertices, this condition is utomticlly stised. If G contins tringle, we my hve tken tht tringle to be the outer fce in the ppliction of Tutte's theorem. Thus, the only cse tht remins to be shown is when G contins no tringle. From Exercise 1.11, we know tht G, the dul grph of G, contins tringle. Clerly, G is plnr, nd it cn be shown tht it is lso 3- connected. We my therefore relize G s the 1-skeleton of convex polytope. Now, known construction, polrity, llows to trnsform 3- polytope into nother one, whose 1-skelet re dul to ech other [75, Sect. 2.3]. So G is the 1-skeleton of convex polytope s well. 1.5 Notes Topology nd combintoril representtions For more informtion on bsic topology, see for exmple Armstrong [3] or Henle [40]; see lso Stillwell [67]. There re mny essentilly equivlent wys of representing plnr grph embeddings [26, 46]; the computtionl geometry librry CGAL implements one of them 3. We will see lter tht (most of) these dt structures generlize to grphs embedded on surfces. There re further generliztions to higher dimensions [7, 50, 51]; this is importnt especilly in geometric modelling. Eppstein provides mny proofs of Euler's formul More properties of plnr grphs Plnr grphs re n extensive subject, both from the combintoril nd the lgorithmic point of view. See for exmple Mohr nd Thomssen [57, Chpter 2] for survey nd references. Let us mention few very centrl fcts. Exercise 1.9 shows tht K 5 nd K 3,3 re not plnr. There is converse sttement: Kurtowski's theorem sserts tht grph G is plnr if nd only if it does not contin K 5 or K 3,3 s subdivision; in other words, if nd only if one cnnot obtin K 5 or K 3,3 from G by removing edges nd isolted vertices nd replcing every degree-two vertex nd its two incident edges with single edge [47,53,69]. Given grph G, Hopcroft nd Trjn [43] (see lso Thoms [68]) prove tht we cn decide whether it is plnr in liner time; if the nswer is positive, we cn nd n embedding in liner time. Therefore, lgorithmiclly, the confusion between plnr grphs nd grph embeddings in S 2 is not n issue. An esier lgorithm works in cubic time [22, Section 3.3]. The vertices of every plnr grph cn be colored using t most four colors such tht ny two djcent vertices hve dierent colors. This is the fmous four-color theorem [2]. There re lso versions for grphs on surfces [39]. 3 Chpter_min.html

16 ALGORITHMS FOR GRAPHS ON SURFACES 1. Plnr grphs Algorithms The minimum spnning tree lgorithm described bove is bsed on Mtsui [55] (see lso Cheriton nd Trjn [15] for more complicted, but more generl, lgorithm). Actully, the sme technique shows tht minimum spnning tree of grph cellulrly embedded on surfce of genus g cn be computed in O(gn) time. (See next chpter for more on surfces.) On rbitrry grphs, things re more complicted: there is rndomized lgorithm with liner time [45], nd deterministic lgorithm with lmost liner time (where lmost mens up to fctor involving the inverse Ackermnn function) [13]. The liner-time seprtor lgorithm is inspired by the originl pper by Lipton nd Trjn [52], but ws quite simplied (in prticulr the weighting scheme of the dul grph in the proof of Lemm 1.13). It might be tht the simplied proof is new, but it is, however, more likely tht such or similr simpliction ws found erlier, nd the uthor welcomes ny informtion on this subject. The interest of seprtors is tht they often llow for divide-nd-conquer strtegies, by cutting the problem into two subproblems of roughly hlf the size of the originl problem, computing solution in these subproblems, nd using them to compute the entire solution. Wht is ctully more powerful is seprtor decomposition of the grph, where seprtors of the subgrphs re recursively computed; such decomposition is lso computble in liner time [36]. There re lternte proofs for the existence of seprtors, see Alon et l. [1] for grphtheoretic pproch nd Miller et l. [56] for geometric pproch using circle pcking with pplictions. Among the numerous pplictions of seprtors, Henzinger et l. [41] proved tht single-source shortest pths cn be computed in liner time in plnr grph with non-negtive lengths. The treewidth (nd its cousin, brnchwidth) of grph is prmeter tht cptures, in some unprecise sense, the fct tht the grph cn be recursively split into subgrphs with smll overlp [63]. It is in prticulr useful in plnr grphs Figure An embedding tht cnnot be lifted to convex polytope. Indeed, ssume every interior edge is n edge on the bottom of the convex polytope. We cn suppose, by dding suitble ne form to ll the z i 's, tht z 4 = z 5 = z 6 = 0. Then z 1 > z 2 > z 3 > z 1, which is impossible. representtions: the vertices re mpped to non-overlpping disks in the plne, two of which re tngent if nd only if n edge between the corresponding vertices exists (see Mohr nd Thomssen [57, Chpter 2] for proof nd references). The proof of Tutte's theorem we described uses rguments from Edelsbrunner nd Hrer [24] other sources [19, 62]; the proof of Steinitz' theorem is lso tken from Richter-Gebert [62]. In ddition to the originl pper proving Tutte's theorem [72], there re mny other proofs [4,16, 19, 33, 37, 62,71]. The correspondence between Tutte embeddings where every vertex is brycenter of its neighbors nd the height function g is the Mxwell-Cremon correspondence (see for exmple Hopcroft nd Khn [44]). There re some stright-line grph embeddings tht cnnot be lifted to convex polytope (Figure 1.12) Stright-line drwings As proved in Theorem 1.16, every plnr grph without loops or multiple edges dmits stright-line embedding; this ws shown few decdes before Tutte's result [31, 66, 74]. Actully, if G is plnr grph without loops or multiple edges with n vertices, stright-line embedding exists where ll vertices lie in the (n 2) (n 2)-grid [32]. Mny other representtions exist, such s circle pcking 16

17 ALGORITHMS FOR GRAPHS ON SURFACES 2. Topology of surfces 11 Chpter Topology of surfces 2.1 Denition nd exmples A surfce is topologicl spce in which ech point hs neighborhood homeomorphic to the unit open disk { (x, y) R 2 x 2 + y 2 < 1 }. We only consider compct surfces in this chpter (nd even lter, unless speciclly noted). Exmples of surfces re the sphere, the torus, nd the double torus: these re compct, connected, orientble (to be dened lter) surfces with zero, one, nd two hndles, respectively (see Figure 2.1). The clssiction of surfces (Theorem 2.5) sserts tht two compct, connected, nd orientble surfces re homeomorphic if nd only if they hve the sme number of hndles. Despite the gures, note tht surfce is bstrct: the only knowledge we hve of it is the neighborhoods of ech point. A surfce is not necessrily embedded in R 3. Actully, the non-orientble surfces cnnot be Figure 2.2. A polygonl schem of grph embedded on sphere (the grph of the cube) is: 2 11 ā 1 ā 12, 3 7 ā 2 ā 8, 4 ā 5 ā 3 6, 1 ā 9 ā 4 10, 9 ā 11 ā 7 5, nd 12 ā 10 ā 6 8. embedded in R Surfce (de)construction Surfce deconstruction A grph embedded on surfce is cellulrly embedded if ll its fces re topologicl disks. As in the cse of the plne, we my consider the combintoril representtion of grph cellulrly embedded on surfce; the dt structures re identicl. The dul grph is dened similrly. The polygonl schem ssocited with cellulr grph embedding is de- ned s follows: ssign n rbitrry orienttion to ech edge; for ech fce, record the cyclic list of edges round the fce, with br if nd only if it ppers in reverse orienttion round the fce. See Figure Surfce construction Figure 2.1. A torus nd double-torus. Conversely, the dt of polygonl schem llows to build up surfce nd the cellulr grph embedding. More precisely, let S be nite set of symbols nd let S = { s s S}. Let R be nite set of reltions, ech reltion being non-empty word in the lphbet S S, so tht for every 17

18 ALGORITHMS FOR GRAPHS ON SURFACES 2. Topology of surfces v () (b) Figure 2.3. The corners incident to some vertex v cn be ordered cycliclly. s S, the totl number of occurrences of s plus the number of occurrences of s in R is exctly two. For ech reltion of size n, build n n-gon; lbel its edges by the elements of R, in order, the presence of br indicting the orienttion of the edge (see Figure 2.2). (Polygons with one or two sides re lso llowed.) Now, identify the twin edges of the polygons corresponding to the sme symbol in S, tking the orienttion into ccount. (As consequence, vertices get identied, too.) Lemm 2.1. The topologicl spce obtined by the bove process is compct surfce. Proof. Let X be the resulting topologicl spce; X is certinly compct. We hve to show tht every point of X hs neighborhood homeomorphic to the unit disk. The only non-obvious cse is tht of vertex v in X, tht is, point corresponding to vertex of some polygons. But it is not hrd to prove tht neighborhood of v is n umbrell: the corners (vertices) of the polygons corresponding to v cn be rrnged into cyclic order; see Figure 2.3. We dmit the following converse: Theorem 2.2 (Kerékjártó-Rdó; see Thomssen [70] or Doyle nd Morn [23]). Any compct surfce is homeomorphic to surfce obtined by the gluing process bove. This mounts to sying tht, on ny compct surfce, there exists cellulr embedding of grph. Equivlently, every surfce cn be tringulted. Figure 2.4. () The orienttions of these two fces (tringles) re comptible. (b) Two non-comptible orienttions of the fces. A surfce is orientble if there exist orienttions of ll fces tht re comptible. 2.3 Clssiction of surfces Euler chrcteristic nd orientbility chrcter Let G be grph cellulrly embedded on compct surfce S. The Euler chrcteristic of G equls v e + f, where v is the number of vertices, e is the number of edges, nd f is the number of fces of the grph. Proposition 2.3. The Euler chrcteristic is topologicl invrint: it only depends on the surfce S, not on the cellulr embedding. Sketch of proof. The Euler chrcteristic is esily seen to be invrint under Euler opertions. The result is then implied by the following clim: ny two cellulr embeddings on given surfce cn be trnsformed one into the other vi nite sequence of Euler opertions. Proving this is not very dicult but requires some work; key property is tht one cn ssume both embeddings to be piecewise liner with respect to given tringultion of the surfce. G is orientble if the boundry of its fces cn be oriented so tht ech edge gets two opposite orienttions by its incident fces (Figure 2.4). The orientbility chrcter is topologicl invrint s well; the sme proof s tht of Proposition 2.3 works, but it cn lso be proven directly: Exercise G is orientble if nd only if no subset of S is Möbius strip. 18