Maximizing Maximal Angles for Plane Straight-Line Graphs

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1 Mximizing Mximl Angles for Plne Stright-Line Grphs Oswin Aihholzer 1, Thoms Hkl 1, Mihel Hoffmnn 2, Clemens Huemer 3, Attil Pór 4, Frniso Sntos 5, Bettin Spekmnn 6, n Birgit Vogtenhuer 1 1 Institute for Softwre Tehnology, Grz University of Tehnology, [oih thkl vogt]@ist.tugrz.t 2 Institute for Theoretil Computer Siene, ETH Zürih, hoffmnn@inf.ethz.h 3 Deprtment e Mtemáti Apli II, Universitt Politéni e Ctluny, lemens.huemer@up.eu 4 Deprtment of Applie Mthemtis n Institute for Theoretil Computer Siene, Chrles University, por@km.mff.uni.z 5 Dept. e Mtemátis, Estísti y Computión, Universi e Cntri, frniso.sntos@unin.es 6 Deprtment of Mthemtis n Computer Siene, TU Einhoven, spekmn@win.tue.nl Astrt. Let G = (S, E) e plne stright-line grph on finite point set S R 2 in generl position. For point p S let the mximum inient ngle of p in G e the mximum ngle etween ny two eges of G tht pper onseutively in the irulr orer of the eges inient to p. A plne stright-line grph is lle ϕ-open if eh vertex hs n inient ngle of size t lest ϕ. In this pper we stuy the following type of uestion: Wht is the mximum ngle ϕ suh tht for ny finite set S R 2 of points in generl position we n fin grph from ertin lss of grphs on S tht is ϕ-open? In prtiulr, we onsier the lsses of tringultions, spnning trees, n pths on S n give tight ouns in most ses. 1 Introution Conitions on ngles in plne stright-line grphs hve een stuie extensively in isrete n omputtionl geometry. It is well known tht Deluny tringultions mximize the minimum ngle over ll tringultions, n tht in (Eulien) minimum weight spnning tree eh ngle is t lest π 3. In this pper we ress the funmentl omintoril uestion, wht is the mximum vlue O.A., T.H., n B.V. were supporte y the Austrin FWF Joint Reserh Projet Inustril Geometry S9205-N12. C.H. ws prtilly supporte y projets MEC MTM n Gen. Ct. 2005SGR A.P. ws prtilly supporte y Hungrin Ntionl Fountion Grnt T F.S. ws prtilly supporte y grnt MTM C02-02 of the Spnish Ministry of Eution n Siene. Preliminry results of this rtile hve een presente in [1].

2 α suh tht for eh finite point set in generl position there exists (ertin type of) plne stright-line grph where eh vertex hs n inient ngle of size t lest α. In other wors, we onsier min mx min mx prolems, where we minimize over ll finite point sets S in generl position in the plne, the mximum over ll plne stright-line grphs G (of the onsiere type), of the minimum over ll p S, of the mximum ngle inient to p in G. We present ouns on α for three lsses of grphs: spnning pths, (generl n oune egree) spnning trees, n tringultions. Most of the ouns we give re tight. In orer to show tht, we esrie fmilies of point sets for whih no grph from the respetive lss n hieve greter inient ngle t eh vertex. Bkgroun. Our motivtion for this reserh stems from the investigtion of pseuo-tringultions, stright-line frmework whih prt from eep omintoril properties hs pplitions in motion plnning, ollision etetion, ry shooting n visiility; see [3, 12, 13, 15, 16] n referenes therein. Pseuotringultions with minimum numer of pseuo-tringles (mong ll pseuotringultions for given point set) re lle minimum (or pointe) pseuotringultions. They n e hrterize s plne stright-line grphs where eh vertex hs n inient ngle greter thn π. Furthermore, the numer of eges in minimum pseuo-tringultion is mximl, in the sense tht the ition of ny ege proues n ege-rossing or negtes the ngle onition. In omprison to these properties, we onsier onnete plne stright-line grphs where eh vertex hs n inient ngle α to e mximize n the numer of eges is miniml (spnning trees) n the vertex egree is oune (spnning trees of oune egree n spnning pths). We further show tht ny plnr point set hs tringultion in whih eh vertex hs n inient ngle whih is t lest 2π 3. Oserve tht perfet mthings n e esrie s plne stright-line grphs where eh vertex hs n inient ngle of 2π n the numer of eges is mximl. Relte Work. There is vst literture on tringultions tht re optiml oring to ertin riteri, f. [2]. Similr to Deluny tringultions whih mximize the smllest ngle over ll tringultions for point set, frthest point Deluny tringultions minimize the smllest ngle over ll tringultions for onvex polygon [9]. If ll ngles in tringultion re π 6 then it ontins the reltive neighorhoo grph s sugrph [14]. The reltive neighorhoo grph for point set onnets ny pir of points whih re mutully losest to eh other (mong ll points from the set). Eelsrunner et l. [10] showe how to onstrut tringultion tht minimizes the mximum ngle mong ll tringultions for set of n points in O(n 2 log n) time. In pplitions where smll ngles hve to e voie y ll mens, Deluny tringultion my not e suffiient in spite of its optimlity euse even there ritrrily smll ngles n our. By ing so-lle Steiner points one n onstrut tringultion on superset of the originl points in whih there is some solute lower oun on the size of the smllest ngle [7]. Di et l. [8] esrie severl heuristis to onstrut minimum weight tringultions (tringu-

3 ltions whih minimize the totl sum of ege lengths) sujet to solute lower or upper ouns on the ourring ngles. Spnning yles with ngle onstrints n e regre s vrition of the trveling slesmn prolem. Fekete n Woeginger [11] showe tht if the yle my ross itself then ny set of t lest five points mits lolly onvex tour, tht is, tour in whih the ngle etween ny three onseutive points is positive. Arkin et l. [5] onsier s mesure for (non-)onvexity of point set S the minimum numer of (interior) reflex ngles (ngles > π) mong ll plne spnning yles for S. Aggrwl et l. [4] prove tht fining spnning yle for point set whih hs miniml totl ngle ost is NP-hr, where the ngle ost is efine s the sum of iretion hnges t the points. Regring spnning pths, it hs een onjeture tht eh plnr point set mits spnning pth with minimum ngle t lest π 6 [11]; reently, lower oun of π 9 hs een presente [6]. Definitions n Nottion. Let S R 2 e finite set of points in generl position, tht is, no three points of S re olliner. In this pper we onsier plne stright-line grphs G = (S, E) on S. The verties of G re preisely the points in S, the eges of G re stright-line segments tht onnet two points in S, n two eges of G o not interset exept possily t their enpoints. For point p S the mximum inient ngle op G (p) of p in G is the mximum ngle etween ny two eges of G tht pper onseutively β α γ p δ Fig. 1. The inient ngles of p. in the irulr orer of the eges inient to p. For point p S of egree t most one we set op G (p) = 2π. We lso refer to op G (p) s the openness of p in G n ll p S ϕ-open in G for some ngle ϕ if op G (p) ϕ. Consier for exmple the grph epite in Fig. 1. The point p hs four inient eges of G n, therefore, four inient ngles. Its openness is op G (p) = α. The point hs only one inient ngle n orresponingly op G () = 2π. Similrly we efine the openness of plne stright-line grph G = (S, E) s op(g) = min p S op G (p) n ll G ϕ-open for some ngle ϕ if op(g) ϕ. In other wors, grph is ϕ-open if n only if every vertex hs n inient ngle of size t lest ϕ. The openness of lss G of grphs is the supremum over ll ngles ϕ suh tht for every finite point set S R 2 in generl position there exists ϕ-open onnete plne stright-line grph G on S n G is n emeing of some grph from G. For exmple, the openness of minimum pseuo-tringultions is π. Oserve tht without the generl position ssumption mny of the uestions eome trivil euse for set of olliner points the non-rossing spnning tree is uniue the pth tht onnets them long the line n its interior points hve no inient ngle greter thn π. The onvex hull of point set S is enote with CH(S). Points of S on CH(S) re lle verties of CH(S). Let,, n e three points in the plne tht re not olliner. With we enote the ounterlokwise ngle etween the segment (, ) n the segment (, ) t.

4 Tringultions Trees Trees with mxeg. 3 Pths (onvex sets) Pths (generl) 2π 3 5π 3 3π 2 Tle 1. Openness of severl lsses of plne stright-line grphs. All given vlues exept for pths on point sets in generl position re tight. Results. In this pper we stuy the openness of severl well-known lsses of plne stright-line grphs, suh s tringultions (Setion 2), (generl n oune egree) trees (Setion 3), n pths (Setion 4). The results re summrize in Tle 1 ove. 2 Tringultions Theorem 1. Every finite point set in generl position in the plne hs tringultion tht is 2π 3 -open n this is the est possile oun. Proof. Consier point set S R 2 in generl position. Clerly, op G (p) > π for every point p CH(S) n every plne stright-line grph G on S. We reursively onstrut 2π 3 -open tringultion T of S y first tringulting CH(S); every reursive suprolem onsists of point set with tringulr onvex hull. Let S e point set with tringulr onvex hull n enote the three points of CH(S) with,, n. If S hs no interior points, then we re one. Otherwise, let, n e (not neessrily istint) interior points of S suh tht the tringles, n re empty (see Fig. 2). Sine the sum of the six exterior ngles of the hexgon euls 8π, the sum of the three ngels,, n is t lest 2π. In prtiulr, one of them, sy, is t lest 2π/3. We then reurse on the two susets of S tht hve n s their respetive onvex hulls. The upper oun is ttine y set S of n points s epite in Fig. 3. S onsists of point p n of three sets S, S, n S tht eh ontin n 1 3 points. S, S, n S re ple t the verties of n euilterl tringle n p is ple t the ryenter of. Any tringultion T of S must onnet p with t lest one point of eh of S, S, n S n hene op T (p) pprohes ritrrily lose. 2π 3 3π 2 S 5π 4 p S S Fig. 2. Construting 2π -open tringultion. 3 Fig. 3. The openness of tringultions of this point set pprohes 2π 3.

5 3 Spnning Trees In this setion we give tight ouns on the ϕ-openness of two si types of spnning trees, nmely generl spnning trees n spnning trees with oune vertex egree. Consier point set S R 2 in generl position n let p n e two ritrry points of S. Assume w.l.o.g. tht p hs smller x-oorinte thn. Let l p n l enote the lines through p n tht re perpeniulr to the ege (p, ). We efine the orthogonl sl of (p, ) to e the open region oune y l p n l. Oservtion 1 Assume tht r S \ {p, } lies in the orthogonl sl of (p, ) n ove (p, ). Then pr π 2 n rp π 2. A symmetri oservtion hols if r lies elow (p, ). Rell tht the imeter of point set is the istne etween pir of points tht re furthest wy from eh other. Let n efine the imeter of S n ssume w.l.o.g. tht hs smller x-oorinte thn. Clerly, ll points in S \ {, } lie in the orthogonl sl of (, ). Oservtion 2 Assume tht r S \ {, } lies ove imetril segment (, ) for S. Then r π 3 n hene t lest one of the ngles r n r is t most π 3. A symmetri oservtion hols if r lies elow (, ). 3.1 Generl Spnning Trees Theorem 2. Every finite point set in generl position in the plne hs spnning tree tht is 5π 3 -open n this is the est possile oun. The upper oun is ttine y the point set epite in Fig. 4. Eh of the sets S i, i 1, 2, 3 onsists of n 3 points. If point p S 1 is onnete to ny other point from S 1 S 2, then it n only e onnete to point of S 3 forming n ngle of t lest π 3 ε. As the sme rgument hols for S 2 n S 3, respetively, ny onnete grph, n thus ny spnning tree on S is t most 5π 3 -open. The proof for the lower oun is eferre to the ppenix. S 3 S 1 S 2 Fig. 4. Every spnning tree of S is t most 5π 3 -open. 3.2 Spnning Trees of Boune Vertex Degree Theorem 3. Let S R 2 e set of n points in generl position. There exists 3π 2 -open spnning tree T of S suh tht every point from S hs vertex egree t most three in T. The ngle oun is est possile, even for the muh roer lss of spnning trees of vertex egree t most n 2.

6 S S + S S S Fig. 5. Construting 3π -open spnning tree with mximum vertex egree four. 2 Proof. We show in ft tht S hs 3π 2 -open spnning tree with mximum vertex egree three. To o so, we first esrie reursive onstrution tht results in 3π 2 -open spnning tree with mximum vertex egree four. We then refine our onstrution to yiel spnning tree of mximum vertex egree three. Let n efine the imeter of S. W.l.o.g. hs smller x-oorinte thn. The ege (, ) prtitions S \ {, } into two (possily empty) susets: the set S of the points ove (, ) n the set S of the points elow (, ). We ssign S to n S to (see Fig 5). Sine ll points of S \ {, } lie in the orthogonl sl of (, ) we n onnet ny point p S to n ny point of S to n y this otin 3π 2 -open pth P = p,,,. Bse on this oservtion we reursively onstrut spnning tree of vertex egree t most four. If S is empty, then we proee with S. If S ontins only one point p then we onnet p to. Otherwise onsier imetril segment (, ) for S. W.l.o.g. hs smller x-oorinte thn n lies ove (, ). Either or must e less thn π 2. W.l.o.g. ssume tht < π 2. Hene we n onnet vi to n otin 3π 2 -open pth P =,,,. The ege (, ) prtitions S into two (possily empty) susets: the set S of the points ove (, ) n the set S of the points elow (, ). The set S is gin prtitione y the ege (, ) into set S + of points tht lie ove (, ) n set S of points tht lie elow (, ). We ssign S to n oth S + n S to n proee reursively. The lgorithm mintins the following two invrints: (i) t most two sets re ssigne to ny point of S, n (ii) if set S p is ssigne to point p then p n e onnete to ny point of S p n op T (p) 3π 2 for ny resulting tree T. We now refine our onstrution to otin 3π 2 -open spnning tree of mximum vertex egree three. If S + is empty then we ssign S to, n vie p S p S + Fig. 6. Construting 3π -open spnning tree with mximum vertex egree three. 2

7 p S 1 Fig. 7. Every spnning tree of this point set with vertex egree t most n 2 is t most 3π 2 -open. Fig. 8. A zigzg pth. vers. Otherwise, onsier the tngents from to S n enote the points of tngeny with p n (see Fig. 6). Let l p n l enote the lines through p n tht re perpeniulr to (, ). W.l.o.g. l is loser to thn l p. We reple the ege (, ) y the three eges (, p), (p, ), n (, ). The resulting pth is 3π 2 -open n prtitions S into three sets whih n e ssigne to p,, n while mintining invrint (ii). The refine reursive onstrution ssigns t most one set to every point of S n hene onstruts 3π 2 -open spnning tree with mximum vertex egree three. The upper oun is ttine y the set S of n points epite in Fig. 7. S onsists of n 1 ner-olliner points lose together n one point p fr wy. In orer to onstrut ny onnete grph with mximum egree n 2, one point of S 1 hs to e onnete to nother point of S 1 n to p. Thus ny spnning tree on S with mximum egree t most n 2 is t most 3π 2 -open. 4 Spnning Pths Spnning pths n e regre s spnning trees with mximum vertex egree two. Therefore, the upper oun onstrution from Fig. 7 pplies to pths s well. We will show elow tht the resulting oun of 3π 2 is tight for points in onvex position, even in very strong sense: There exists 3π 2 -open spnning pth strting from ny point. 4.1 Point Sets in Convex Position Consier set S R 2 of n points in onvex position. We n onstrut spnning pth for S y strting t n ritrry point p S n reursively tking one of the tngents from p to CH(S \ {p}). As long s S > 2, there re two tngents from p to CH(S \ {p}): the left tngent is the oriente line t l through p n point p l S \ {p} (oriente in iretion from p to p l ) suh tht no point from S is to the left of t l. Similrly, the right tngent is the oriente line t r through p n point p r S \ {p} (oriente in iretion from p to p r ) suh tht no point from S is to the right of t r. If we tke the left n the right tngent lterntely, see Fig. 8, we ll the resulting pth zigzg pth for S.

8 Theorem 4. Every finite point set in onvex position in the plne mits spnning pth tht is 3π 2 -open n this is the est possile oun. Proof. As zigzg pth is ompletely etermine y one of its enpoints n the iretion of the inient ege, there re extly n zigzg pths for S. (Count irete zigzg pths: There re n hoies for the strtpoint n two possile iretions to ontinue in eh se, tht is, 2n irete zigzg pths n, therefore, n (unirete) zigzg pths.) Now onsier point p S n sort ll other points of S rilly roun p, strting with one of the neighors of p long CH(S). Any ngle tht ours t p in some zigzg pth for S is spnne y two points tht re onseutive in this ril orer. Moreover, ny suh ngle ours in extly one zigzg pth euse it etermines the zigzg pth ompletely. Sine the sum of ll these ngles t p is less thn π, for eh point p t most one ngle n e π 2. Furthermore, if p is n enpoint of imetril segment for S then ll ngles t p re < π 2. Sine there is t lest one imetril segment for S, there re t most n 2 ngles > π 2 in ll zigzg pths together. Thus, there exist t lest two spnning zigzg pths tht hve no ngle > π 2, tht is, they re 3π 2 -open. To see tht the oun of 3π 2 is tight, onsier gin the point set shown in Fig. 7. A onstrutive proof for Theorem 4 is eferre to the ppenix. There we lso proof the following stronger sttement. Corollry 1 For ny finite set S R 2 of points in onvex position n ny p S there exists 3π 2 -open spnning pth for S whih hs p s n enpoint. 4.2 Generl Point Sets The min result of this setion is the following theorem out spnning pths of generl point sets. Theorem 5. Every finite point set in generl position in the plne hs 5π 4 -open spnning pth. Let S R 2 e set of n points in generl position. For suitle leling of the points of S we enote spnning pth for ( suset of k points of) S with p 1,..., p k, where we ll p 1 the strting point of the pth. Then Theorem 5 iretly follows from the following, stronger result. Theorem 6. Let S e finite point set in generl position in the plne. Then (1) For every vertex of the onvex hull of S, there exists 5π 4 -open spnning pth, p 1,..., p k on S strting t. (2) For every ege 1 2 of the onvex hull of S there exists 5π 4 -open spnning pth strting t either 1 or 2 n using the ege 1 2, tht is, spnning pth 1, 2, p 1,..., p k or 2, 1, p 1,..., p k.

9 Proof. For eh vertex p in pth G the mximum inient ngle op G (p) is the lrger of the two inient ngles (exept for strt- n enpoint of the pth). To simplify the se nlysis we will onsier the smller ngle t eh point n prove tht we n onstrut spnning pth suh tht it is t most 3π 4. We enote with (, S) spnning pth for S strting t, n with ( 1 2, S) spnning pth for S strting with the ege onneting 1 n 2. The outer norml one of vertex y of onvex polygon is the region etween two hlflines tht strt t y, re respetively perpeniulr to the two eges inient t y, n re oth in the exterior of the polygon. We prove the sttements (1) n (2) of Theorem 6 y inution on S. The se ses S = 3 re oviously true. Inution for (1): Let K = CH(S \ {}). Cse 1.1 lies etween the outer norml ones of two onseutive verties y n z of K, where z lies to the right of the ry y. Inution on (yz, S\{}) results in 5π 4 -open spnning pth y, z, p 1,..., p k or z, y, p 1,..., p k of S \ {}. Oviously yz π 2 < 3π 4 n yz π 2 < 3π 4, n thus we get 5π 4 -open spnning pth, y, z, p 1,..., p k or, z, y, p 1,..., p k for S (see Fig. 9). Cse 1.2 lies in the outer norml one of vertex of K. Let p e tht vertex n let y n z e the two verties of K jent to p, z eing to the right of the ry py. The three ngles pz, zpy n yp roun p oviously up to 2π. We onsier suses oring to whih of the three ngles is the smllest, the ses of pz n yp eing symmetri (see Fig. 10). Cse zpy is the smllest of the three ngles. Then, in prtiulr, zpy < 3π 4. Assume without loss of generlity tht pz is smller thn yp n, in prtiulr, tht it is smller thn π. Sine is in the norml one of p, pz is t lest π 2, hene pz is t most π 2 < 3π 4. Let S = S \ {, z} n onsier the pth tht strts with n z followe y (p, S ), tht is, z, p, p 1,..., p k. Note tht zpp 1 zpy. Cse yp is the smllest of the three ngles. Then yp < 3π 4. Moreover, in this se ll three ngles pz, yp n zpy re t lest π 2, the first two euse lies in the norml one of p, the ltter euse it is is not the smllest of the three ngles. We hve yp < π 2 euse this ngle lies in the tringle ontining yp π 2, n yp < 3π 4 y ssumption. We iterte on (py, S \ {}) n get 5π 4 -open spnning pth y z Fig. 9. Cse 1.1 y p z Fig. 10. Cse α T ω l 1 l 2 Fig. 11. Cse 2

10 1 2 y p z l 1 l 2 Fig. 12. Cse α 1 α 2 α β 2 β ω γ δ γ 1 γ 2 y p ɛ η Fig. 13. Cse [1,2] z 1 y 2 z l 1 l 2 Fig. 14. Cse on S \ {} y inution, whih n e extene to 5π 4 -open spnning pth on S,, p, y, p 1,..., p k or, y, p, p 1,..., p k, respetively. Inution for (2): Let n e the neighoring verties of 1 n 2 on CH(S), suh tht CH(S) res...,, 2, 1,,... in lokwise orer (see Fig. 11). Cse 2.1 α < 3π 4 or ω < 3π 4 (see Fig. 11). Without loss of generlity ssume tht α < 3π 4. By inution on ( 1, S \{ 2 }) we get 5π 4 -open spnning pth 1, p 1,..., p k on S \ { 2 }. As 2 1 p 1 α < 3π 4 we get 5π 4 -open spnning pth 2, 1, p 1,..., p k on S. Cse 2.2 Both α n ω re t lest 3π 4. Let l 1 n l 2 e the lines through 1 n 2, respetively, n orthogonl to 1 2. Further let K = CH(S \ { 1, 2 }) n with T we enote the region oune y 1 2, l 1, l 2 n K (see Fig. 11). Cse At lest one vertex p of K exists in T. If there exist severl verties of K in T, then we hoose p s the one with smllest istne to 1 2. Oviously the eges 1 p n 2 p interset K only in p n the ngles α 1 n β re eh t most π 2 (see Fig. 12). Cse γ 2 > π 2 (see Fig.13). By inution on (p, S\{ 1, 2 }) we get 5π 4 -open spnning pth p, p 1,..., p k for S \ { 1, 2 }. Moreover the smller of 2 pp 1 n p 1 p 1 is t most 2π π 2 2 = 3π 4. Thus we get 5π 4 -open spnning pth 1, 2, p, p 1,..., p k or 2, 1, p, p 1,..., p k for S. Cse γ 2 π 2 (see Fig.13). Let y n z e verties of K, with y eing the lok-wise neighor of p n z eing the ounterlokwise one ( might eul y n might eul z). At lest one of α 1 or β is π 4. Without loss of generlity ssume tht β π 4, the other se is symmetri. Then 1, 2, p, y form onvex four-gon euse α 3π 4 n β π 4 imply tht p 2 in the four-gon, 1, 2, p is less thn π. Therefore lso γ p 2 < π. We will show tht ll four ngles α 1, γ 1, β 2 n δ re t most 3π 4. Then we pply inution on (py, S \ { 1, 2 }) n get 5π 4 -open spnning pth on S \ { 1, 2 }, whih n e omplete to 5π 4 -open spnning pth for S, 2, 1, p, y, p 1,..., p k or 1, 2, y, p, p 1,..., p k, respetively. Both α 1 n β 2 < β re lerly smller thn π 2, hene smller thn 3π 4.

11 For γ 1, oserve tht the supporting line of yp must ross the segment 1, so tht we hve α 2 + γ 1 < π (they re two ngles of tringle). Also, α 2 = α α 1 3π 4 π 2 = π 4, so γ 1 < 3π 4. Finlly, for δ we look t the ngles roun vertex p. By the sme rguments use in γ 1, we onlue ε < 3π 4. Sine η < π s the three verties z, p n y re on K, we hve γ > 2π π 3π 4 = π 4. Consiering the ngles insie the tringle 2 yp we get δ < π γ < π π 4 = 3π 4. Cse No vertex of K exists in T. Both, l 1 n l 2, interset the sme ege yz of K (in T ), with y loser to l 1 thn to l 2 (see fig. 14). We will show tht the four ngles yz 1, 2 1 z, y 2 1 n 2 yz re ll smller thn 3π 4. Then inution on (yz, S\{ 1, 2 }) yiels pth tht n e extene to 5π 4 -open pth 2, 1, z, y, p 1,..., p k or 1, 2, y, z, p 1,..., p k. Clerly, the ngles 2 1 z n y 2 1 re oth smller thn π 2. The sum of 2yz + 2 y is smller thn π euse the supporting line of yz intersets the segment 2. Now, 2 y is t lest π 4 y the ssumption tht 2 1 3π 4. So, 2yz < 3π 4. The symmetri rgument shows tht yz 1 < 3π 4. Note tht for Theorem 6 it is essentil tht the preefine strting point of 5π 4 -open pth is n extreme point of S, s n euivlent result is in generl not true for interior points. As ounter exmple onsier regulr n-gon with n itionl point in its enter. It is esy to see tht for suffiiently lrge n strting t the entrl point uses pth to e t most π + ε-open for smll onstnt ε. Similr, non-symmetri exmples lrey exist for n 6 points, n nlogously, if we reuire n interior ege to e prt of the pth, there exist exmples ouning the openness y 4π 3 + ε [17]. Despite these exmples we onlue this setion with the following onjeture. Conjeture 1 Every finite point set in generl position in the plne hs 3π 2 - open spnning pth. Aknowlegments. Reserh on this topi ws initite t the thir Europen Pseuo-Tringultion working week in Berlin, orgnize y Günter Rote n Anré Shulz. We thnk Srh Kppes, Hnnes Krsser, Dvi Oren, Günter Rote, Anré Shulz, Ilen Streinu, n Louis Thern for mny vlule isussions. We lso thnk Sonj Čukić n Günter Rote for helpful omments on the mnusript. Referenes 1. O. Aihholzer, T. Hkl, M. Hoffmnn, C. Huemer, F. Sntos, B. Spekmnn, B. Vogtenhuer. Mximizing Mximl Angles for Plne Stright Line Grphs - Extene Astrt. in: Astrts 23r Europen Workshop Comput. Geom., 2007, F. Aurenhmmer n Y.-F. Xu. Optiml Tringultions. Kluwer Aemi Pulishing, Enylopei of Optimiztion 4, , 2000.

12 3. O. Aihholzer, F. Aurenhmmer, H. Krsser, n P. Brss. Pseuo-Tringultions from Surfes n Novel Type of Ege Flip. SIAM J. Comput. 32, 6 (2003), A. Aggrwl, D. Coppersmith, S. Khnn, R. Motwni, n B. Shieer. The Angulr-Metri Trveling Slesmn Prolem. SIAM J. Comput. 29, 3 (1999), E. Arkin, S. Fekete, F. Hurto, J. Mithell, M. Noy, V. Sristán, n S. Sethi. On the Reflexivity of Point Sets. Disrete n Computtionl Geometry: The Goomn-Pollk Festshrift. Springer-Verlg, Series Algorithms n Comintoris 25, , I. Bárány, A. Pór, n P. Vltr. Pths with no Smll Angles. Mnusript in preprtion, M. Bern, D. Eppstein, n J. Gilert. Provly Goo Mesh Genertion. J. Comput. Syst. Si. 48, 3 (1994), Y. Di, N. Ktoh, n S.-W. Cheng. LMT-Skeleton Heuristis for Severl New Clsses of Optiml Tringultions. Comput. Geom. Theory Appl. 17, 1 2 (2000), D. Eppstein. The Frthest Point Deluny Tringultion Minimizes Angles. Comput. Geom. Theory Appl. 1, 3 (1992), H. Eelsrunner, T. S. Tn, n R. Wupotitsh. An O(n 2 log n) Time Algorithm for the Minmx Angle Tringultion. SIAM J. Si. Stt. Comput. 13, 4 (1992), S. P. Fekete n G. J. Woeginger. Angle-Restrite Tours in the Plne. Comput. Geom. Theory Appl. 8, 4 (1997), R. Hs, D. Oren, G. Rote, F. Sntos, B. Servtius, H. Servtius, D. Souvine, I. Streinu, n W. Whiteley. Plnr Minimlly Rigi Grphs n Pseuo- Tringultions. Comput. Geom. Theory Appl. 31, 1 2 (2005), D. Kirkptrik, J. Snoeyink, n B. Spekmnn. Kineti Collision Detetion for Simple Polygons. Internt. J. Comput. Geom. Appl. 12, 1 2 (2002), J. M. Keil n T. S. Vssilev. The Reltive Neighourhoo Grph is Prt of Every 30eg-Tringultion. in: Astrts 21st Europen Workshop Comput. Geom., 2005, G. Rote, F. Sntos, n I. Streinu. Pseuo-Tringultions Survey. Mnusript, I. Streinu. Pseuo-Tringultions, Rigiity n Motion Plnning. Disrete Comput. Geom. 34, 4 (2005), B. Vogtenhuer. On Plne Stright Line Grphs. Mster Thesis, Grz University of Tehnology, Grz, Austri, Jnury 2007.

13 5 Appenix 5.1 Proof of Theorem 2 We sy tht n ngle ϕ is lrge if ϕ > π 3. Corresponingly, if ϕ π 3 tht ϕ is smll. then we sy Theorem 2. Every finite point set in generl position in the plne hs spnning tree tht is 5π 3 -open n this is the est possile oun. Proof. Consier point set S R 2 in generl position n let n efine the imeter of S. W.l.o.g. hs smller x-oorinte thn. Let S \{, } e the point ove (, ) tht is furthest wy from (, ) n let S \ {, } e the point elow (, ) tht is furthest wy from (, ). (The speil se tht (, ) is n ege of the onvex hull of S n hene either or oes not exist is hnle t the en of the proof.) All points of S lie within the ouning ox efine y the orthogonl sl of (, ) n two lines through n prllel to (, ). To onstrut 5π 3 -open spnning tree, we first onstrut speil 5π 3 -open pth P whose enpoints re either n or n. P hs the itionl property tht the smller ngle t its enpoints etween the pth n the ouning ox is lso smll. We exten P to spnning tree in the following mnner. Every point p i of P hs smll inient ngle. Consier the one C i with pex p i efine y the eges of P (n the ouning ox if p i is n enpoint) enlosing the smll ngle t p i. When onstruting P we ensure tht every point p of S \ P is ontine in extly one one C i. We ssemle the spnning tree y onneting eh point in S \ P to the pex of its ontining wege (see Fig. 15). It remins to show tht we n lwys fin pth P with the properties esrie ove. We will prove this through se istintion on the size of the ngles tht re epite in Fig. 16. Sine (, ) is imetril for S, Oservtion 2 implies tht γ π 3 n δ π 3. Furthermore, t lest one of α 1 n β 1 n one of α 2 n β 2 is smll. Fig. 15. The pth P (thik eges), the ones of n (left), the spnning tree onstrute from P (right).

14 α 1 α 1 α 2 α α 2 δ δ 1 δ 2 γ 1 γ 2 γ β 1 β β 1 β 2 β 2 Fig. 16. The ouning ox of S with ll relevnt ngles lele. Cse 1 Neither t nor t oth ngles (α 1 n α 2 or β 1 n β 2, respetively) re lrge. This mens tht either α 1 n β 2 or α 2 n β 1 re smll. If α 1 n β 2 re smll, then we hoose P =,,,. P is 5π 3 -open n the smller ngles t n etween P n the ouning ox re t most π 3. Furthermore, P prtitions S \ {,,, } into four susets n eh suset is ontine in extly one of the four ones with pex,,, n. Symmetrilly, if α 2 n β 1 re smll, then P =,,,. Cse 2 Either t or t oth ngles re lrge. W.l.o.g. ssume tht α 1 n α 2 re lrge n hene β 1 n β 2 re oth smll. Cse 2.1 β = β 1 + β 2 is smll. We hoose P =,, (see Fig. 17). P is 5π 3 -open n the smller ngles t n etween P n the ouning ox re t most π 3. Furthermore, P prtitions S \ {,, } into three susets n eh suset is ontine in extly one of the three ones with pex,, n. Cse 2.2 β = β 1 + β 2 is lrge. Sine α 1 n α 2 re oth lrge it follows tht oth γ 1 n δ 1 re smll (s even their sum is smll). Aitionlly, oth α 1 = π 2 α 1 n α 2 = π 2 α 2 re smll. Furthermore, sine β = β 1 + β 2 is lrge it follows tht t lest one of γ 2 n δ 2 n t lest one of β 1 = π 2 β 1 n β 2 = π 2 β 2 is smll. Cse Either oth β 1 n γ 2 re smll or oth β 2 n δ 2 re smll. If oth β 1 n γ 2 re smll then we hoose P =,,, (see Fig. 18). P is 5π 3 -open n prtitions S \ {,,, } into four susets whih eh re ontine in extly one of the four ones with pex,,, n. Symmetrilly, if oth β 2 n δ 2 re smll, then P =,,,. Cse Either γ 2 is smll n β 1 is lrge or δ 2 is smll n β 2 is lrge. β 1 β Fig. 17. Cse 2.1 α 2 δ 1 γ 2 Fig. 18. Cse β 1 ω β 2 β 1 ν e ρ ε ϕ β 1 β 2 Fig. 19. Cse

15 Cse If γ 2 is smll n β 1 is lrge, onsier the suset S of S tht onsists of the points ove (, ). If the ngle r is smll for ll points r S then we n still use the onstrution from Cse If δ 2 is smll n β 2 is lrge, onsier the suset S of S tht onsists of the points elow (, ). If the ngle r is smll for ll points r S then we n gin use the onstrution from Cse Cse γ 2 is smll, β 1 is lrge, n there is t lest one point p S suh tht the ngle p is lrge. Let e S e the point suh tht ϕ = e is lrgest mong the points in S. We hoose P =, e,,, (see Fig. 19). The ngle ν is smll sine it is smller thn β 1, n β 1 is smll. Furthermore, ϕ is lrge y efinition of e n Oservtion 2 implies tht e = ε is t lest π 3. Summing the ngles within e yiels ϱ+β 1 +ϕ+ε = π n therefore ϱ+β 1 is smll. Similrly, the ngle sum within e is ω + β 1 + ϕ + ε = π n therefore ω + β 1 is smll. In summry, ll of β 2, ω, ϱ, n ν re smll n hene P is 5π 3 -open. Sine the gry-she region in Fig. 19 oes not ontin ny points of S y onstrution, P prtitions S \ {,,,, e} into five susets n eh suset is ontine in extly one of the five ones with pex,,,, n e. If δ 2 is smll, β 2 is lrge, n there is t lest one point r S suh tht the ngle r is lrge, then P n e onstrute similrly. Finlly, if (, ) is n ege of the onvex hull then either or oes not exist. If oes not exist then we n hoose either P =,, or P =,,. A symmetri rgument hols if oes not exist. 5.2 Construtive Proof for Theorem 4 n Corollry 1 For two istint points p, r R 2 enote y H (p, r) the set of points on or to the right of the ry pr, tht is, those t R 2 for whih prt π. Corresponingly, enote y H + (p, r) the set of points on or to the left of the ry pr, tht is, those t R 2 for whih prt π. Let S + (p, r) := S H + (p, r) n S (p, r) := S H (p, r). Consier irete segment (p, r), for some p, r S, n iretion τ {+, }. Denote y n s the neighors of p n r, respetively, long CH(S) tht p H (p, r) H + (p, r) Fig. 20. (p, r) is expning in iretion +. re in S τ (p, r) (possily, = s or even = r n s = p). We ll (p, r) expning in iretion τ if the two rys p n sr interset outsie H τ (p, r); otherwise, (p, r) is lle non-expning in iretion τ. Oserve tht if S τ (p, r) 3 then (p, r) is non-expning in iretion τ. Theorem 4. Every finite point set in onvex position in the plne mits spnning pth tht is 3π 2 -open n this is the est possile oun. Proof (Construtive proof for Theorem 4). The proof is se on the following more generl sttement. r s

16 Clim 1 Consier irete segment (p, r), for some p, r S, n iretion τ {+, }. Denote y n s the neighors of p n r, respetively, long CH(S) tht re in S τ (p, r) (possily, = s or even = r n s = p). Suppose tht (p, r) is non-expning in iretion τ n tht if τ = + then trp π 2 for ll t S+ (p, r) \ {p, r}; if τ = then prt π 2 for ll t S (p, r) \ {p, r}. Then there is 3π 2 -open spnning pth for Sτ (p, r) tht strts with p, r. Oserve tht the onition out the ngles ove sttes extly tht p, r n e extene to 3π 2 -open pth y ny single point from Sτ (p, r) \ {p, r}. In prtiulr, ll onitions from the lim re fulfille y ny imetril segment (p, r) of S, for oth of its two possile orienttions. Therefore, pplying the lim to oth (p, r) n iretion + s well s (r, p) n iretion + yiels Theorem 4. Proof (of Clim 1). We use inution on S τ (p, r). The sttement is trivil if S τ (p, r) {2, 3}. Therefore let S τ (p, r) 4 n onsier the segment (, s). Oserve tht y onvexity of S the segment (, s) is non-expning in iretion τ n S τ (, s) = S τ (p, r) \ {p, r}. From now on, ssume tht τ = +; the se τ = is symmetri. Cse 1 sr π 2. Illustrte in Fig. 21, (, s) fulfills the ngle onition, sine for every t S + (, s) \ {, s} ts = tsr sr tsr π 2 n tsr π y onvexity of S. Thus, we n exten, s to 3π 2 -open spnning pth for S + (, s) inutively n tht pth together with p, r, forms 3π 2 -open spnning pth for S. Cse 2 sr < π 2. Illustrte in Fig. 21, s (p, r) is non-expning in iretion +, we hve srp + rp π. Summing the ngles within the urilterl (p, r, s, ) yiels 2π = srp + rp + ps + sr < 3π 2 + ps, t s t s t s t s p r p r p r p r () Cse 1: Angles. () Cse 1: Pth. () Cse 2: Angles. () Cse 2: Pth. Fig. 21. Construting 3π -open spnning pth. 2

17 tht is, ps > π 2. We onlue tht for every t S (s, ) \ {, s} st = pt ps < pt π 2 π 2 where pt π y onvexity of S. Thus, we n exten s, to 3π 2 -open spnning pth for S (s, ) inutively n tht pth together with p, r, s forms 3π 2 -open spnning pth for S. Corollry 1. For ny finite set S R 2 of points in onvex position n ny p S there exists 3π 2 -open spnning pth for S whih hs p s n enpoint. Proof. For S 3 the sttement is trivil. Hene suppose S 4. Denote y (p = p 0, p 1,..., p n 1 ) the seuene of points long CH(S) in ounterlokwise orer n onsier the seuene (s i = (p i, p n i )) i=1... (n 1)/2 p (n+1)/2 p n 1... p (n 1)/2 p p 1 Fig. 22. Segments prllel to p. of segments prllel to p, s epite in Fig. 22. Oserve tht s (n 1)/2 is non-expning in iretion euse there re no more thn three points in S (p (n 1)/2, p (n+1)/2 ). Anlogously, s 1 is non-expning in iretion +. Therefore, the minimum inex k, 1 k (n 1)/2, for whih s k is nonexpning in iretion is well efine. If k = 1 then s 1 is segment tht is non-expning for oth iretions. Otherwise, y the minimlity of k the segment s k 1 is expning for iretion. By efinition if s i is expning in iretion then s i+1 is non-expning in iretion +, for 1 i < (n 1)/2. Thus, in ny se s k is segment tht is non-expning for oth iretions. Suppose there is point S (p k, p n k )\{p k, p n k } for whih p k p n k > π 2. Then the onvexity of S implies rp n kp k < π 2 for ll r S+ (p k, p n k ) \ {p k, p n k }. Moreover, s s k is non-expning in iretion we hve rp k p n k < π 2. Applition of Clim 1 to (p k, p n k ) n τ = + yiels 3π 2 -open spnning pth for S + (p k, p n k ) strting with p k, p n k. Similrly, pplying Clim 1 to (p n k, p k ) n τ = + we otin 3π 2 -open spnning pth for S+ (p n k, p k ) strting with p n k, p k. Comining oth pths provies the esire 3π 2 -open spnning pth for S. This pth hs p s one of its enpoints y onstrution. In symmetri wy, we n hnle the se tht there is point s S + (p k, p n k ) \ {p k, p n k } for whih p n k p k s > π 2. Finlly, if neither of the points n s exist, we n pply Clim 1 to (p k, p n k ) n τ = s well s to (p n k, p k ) n τ = n in this wy otin 3π 2 -open spnning pth for S whih hs p s one of its enpoints.