Minimum-Area Drawings of Plane 3-Trees

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1 Journl o Grh Algorithms nd Alitions htt://jg.ino/ vol. 15, no. 2, (2011) Minimum-Are Drwings o Plne -Trees Dejyoti Mondl Rhnum Islm Nisht Md. Sidur Rhmn Muhmmd Jwherul Alm Grh Drwing nd Inormtion Visuliztion Lortory, Dertment o Comuter Siene nd Engineering, Bngldesh University o Engineering nd Tehnology (BUET), Dhk , Bngldesh. Astrt A stright-line grid drwing o lne grh G is lnr drwing o G, where eh vertex is drwn t grid oint o n integer grid nd eh edge is drwn s stright-line segment. The height, width nd re o suh drwing re resetively the height, width nd re o the smllest xis-ligned retngle on the grid whih enloses the drwing. A minimum-re drwing o lne grh G is stright-line grid drwing o G where the re is the minimum. It is NP-omlete to determine whether lne grh G hs stright-line grid drwing with given re or not. In this er we give olynomil-time lgorithm or inding minimum-re drwing o lne -tree. Furthermore, we show 2n 1 2 n lower ound or the re o stright-line grid drwing o lne -tree with n 6 verties, whih imroves the reviously known lower ound 2(n 1) 2(n 1) or lne grhs. We lso exlore severl interesting roerties o lne -trees. Keywords. Grh drwing, Minimum re, Minimum lyer, Plne -tree, Lower ound. Sumitted: July 2010 Artile tye: Regulr Reviewed: Otoer 2010 Finl: Deemer 2010 Revised: Novemer 2010 Pulished: Ferury 2011 Aeted: Novemer 2010 Communited y: G. Liott E-mil ddresses: dejyoti mondl se@yhoo.om (Dejyoti Mondl) nisht.uet@gmil.om (Rhnum Islm Nisht) sidurrhmn@se.uet..d (Md. Sidur Rhmn) jwherul@gmil.om (Muhmmd Jwherul Alm)

2 178 Mondl et l. Minimum-Are Drwings o Plne -Trees 1 Introdution A lne grh is lnr grh with ixed lnr emedding. In stright-line grid drwing Γ o lne grh G, eh vertex o G is drwn t grid oint o n integer grid nd eh edge ogis drwn s stright-linesegment. The re o Γ is mesured y the size o the smllest retngle with sides rllel to the xes whih enloses Γ. The width W o Γ is the width o suh retngle nd the heighth oγis the height o suh retngle. The reis usully desried s W H. A minimum-re drwing o lne grh G is stright-line grid drwing o G where the re is the minimum. Figure 1() deits lne grh G nd Figure 1() deits minimum-re drwing o G. Wgner [28], Fry [18] nd Stein [26] indeendently roved tht every lnr grh G hs stright-line drwing. A nturl question rises: wht is the minimum size o grid required or stright-line grid drwing? For given lne grh G with n verties, de Frysseix et l. [12] nd Shnyder [25] indeendently showed tht G hs stright-line grid drwing on re (2n 4) (n 2) nd(n 2) (n 2), resetively. Reently, Brndenurg[7] hs imroved the uer ound o stright-line grid drwing to 4 n 2 n re. The rolem o inding minimum-re drwings or lne grhs hs een shown to e NP-hrd y Krug nd Wgner [22]. Furthermore, they resented n itertive roh to omtiy lnr stright-line grid drwings. Frti nd Ptrignni [20] roved tht 2n 2 /9+O(n) re is suiient nd n 2 /9+Ω(n) re is neessry or lnr stright-line grid drwings o nested tringles grhs. Reserhers hve lso onentrted their ttention on minimizing one dimension o the drwing where the other dimension o the drwing is unounded [1, 10, 15, 19, 27]. Suh drwings re known s lyered drwings. A lyered drwing o lne grh G is lnr drwing o G, where the verties re drwn on set o horizontl lines lled lyers nd the edges re drwn s stright line segments. A minimum-lyer drwing o lne grh G is lyered drwing o G where the numer o lyers is the minimum. Figure 1() deits lne grh G nd Figure 1() deits minimum-lyer drwing o G. Chrok nd Nkno [8] gve liner-time lgorithm to otin stright-line grid drwing o lne grh G with n verties where one dimension o the drwing is ounded y 2n 1. So, it is ovious tht ny lne grh G dmits lyered drwing on 2n 1 lyers. () () () Figure 1: () A lne grh G, () minimum-lyer drwing o G nd () minimum-re drwing o G.

3 JGAA, 15(2) (2011) 179 In this er, we onsider the rolem o inding minimum-re drwings o sulss o lnr grhs lled lne -trees. A lne -tree G n with n verties is lne grh or whih the ollowing () nd () hold: () G n is tringulted lne grh; () i n >, then G n hs vertex whose deletion gives lne -tree G n 1. Mny reserhers hve shown their interest on lne -trees or long time [2, 4, 14, 16]. In this er, we exlore some interesting roerties o lne -trees whih leds to olynomil-time lgorithm to otin their minimum-re drwings. We lso show tht, there exists lne -tree with n 6 verties or whih 2n 1 2 n re is neessry or ny lnr stright-line grid drwing. As side result, we give n O(nh 4 m) time lgorithm to omute minimum-lyer drwing o lne -tree G, where h m is the minimum numer o lyers required or ny lyered drwing o G. Note tht, Dujmović et l. gve (h) n time lgorithm tht n deide whether given grh G with n verties dmits lnr drwing in h lyers or not [15]. The running time o their lgorithm is dominted y the ost o inding th deomosition o G. To the est o our knowledge, the lgorithm urrently known to otin th deomosition o grh with treewidth l, tkes t lest Ω(n 4l+ ) time [5]. Clerly, one n otin minimum-lyer drwings or lne -trees using the tehnique resented in [15] ut it tkes t lest Ω(n 15 ) time, sine the treewidth o lne -trees is three. An outline o our lgorithm to omute minimum-lyer drwing o lne -tree is resented here. Let G n e lne -tree with n verties nd h e ositive integer. Sine ny lne grh dmits lyered drwing on 2n 1 lyers [8], we test whether G n n e drwn on h lyers or not, y iterting h rom 1 to 2n 1. For eh h rom 1 to 2n 1, we use dynmi rogrmming to test whether G n hs drwing on h lyers. We show tht ny lne -tree G n with n > verties hs n inner vertex whih is the ommon neighor o ll the three outer verties o G n. The vertex, long with the three outer verties o G n, divides the interior region o G n into three new regions. We rove tht the sugrhs enlosed y those three regions re lso lne -trees. For eh esile y-oordinte ssignment o the outer verties o G n, these sugrhs re the three surolems o our testing rolem. We deine the result o the testing rolem in terms o the test results o the surolems. Figure 2() deits lne -tree G where is the ommon neighor o the three outer verties,, o G. Figures 2() nd () show the surolems o the inut grh G or two dierent lements o. We divide nd test the surolems reursively nd store the test results o the surolems in tle to omute the minimum numer o lyers h m mong ll the ossile lyered drwings o G. Figure 2(d) illustrtes tht G does not dmit lyered drwing or the lyer ssignment o the vertex s in Figure 2(). Figure 2(e) is the drwing o G orresonding to the drwings o the surolems illustrted in Figure 2(). We n otin minimum-re drwing o G in similr method. The rest o the er is orgnized s ollows. Setion 2 desries some deinitions nd resents reliminry results. Setion introdues some interesting roerties o lne -trees. Setion 4 resents n O(nh 4 m ) time lgorithm to

4 180 Mondl et l. Minimum-Are Drwings o Plne -Trees G () Flse Flse () (d) () (e) Figure 2: Illustrtion o the lgorithm or minimum-lyer drwings. omute minimum-lyer drwing o lne -tree G n with n verties where h m is the minimum numer o lyers required or ny lyered drwing o G. Setion 5 illustrtes n O(n 9 logn) time lgorithm to otin minimum-re drwing o G n. Setion 6 gives lower ound on the re requirements or stright-line drwings o lne -trees. Finlly, Setion 7 onludes with disussions suggesting uture works. An erly version o this er hs een resented t [2]. 2 Preliminries In this setion we give some relevnt deinitions tht will e used throughout the er nd resent some reliminry results. Let G = (V,E) e onneted simle grh with vertex set V nd edge set E. The degree o vertex v is the numer o neighors o v in G. We denote y degree(v) the degree o the vertex v. A sugrh o grh G = (V,E) is grh G = (V,E ) suh tht V V nd E E. I G ontins ll the edges o G tht join verties in V, then G is lled the sugrh indued y V. A grh G is onneted i or ny two distint verties u nd v there is th etween u nd v in G. A grh whih is not onneted is lled disonneted grh. The onnetivity κ(g) o grh G is the minimum numer o verties whose removl results in disonneted grh or single-vertex grh. We sy tht G is k-onneted i κ(g) k. We ll set o verties in onneted grh G sertor or vertex-ut i the removl o the verties in the set results in disonneted or single-vertex grh. A tree is onneted grh without ny yle. A rooted tree T is tree in whih one o the verties is distinguished rom the others. The distinguished vertex is lled the root o the tree T nd every edge o T is direted wy rom

5 JGAA, 15(2) (2011) 181 the root. I v is vertex in T other thn the root, the rent o v is the vertex u suh tht there is direted edge rom u to v. When u is the rent o v, v is lled hild o u. A vertex in T, whih hs no hildren, is lled le. Any vertex whih is not le in T is n internl vertex. A desendnt o u is vertex v other thn u suh tht there is direted th rom u to v. Let i e ny vertex o T. Then we deine sutree T(i) rooted t i s sugrh o T indued y vertex i nd ll the desendnts o i. An ordered rooted tree is rooted tree where the hildren o ny vertex re ordered ounter-lokwise. A grh is lnr i it n e emedded in the lne without edge rossing exet t the verties where the edges re inident. A lne grh is lnr grh with ixed lnr emedding. A lne grh divides the lne into some onneted regions lled the es. The unounded region is lled the outer e nd ll the other es re lled the inner es. The verties on the outer e re lled the outer verties nd ll the other verties re lled the inner verties. I ll the es o lne grh G re tringles, then G is lled tringulted lne grh. For yle C in lne grh G, we denote y G(C) the lne sugrh o G inside C (inluding C). A lne grh G with n verties is lled lne -tree i the ollowing () nd () hold: () G is tringulted lne grh; () i n >, then G hs vertex x whose deletion gives lne -tree G o n 1 verties. Note tht, vertex x my e n inner vertex or n outer vertex o G. We denote lne -tree o n verties y G n. Exmles o lne -trees re shown in Figure ; G 6 is otined rom G 7 y removing the inner vertex o degree three. Then G 5 is otined rom G 6 y deleting the inner vertex o degree three. G 4 is otined rom G 5 y deleting the outer vertex g o degree three nd G is otined in similr wy. e e e e g g g d d d G 7 G 6 G 5 G 4 d G d Figure : Exmles o lne -trees. Proerties o Plne -Trees In this setion we introdue some roerties o lne -trees. The ollowing results re known on lne -trees.

6 182 Mondl et l. Minimum-Are Drwings o Plne -Trees Lemm 1[4] Let G n e lne -tree with n verties where n >. Then the ollowing () nd () hold. () G n hs n inner vertex x o degree three suh tht the removl o x gives the lne -tree G n 1. () G n hs extly one inner vertex y suh tht y is the neighor o ll the three outer verties o G n. By Lemm 1() or ny lne -tree G n, n >, there is extly one inner vertex y whih is the ommon neighor o ll the outer verties o G n. We ll vertex y the reresenttive vertex o G n. Aserting tringle otringultedlnegrhgistringlein Gwhose interior nd exterior ontin t lest one vertex eh. Let G n e lne -tree nd C e tringle in G n, then we rove tht G n (C) is lso lne -tree s in the ollowing lemm. Lemm 2 Let G n e lne -tree with n > verties nd C e ny tringle o G n. Then the sugrh G n (C) is lne -tree. We use the ollowing two ts to rove Lemm 2. Ft [17] Any tringulted grh with more thn three verties is trionneted grh. Ft 4 Let G n e tringulted lne grh nd C e serting tringle o G n where n >. Then eh o the three verties on C must hve degree t lest our in G n. Proo. Sine G n nd G n (C) re tringulted nd n >, they re trionneted y Ft. Thereore eh o the three verties on C hs degree t lest three in G n (C). Suose or ontrdition tht t lest one o the verties w on C hs degree extly three in G n s illustrted in Figure 4. Sine G n (C) is trionneted with more thn three verties, two o the neighors o w re on C nd the other neighor is inside C. Sine w hs no neighor outside C, we n disonnet the exterior verties o C rom the interior verties o C y deleting the other two verties on C exet w. This imlies tht G n hs vertex-ut o two verties, nd hene G n would not e trionneted, ontrdition. We re now redy to give roo o Lemm 2. w C G n Figure 4: Illustrtion or the roo o Ft 4. Proo o Lemm 2. The roo is trivil or the se when the tringle C is

7 JGAA, 15(2) (2011) 18 not serting tringle. I C is the outer e o G, then G n (C) is itsel lne -tree; otherwise C is tringle whose interior ontins no vertex nd G n (C) is lne -tree y deinition. We now onsider the se when C is serting tringle in G. Sine G n is tringulted, G n (C) is tringulted. Then, it is suiient to rove tht we n delete inner verties o degree three reursively rom G n (C) to otin the yle C. By Lemm 1, G n hs n inner vertex o degree three whose deletion gives lne -tree G n 1. We delete suh inner verties o G n reursively whih re outside o G n (C). Assume tht ter deleting k vertieswe hveno inner vertexodegreethree outside G n (C) nd let the resulting lne -tree e G n k. As we never deleted the outer verties o G n nd the inner verties o G n (C), C is lso serting tringle o G n k. There must e n inner vertex o degree three in G n k y Lemm 1. Tht vertex must e n inner vertex o G n k (C) sine eh o the three outer verties o G n k (C) hs degree t lest our in G n k y Ft 4. We now delete ll the inner verties o degree three o G n k (C) reursively in suh wy tht t eh deletion the resulting grh remins lne -tree. By deinition ter deleting m suh inner verties o G n k (C) reursively we get lne -tree G n k m. Suose or ontrdition tht there is no inner vertex o degree three in G n k m (C). We irst onsider the se when C hs no interior vertex whih imlies tht we hve reursivelydeleted the inner verties odegree three o G n (C) to get the tringle C nd G n (C) is ertinly lne -tree. We next onsider the se where C still ontinst lest one interiorvertex. Then G n k m (C) hs morethn three verties nd there is no inner vertex o degree three in G n k m. Hene G n k m would not e lne -tree y Lemm 1(), ontrdition. Let e the reresenttive vertex nd,, e the outer verties o G n. The vertex, long with the three outer verties, nd, orm three tringles {,,}, {,,} nd {,,} s illustrted in Figure 5. We ll those three tringles the nested tringles round. C C 1 C 2 Gn Figure 5: Nested tringles round. We now deine the reresenttive tree o G n s n ordered rooted tree T n stisying the ollowing two onditions () nd (). () i n =, T n onsists o single vertex. () i n >, then the root o T n is the reresenttive vertex o G n nd the sutrees rooted t the three ounter-lokwise ordered hildren q 1, q 2 nd q o in T n re the reresenttive trees o G n (C 1 ), G n (C 2 ) nd G n (C ), resetively, where C 1, C 2 nd C re the three nested tringles round in ounter-lokwise order.

8 184 Mondl et l. Minimum-Are Drwings o Plne -Trees Figure 6 illustrtes the reresenttive tree T n o the lne -tree G n. Note tht the 4-lok trees [21] nd the tree o the tree deomosition [5] re quite similr to the reresenttive trees or the lne -trees. d g e G n g d e T n Figure 6: Reresenttive tree T n o G n. We now rove tht T n is unique or G n in the ollowing lemm. Lemm 5 Let G n e ny lne -tree with n verties. Then G n hs unique reresenttive tree T n with extly n internl verties nd 2n 5 leves. Proo. The se n = is trivil sine the reresenttive tree o G is single vertex. We my thus ssume tht G hs our or more verties. By Lemm 1() G n hs extly one reresenttive vertex. Let e tht reresenttive vertex o G n nd C 1, C 2, C e the three nested tringles round. By Lemm 2, G n (C 1 ), G n (C 2 ) nd G n (C ) relne-trees. Let n 1, n 2 nd n e the numer o verties in G n (C 1 ), G n (C 2 ) nd G n (C ), resetively. Then y the indution hyothesis, T n1, T n2 nd T n re the unique reresenttive trees o G n (C 1 ), G n (C 2 ) nd G n (C ), resetively. We now ssign s the rent o q 1, q 2 nd q, where q 1, q 2 nd q re the roots o T n1, T n2 nd T n, resetively. Sine is the unique reresenttive vertex o G n, the hoie or the root o T n is unique. Sine G n hs n verties nd ny inner vertex o G n exet elongs to extly one o G n (C 1 ), G n (C 2 ) nd G n (C ), the totl numer o verties in T n1, T n2 nd T n is n 1 +n 2 +n = n 4. Thus the new tree T n with root hs n = n internl verties. Sine T n1, T n2 nd T n re ordered trees nd q 1, q 2 nd q re ordered ounter-lokwise round, T n is lso n ordered tree. Furthermore one n esily oserve tht, the leves reresent only the internl es o G n. Sine the numer o internl es o G n is 2n 5 y Euler s Theorem, T n hs 2n 5 leves. Now we hve the ollowing lemm whose roo is immedite rom the deinition o the reresenttive tree nd rom Lemm 5. Lemm 6 Let T n e the reresenttive tree o lne -tree G n with n verties nd let T(i) e sutree rooted t vertex i o T n. Then there exists unique tringle C in G n suh tht T(i) is the reresenttive tree o G n (C). By Lemm 6, or ny vertex o T n, there is unique tringle in G n whih we denote s C or the rest o this rtile. Furthermore, i is the root o T n,

9 JGAA, 15(2) (2011) 185 then C is the outer e o G n ; i is le o T n, then C is n inner e o G n nd i is n internl vertex in T n, then C is serting tringle in G n. Let L e the set o leves in T n nd let, nd e the outer verties o G n. Then T n L is snning tree o G n {,,} where eh vertex o T n L is med to the reresenttive vertex o G n (C ), s illustrted in Figure 6. Thus or the rest o this rtile, we shll oten use n internl vertex o T n nd the reresenttive vertex o G n (C ) interhngely. We shll lso denote y T() the reresenttive tree o G n (C ). Figures 7() nd () illustrte G n (C ) nd its reresenttive tree T(), resetively. q q 1 q q () 2 G n ( C ) q 1 C q 2 T() () Figure 7: () Illustrtion o G n (C ) nd () the reresenttive tree T(). We now hve the ollowing lemm. Lemm 7 For ny lne -tree G n with n verties, the reresenttive tree T n o G n n e ound in time O(n). Proo. To onstrut T n we irst ind the reresenttive vertex o G n. We kee list or eh inner vertex u o G n. For eh outer vertex v i o G n, i {1,2,}, we dd v i in the list o u i u is djent to v. One n esily oserve tht, only the list o the reresenttive vertex will ontin the three outer verties o G n. Thus we n ind in time O( i=1 degree(v i)). Let C q1, C q2, C q e the nested tringles round. We n ind the three hildren q 1, q 2 nd q o y udting the lists s ollows. Sine the lists re lredy udted or ll the outer verties o G n (C q1 ), G n (C q2 ) nd G n (C q ) exet, we only need to udte the lists y dding to the list o u i u is djent to. Thus the three hildren o n e ound in time O(degree()). We then ontinue udting the lists reursively to ind the other verties o T n. One the lists re udted y vertex, we do not onsider tht vertex lter to udte the lists. The roess o udting the lists or eh vertex v tkes O(degree(v)) time nd hene the totl time o onstruting the reresenttive tree is O( v V degree(v)) = O(n) sine G n is lnr.

10 186 Mondl et l. Minimum-Are Drwings o Plne -Trees The roo o Lemm 7 leds to liner-time lgorithm to onstrut the reresenttive tree o lne -tree. 4 Minimum-Lyer Drwings In this setion we onsider the rolem o inding minimum-lyer drwings o lne -trees. In lyered drwing o lne grh G, the verties re drwn on set o horizontl lines lled lyers nd the edges re drwn s stright line segments. We ssume tht the lyers re ligned rllel to the x-xes with dierent y- oordintes nd the y-oordintes o the lyers re deined s ollows. We denote y y(l) the y-oordinte o lyer l. Let {l 1,l 2,...,l n } e set o n lyers where y(l 1 ) < y(l 2 ) <... < y(l n ), then y(l i ) = i, 1 i n. Thus or the rest o this rtile, we denote lyer ssignment o vertex v y y-oordinte ssignment o v. Chrok et l. [8] showed tht the uer ound or one dimension o stright-line grid drwing o ny lne grh G with n verties is 2n 1. So, it is ovious tht ny lne -tree G dmits lyered drwing on 2n 1 lyers. Thereore we ssume tht, G dmits lyered drwing on h lyers nd iterte h rom 1 to 2n 1. For eh itertion, we hek whether G is drwle on h lyers or not. The irst h within whih G is drwle is the minimum numer o lyers h m required to drw G. A rute ore roh to solve this rolem is to ssign ll ossile omintions oy-oordintesto the verties ognd hekwhether there is ny edge rossing. However,ithe totlnumerovertiesisnndthenumerolyersis h, there re n h dierent ssignments ossile. This exonentil time mkes the roh imrtil or lrge n nd h. We now resent dynmi rogrmming roh to solve the rolem. We irst give n lgorithm Minimum-Lyer to generte ll the esile y-oordinte ssignments o the verties o G iterting h rom 1 to 2n 1. Then we give n lgorithm Fesiility-Cheking to hek whether G dmits lyered drwing on h lyers or rtiulr y- oordinte ssignment o its outer verties. For onveniene, we desrie Algorithm Fesiility-Cheking eore Algorithm Minimum-Lyer. At the end o this setion we give seudoodes or oth o the lgorithms. We now ormlly deine the inut nd the outut o the deision rolem Fesiility Cheking. Inut: A lne -tree G nd y-oordinte ssignments o the three outer verties, nd o G. Outut: I G dmits lyered drwing with the given y-oordintes o, nd, the outut is True, nd Flse otherwise. Let T e the reresenttive tree o lne -tree G nd v y e the y- oordinte o ny vertex v. For ny vertex o T, we denote y Γ lyered drwing o G(C ) nd y F ( y, y, y ) the Fesiility Cheking rolem o where y, y, y re the y-oordintes o the three outer verties,, o G(C ), resetively. We solve this Fesiility Cheking rolem using dynmi rogrmming y hrterizing the otiml sustruture nd overling

11 JGAA, 15(2) (2011) 187 surolems roerties o the rolem whih re the two key ingredients or the dynmi rogrmming to e lile [9]. Chrterizing otiml sustruture mens showing tht the otiml solution o the rolem onsists o the otiml solutions o the surolems. To show the otiml sustruture roerty o the Fesiility Cheking rolem, we need the ollowing two lemms. Lemm 8 Let G e lne -tree with reresenttive vertex. Let Γ e lyered drwing o G nd let Γ(C ) e the lyered drwing o C in Γ. Let Γ (C ) e nother lyered drwing o C where, nd hve the sme y- oordintes s in Γ(C ). Then G hs lyered drwing Γ hving Γ (C ) s the drwing o C. Proo. The se or n = is trivil sine or this se Γ oinides with Γ (C ). We my thus ssume tht n is greter thn three nd the lim holds or ny lne -tree o less thn n verties. Let l e the lyer tht ontins vertex nd let y e the y-oordinte o in Γ. The lyer l intersets Γ (C ) t two oints (x 1, y ) nd (x 2, y ), x 1 x 2. We le on l in etween x 1 nd x 2 to otin Γ (C q1 ), Γ (C q2 ) nd Γ (C q ) where C q1, C q2 nd C q re the nested tringles round. By indution hyothesis G(C q1 ), G(C q2 ) nd G(C q ) dmit lyered drwings Γ q 1, Γ q 2 nd Γ q whih ontin the drwings Γ (C q1 ), Γ (C q2 ) nd Γ (C q ), resetively. Clerly, one n otin Γ y thing Γ q 1, Γ q 2 nd Γ q inside Γ (C q1 ), Γ (C q2 ) nd Γ (C q ), resetively, s illustrted in Figure 8. (C q ) 1 (C q ) (C q ) 2 (C q ) 1 (C q ) (C q ) 2 () () Figure 8: Illustrtion or the roo o Lemm 8. () Lyered drwing Γ o G nd () lyered drwing Γ o G. Now we hve the ollowing lemm. Lemm 9 Let G e lne -tree with the reresenttive tree T. Let e ny internl vertex o T with the three hildren q 1, q 2, q in T nd let,, e the three outer verties o G(C ). Then G(C ) dmits lyered drwing Γ or the ssignment ( y, y, y ) i nd only i Γ q1, Γ q2 nd Γ q dmit lyered drwings or the ssignments ( y, y, y ), ( y, y, y ) nd ( y, y, y ), resetively, where min( y, y, y ) < y < mx( y, y, y ). Proo. The neessity is trivil, nd roo o the suiieny n e otined in similr tehnique s desried in the roo o Lemm 8.

12 188 Mondl et l. Minimum-Are Drwings o Plne -Trees We n redily ind the overling surolems roerty o the Fesiility Cheking rolem. Overling surolem ours when reursive lgorithm visits the sme rolem more thn one. Figure 9 illustrtes this roerty or the Fesiility Cheking rolem where the overling surolems re shown y dotted retngles nd old retngles. We now hve the ollowing theorem e g e g ( e, g 4 1, 4 ) ( e 4, 1, ) ( 1, g 4, ) ( g 4, e 4, ) ( e 4, 1, 2 ) ( 1, g 4, 2 ) ( g 4, e 4, 2 ) ( e 4, 1, ) ( 1,, ) (, e 4, ) ( e 4, 1, 2 ) ( 1, 2, 2 ) ( 2, e 4, 2 ) ( e 4, 1, 2 ) ( 1,, 2 ) (, e 4, 2 ) ( e 4, 1, ) ( 1, 2, ) ( 2, e 4, ) Figure 9: Overling Surolems. whih leds to reursive solution o the Fesiility Cheking rolem. Theorem 4.1 Let G e lne -tree nd let e ny vertex o the reresenttive tree T o G. Let,, e the three outer verties o G(C ) nd q 1, q 2, q e the three hildren o i is n internl vertex o T. Let F ( y, y, y ) denote the Fesiility Cheking rolem o where y, y, y re the y-oordintes o,,. Then F ( y, y, y ) hs the ollowing reursive ormul. F ( y, y, y ) = Flse i {mx{ y, y, y } min{ y, y, y } = 0}; True i {mx{ y, y, y } min{ y, y, y } 1} where is le; Flse i {mx{ y, y, y } min{ y, y, y } 1} where is n internl vertex; y {F q1 ( y, y, y ) F q2 ( y, y, y ) F q ( y, y, y )} where {min{ y, y, y } < y < mx{ y, y, y }}, otherwise. Proo. Considerthesewhenmx{ y, y, y } min{ y, y, y } = 0. Thenwe ssign F ( y, y, y ) = Flse sine tringle nnot e drwn on single lyer. The next se is mx{ y, y, y } min{ y, y, y } 1 when is le. Then

13 JGAA, 15(2) (2011) 189 we ssign F ( y, y, y ) = True sine two lyers re suiient to drw tringle. The next se is mx{ y, y, y } min{ y, y, y } 1 when is n internl vertex. Then we ssign F ( y, y, y ) = Flse or this se sine the outer e needs two lyers to e drwn nd the inner vertex nnot e led on ny o them. The remining se is mx{ y, y, y } min{ y, y, y } > 1 when is n internl vertex. Then we deine F ( y, y, y ) reursively y Lemm 9. We ssoite tle FC i [1: 2n+2,1: 2n+2,1: 2n+2 ] or eh vertex i o the reresenttive tree T o G, where the solution o F i ( y, y, y ) is stored in FC i [ y, y, y ]. To store the omuted y-oordintes o the verties o G, we mintin nother tle Y i [1: 2n+2,1: 2n+2, 1: 2n+2 ] or eh vertex i o T. Eh entry Y i [ y, y, y ] is omuted s ollows. Flse i FC i [ y, y, y ] = Flse; Y i [ y, y, y ] = True i i is le nd FC i [ y, y, y ] = True; i i is n internl vertex nd FC i [ y, y, y ] = True. i y Let G e lne -tree with the outer verties,, nd e the reresenttive vertex o G. I Y [ y, y, y ] is Flse, then G hs no lyered drwing or the given y-oordinte ssignment y, y, y. I the entry is True, then G hs no inner vertex nd G hs lyered drwing or the given y-oordinte ssignment y, y, y. Otherwise, G hs lyered drwing or the given y-oordinte ssignment y, y, y nd the entry Y [ y, y, y ] ontins the y-oordinte o the reresenttive vertex. To otin the y-oordinte ssignment o eh internl vertex o G, we hek the entryy [ y, y, y ]. Ithe entryontinsy-oordinteothe reresenttive vertex, we hek the entries Y q1 [ y, y, y ], Y q2 [ y, y, y ] nd Y q [ y, y, y ] to get the y-oordintes o the three hildren o. We ush Y q1 [ y, y, y ], Y q2 [ y, y, y ] nd Y q [ y, y, y ] on stk nd o one entry or urther exlortion reursively. This is similr to the trversl o the reresenttive tree T o G in reorder, tht is, irst trversing the root o T, then trversing the let, middle nd right sutrees one ter nother. When the stk is emty, y-oordintes or ll the verties o G re otined. Sine T hs n internl verties y Lemm 5, this roess tkes O(n) time. We now desrie Algorithm Minimum-Lyer whih omutes the minimum numer o lyers required to drw G using Algorithm Fesiility- Cheking. Let T e the reresenttive tree o the lne -tree G. We ssume tht G dmits lyered drwing on h lyers nd iterte h rom 1 to 2n 1. At eh itertion we trverse T in reorder nd or eh vertex i o T, Algorithm Minimum-Lyer genertes ll ossile y-oordinte ssignments or the outer verties, nd o G(C i ) within h lyers. For eh suh ssignment y, y nd y, Algorithm Fesiility-Cheking is lled to hek whether G(C i ) is drwle. The irst h within whih G is drwle is the minimum numer o lyers h m required to drw G. At the end o this setion, orml desritions o Algorithm Minimum-Lyer nd Algorithm Fesiility-Cheking re given in Algorithm 1 nd Algorithm 2, resetively.

14 190 Mondl et l. Minimum-Are Drwings o Plne -Trees Lemm 10 Let T e the reresenttive tree o lne -tree G nd i e ny internl vertex o T. Let, nd e the outer verties o G(C i ). Then Algorithm Minimum-Lyer genertes ll ossile y-oordinte ssignments or, nd within h lyers ter the hth itertion. Proo. We rove the orretnesso the lgorithm y indution. For h = 1, the ssignment is ovious rom Line. We my thus ssume tht h > 1 nd ll the y-oordinte ssignments within lyer 1 to h 1 hve een generted nd the results hve een lulted within h 1 itertions. Now we onsider the hth itertion. In Line 8, y is ssigned lyer h nd in Line 9 y nd y re ssigned ll ossile y-oordintes within h. Next, y is ssigned lyer h in Line 17 nd in Line 18, y nd y re ssigned ll ossile y-oordintes within h 1 nd h, resetively. Similrly, y is ssigned lyer h in Line 26 nd in Line 27, y nd y re ssigned ll ossile y-oordintes within h 1. Suose or ontrdition tht the y-oordinte ssignments y, y nd y hve not een generted ter the hth itertion. Clerly mx{ y, y, y } nnot e less thn h, sine ll the y-oordintessignments within lyer1to h 1 hve een generted y indution. We my thus ssume tht mx{ y, y, y } = h. One n oserve tht the hth itertion ensures the genertion o ll ossile y-oordinte ssignments suh tht mx{ y, y, y } = h, ontrdition. We now nlyze the omlexity o Algorithm Minimum-Lyer. Theorem 4.2 Given lne -tree G with n verties, Algorithm Minimum- Lyer omutes the minimum numer o lyers h m required to drw G on lyers in O(nh 4 m ) time. Proo. To rove the lim we irst lulte the numer o times Algorithm Fesiility-Cheking is lled. Sine we iterte the numer o lyers h rom 1 to 2n 1 +1 nd t eh itertion we trverse T in reorder, the numer o times ll the verties o T is onsidered is h m n. For eh internl vertex, Algorithm Fesiility-Cheking is lled or h h times in Line 11, h (h 1) times in Line 20 nd (h 1) (h 1) times in Line 29. For ll the n internl verties o T, in eh itertion the totl numer o lls to Algorithm Fesiility-Cheking y Algorithm Minimum-Lyer is h m n(h 2 +h(h 1)+(h 1) 2 ) = h m n(h 2 +h 2 h+h 2 2h+1) = h m n(h 2 h+1) = O(nh m ) We store the solutions o the surolems in the FC tles where eh entry o the tles initilly ontins null to denote tht the entry is yet to e illed in. When the surolem is irst enountered during the exeution o the reursive lgorithm Fesiility-Cheking, its solution is omuted nd stored in the tle. Eh susequent time the surolem is enountered, the vlue stored in the tle is looked u nd returned. The solutions o the surolems re omuted ottom u nd eh looku tkes O(1) time. Moreover, y n tke t most h m vlues in Line 5 o Algorithm Fesiility-Cheking. Thereore, eh

15 JGAA, 15(2) (2011) 191 ll to Algorithm Fesiility-Cheking tkes O(h m ) O(1) = O(h m ) time. Sine the totl numer o times Algorithm Fesiility-Cheking is lled, inluding the reursive lls, is O(nh m ) the totl running time o this lgorithm is O(nh m) O(h m ) = O(nh 4 m). We now rell tht the onstrution o the reresenttive tree tkes O(n) time y Lemm 7. Thus Algorithm Minimum-Lyer tkes O(n) + O(nh 4 m ) = O(nh4 m ) time in totl. Algorithm 1 Minimum-Lyer(G) 1: Construt the reresenttive tree T o G 2: or eh vertex i o T do : FC i [1,1,1] = Flse 4: end or 5: {The outer verties o G(C i ) re, nd } 6: or eh h rom 2 to 2n 1 +1 do 7: or eh internl vertex i o T in reorder do 8: y = h 9: or y rom 1 to h nd y rom 1 to h do 10: i FC i [ y, y, y ] = null then 11: Fesiility-Cheking (,,) 12: end i 1: i i = root && FC i [ y, y, y ] = true then 14: return 15: end i 16: end or 17: y = h 18: or y rom 1 to h 1 nd y rom 1 to h do 19: i FC i [ y, y, y ] = null then 20: Fesiility-Cheking (,,) 21: end i 22: i i = root && FC i [ y, y, y ] = true then 2: return 24: end i 25: end or 26: y = h 27: or y rom 1 to h 1 nd y rom 1 to h 1 do 28: i FC i [ y, y, y ] = null then 29: Fesiility-Cheking (,,) 0: end i 1: i i = root && FC i [ y, y, y ] = true then 2: return : end i 4: end or 5: end or 6: end or

16 192 Mondl et l. Minimum-Are Drwings o Plne -Trees Algorithm 2 Fesiility-Cheking(,,) 1: {The outer verties o G re, nd nd is its reresenttive vertex} 2: i FC [ y, y, y ] null then : return FC [ y, y, y ] 4: else i (mx{ y, y, y } min{ y, y, y } > 1) & ( is n internl vertex) then 5: or min{ y, y, y } < y < mx{ y, y, y } do 6: i (Fesiility-Cheking(,, ) & Fesiility-Cheking(,, ) & 7: Fesiility-Cheking(,, )) then 8: FC [ y, y, y ] = True, Y [ y, y, y ] = y, rek 9: end i 10: end or 11: else 12: Comute FC [ y, y, y ] nd Y [ y, y, y ] y Theorem 4.1 1: end i 5 Minimum-Are Drwings In this setion we extend the onet o the dynmi rogrmming tehnique o Setion 4 to give n lgorithm Minimum-Are to otin minimum-re drwing o lne -tree G. We now resent n outline o the lgorithm. Sine the uer ound o the re o stright-line grid drwings o lnr grhs is kn 2 with k 1, it is ovious tht the uer ound or the re o minimum-re drwing o lne -tree G is t most kn 2 with k 1. Sine the minimum numer o lyers required or ny stright-line grid drwing o G is h m, the uer ound or width is n 2 /h m. Thereore, we ssume height h nd width w nd iterte rom 1 to n nd 1 to min( n2 n2 h, h m ), resetively. At eh itertion o h nd w we hekwhethergisdrwleonw hgridornot. AlgorithmMinimum-Are genertes ll the ossile (x, y)-oordinte ssignments or the outer verties o G nd heks the drwility o G or eh suh ssignment using Algorithm Are-Cheking. For onveniene, we desrie Algorithm Are Cheking eore Algorithm Minimum-Are. At the end o this setion we give seudoodes or oth o the lgorithms. Here we ormlly deine the inut nd outut o the rolem Are Cheking. Inut: A lne -tree G nd (x, y)-oordinte ssignments o the three outer verties, nd o G. Outut: I G dmits drwing with the given (x,y)-oordintes o, nd, the outut is True nd otherwise it is Flse. Like the Fesiility Cheking rolem or minimum-lyer drwing, we n hrterize the otiml sustruture or the rolem Are Cheking. Let G e

17 JGAA, 15(2) (2011) 19 lne -treewith the reresenttivetreet. We denote the x-oordintend y- oordinte o vertex v y v x nd v y, resetively. We denote y A ( x y, x y, x y) the Are Cheking rolem o ny vertex o T where x y,x y,x y re the (x,y)- oordintes o the three outer verties, nd o G(C ). We denote y Γ minimum-re drwing o G(C ). We now rove tht the Are Cheking rolem hs the ollowing otiml sustruture roerty. Lemm 11 Let G e lne -tree with the reresenttive tree T. Let e ny internl vertex o T with the three hildren q 1, q 2, q in T nd,, e the outer verties o G(C ). Then the Are Cheking rolems o q 1, q 2 nd q re the three surolems o the Are Cheking rolem o. Proo. The vertex is n inner vertex o G nd thereore, must e led inside the outer e o G. Sine the (x,y)-oordintes o,, re ressigned nd x, y re the sme or the drwings Γ q 1, Γ q 2 nd Γ q, those three drwings neominedtogetthedrwingγ og(c )sillustrtedinfigure10. Thus the solution o the Are Cheking rolem o onsists o the solutions o the AreChekingrolemsoq 1, q 2 ndq ; nd henethe AreChekingrolems o q 1, q 2 nd q re the three surolems o the Are Cheking rolem o. G( C q ) 1 G( C q ) 2 G( C q ) G( C q ) G( C q ) 1 G( C q ) 2 () () Figure 10: Illustrtion or the roo o Lemm 11. One n esily oserve the overling surolem roerty or the Are Cheking rolem in similr wy tht we used to show the overling surolem roerty o the Fesiility Cheking rolem. By method similr to the roo o Lemm 10 one n see tht Algorithm Minimum-Are genertes ll ossile (x, y)-oordinte ssignments o the outer verties o G within w min( n2 n2 h, h m ) re. We now rove Theorem 5.1 whih sttes the reursive solution o Are Cheking rolem. Theorem 5.1 Let G e lne -tree with the reresenttive tree T nd e ny vertex o T. Let,, e the three outer verties o G(C ) nd q 1, q 2, q e the three hildren o when is n internl vertex o T. Let A ( x y, x y, x y) e the Are Cheking rolem o where, nd hve distint (x,y)-oordintes.

18 194 Mondl et l. Minimum-Are Drwings o Plne -Trees Then A ( x y,x y,x y ) hs the ollowing reursive ormul. Flse i {mx{ x, x, x } min{ x, x, x } = 0} {mx{ y, y, y } min{ y, y, y } = 0}; True i {{mx{ x, x, x } min{ x, x, x } 1} {mx{ y, y, y } min{ y, y, y } 1}} is le; A ( x y, x y, x y) = Flse i {{mx{ x, x, x } min{ x, x, x } 1} {mx{ y, y, y } min{ y, y, y } 1}} is n internl vertex; x, y {A q1 ( x y, x y, x y) A q2 ( x y, x y, x y) A q ( x y, x y, x y)} where ( x, y ) is inside the tringle with the verties,,, otherwise. Proo. First we onsider the se when mx{ x, x, x } min{ x, x, x } = 0 mx{ y, y, y } min{ y, y, y } = 0. Then we ssigna ( x y, x y, x y) = Flse eusegridotlestre1 1is neessrytodrwtringle. The next se is mx{ x, x, x } min{ x, x, x } 1 mx{ y, y, y } min{ y, y, y } 1 when is le. Then we ssign A ( x y,x y,x y ) = True sine re 1 1 is suiient to drw tringle. The next se is mx{ x, x, x } min{ x, x, x } 1 mx{ y, y, y } min{ y, y, y } 1 when is n internl vertex. We ssign A ( x y,x y,x y ) = Flse sine the width nd height o G(C ) is t most 1 nd nnot e led inside C. The remining se is mx{ x, x, x } min{ x, x, x } > 1 mx{ y, y, y } min{ y, y, y } > 1 when is n internl vertex. Then we deine A ( x y,x y,x y ) reursively ording to Lemm 11. We ssoite tle AC i [1: n2 h m, 1: n2 h m, 1: n2 h m, 1:n, 1:n, 1:n] or eh vertex i o the reresenttive tree T o G, where the solution o A i ( x y, x y, x y) is stored in AC i [ x y,x y,x y ]. To store the omuted (x,y)-oordintes o the verties o G, we mintin two tles X i [1: n2 h m, 1: n2 h m, 1: n2 h m, 1:n, 1:n, 1:n] nd Y i [1: n2 h m, 1: n2 h m, 1: n2 h m, 1:n, 1:n, 1:n] or eh vertex i o T. Eh entry o the two tle X i nd Y i is omuted s ollows. Flse i AC i [ x, x, x, y, y, y ] = Flse; True i i is le nd X i [ x, x, x, y, y, y ] = AC i [ x, x, x, y, y, y ] = True; i x i i is n internl vertex nd AC i [ x, x, x, y, y, y ] = True. Flse i AC i [ x, x, x, y, y, y ] = Flse; True i i is le nd Y i [ x, x, x, y, y, y ] = AC i [ x, x, x, y, y, y ] = True; i y i i is n internl vertex nd AC i [ x, x, x, y, y, y ] = True. Let,, e the outer verties nd e the reresenttive vertex o G. I X [ x, x, x, y, y, y ] or Y [ x, x, x, y, y, y ] is Flse, then G hs no

19 JGAA, 15(2) (2011) 195 stright-line grid drwing or the given (x,y)-oordinte ssignments x y, x y, x y. I the entries re True, then G hs stright-line grid drwing with the given (x,y)-oordinte ssignments x y, x y, x y. Otherwise, the two entries ontin x-oordinte nd y-oordinte o the reresenttive vertex, resetively. We now desrie Algorithm Minimum-Are whih gives drwing o G with the minimum re, using Algorithm Are-Cheking. Let T e the reresenttive tree o the lne -tree G. We ssume width w nd height h or G. We iterte h rom 1 to n nd oreh h, we iterte w rom1to min( n2 n2 h, h m ). At eh itertion we trverse T in reorder. For eh internl vertex i o T, Minimum-Are genertes ll ossile (x, y)-oordinte ssignments or the outer verties, nd o G(C i ) within re w h. For eh suh (x,y)- oordinte ssignment o, nd, Algorithm Are-Cheking is lled to hek whether G(C i ) is drwle. Eh time drwing o G with smller re is ound, the stored re is reled y the smller re nd t the end o the lgorithm, the stored re is the minimum. At the end o this setion, orml desritions o Algorithm Minimum-Are nd Algorithm Are-Cheking re given in Algorithm nd Algorithm 4, resetively. We now nlyze the omlexity o Algorithm Minimum-Are. Theorem 5.2 Given lne -tree G with n verties, Algorithm Minimum- Are gives minimum-re drwing o G in O(n 9 logn) time. Proo. We iterte height h rom 2 to n nd or eh h, width w is iterted rom 2 to min( n2 n2 h, h m ) where h m is the minimum numer o lyers required to drw G. So the totl numer o itertions in Line 7 is n h 2 m h m + n2 n2 h m n = n 2 (1+ 1 h m+1 n ) = n 2 (1+ n k=h m+1 1 k ) n 2 +n 2 log n h m = O(n 2 logn) The irst time esile drwing is ound, we store the re w h or tht drwing. Eh susequent time esile drwing is ound, we rele the stored re only i the re w h or the urrent vlues o w nd h is smller or equl to the stored re. Ater the lgorithm is terminted, the minimum re required to drw G is returned. Let the reresenttive tree o G e T. For eh itertion we trverse T in reorderin Line 8 nd oreh internlvertex ot, Algorithm Are-Cheking is lled w 2 h 2, w 2 h(h 1) nd w 2 (h 1) 2 times in Line 12, Line 2 nd Line 4, resetively. Sine there re n internl verties in T, the totl numer o lls to Algorithm Are-Cheking y Algorithm Minimum-Are in eh itertion is n(w 2 h 2 +w 2 h(h 1)+w 2 (h 1) 2 ) = nw 2 (h 2 h+1) = O(nw 2 h 2 ) We store the solutions o the surolems in the AC tles where eh entry o the tles initilly ontins null to denote tht the entry is yet to e illed

20 196 Mondl et l. Minimum-Are Drwings o Plne -Trees in. When the surolem is irst enountered during the exeution o the reursive lgorithm Are-Cheking, its solution is omuted nd stored in the tle. Eh susequent time the surolem is enountered, the vlue stored in the tle is looked u nd returned. The solutions o these surolems re omuted ottom u nd eh looku tkes O(1) time. Moreover, x y n tke t most w h vlues in Line 9 o Algorithm Are-Cheking. Thereore, eh ll to Algorithm Are-Cheking tkes O(1) O(wh) = O(wh) time. Hene or eh itertion, the numer o times Algorithm Are-Cheking is lled inluding ll the reursive lls is O(nw 2 h 2 ). Thereore, the totl running time oalgorithm Are-Cheking is O(nw 2 h 2 ) O(wh) = O(nw h ) = O(n 7 ) sine wh = O(n 2 ). Thus the totl time required or ll the O(n 2 logn) itertions is O(n 2 logn) O(n 7 ) = O(n 9 logn). We now rell tht the onstrution o the reresenttive tree tkes O(n) time y Lemm 7. Thus AlgorithmMinimum-Lyer tkeso(n) + O(n 9 logn) = O(n 9 logn) time in totl. 6 Lower Bound In this setion we imrove the lower ound on re or stright-line grid drwings o lne grhs. We show tht there exist lne -trees, or whih the imroved ound holds. One o the most mous nd long stnding onjetures sttes tht ny lne grh G with n verties n e drwn in 2n 1 2n 1 re [20]. Frti nd Ptrignni [20] showed tht this ound neglets t lest liner term. They showed tht there exists lne grh with n verties whih requires t lest ( 2n 1) (2n ) rewhere n is multile othree. This indites tht the known ( 2n 1) (2n 1) lower ound on re or the stright-line grid drwings o lne grhs n e imroved urther. The lower ound on re is known to e 2(n 1) 2(n 1) re [8] whih we imrove to 2n 1 2 n re or n 6. Beore showing the grhs or whih the imroved lower ound on re holds, we desrie the nested tringles grhs. Dolev et l. irst exhiited the nested tringles grhs in 1984, to otin lower ound on re ( 2n 1) ( 2n 1) or stright-line grid drwings o lne grhs where the outer e is ixed [1]. Let t 1, t 2 e two disjoint -yles in grh G nd Γ e lnr drwing o G. Then t 1 is nested in t 2 in Γ, i t 1 is drwn in the region enlosed y t 2. This reltionshi is shown y t 2 > t 1. We ll lnr grh G t with n verties nested tringles grh i the ollowing () nd () hold: () i n =, then G t is -yle; () i n >, then G t is tringulted lne grh with extly n/ nested tringles suh tht t n/ >... > t 2 > t 1.

21 JGAA, 15(2) (2011) 197 Algorithm Minimum-Are(G) 1: Construt the reresenttive tree T o G 2: or eh vertex i o T n in reorder do : AC i [1,1,1,1,1,1] = Flse 4: end or 5: {The outer verties o G(C i ) re, nd ; re stores the minimum re} 6: re=n 2 7: or eh h rom 2 to n nd eh w rom 2 to min( n2 h, n2 h m ) do 8: or eh vertex i o T n in reorder do 9: x = w, y = h 10: or 1 x w, 1 y h, 1 x w, 1 y h do 11: i AC i [ x, x, x, y, y, y ] = null then 12: Are-Cheking (,,) 1: end i 14: i i = root && AC i [ x, x, x, y, y, y ] = true then 15: i re w h then 16: re=wh 17: end i 18: end i 19: end or 20: x = w, y = h 21: or 1 x w, 1 y h 1, 1 x w, 1 y h do 22: i AC i [ x, x, x, y, y, y ] = null then 2: Are-Cheking (,,) 24: end i 25: i i = root && AC i [ x, x, x, y, y, y ] = true then 26: i re w h then 27: re=wh 28: end i 29: end i 0: end or 1: x = w, y = h 2: or 1 x w, 1 y h 1, 1 x w, 1 y h 1 do : i AC i [ x, x, x, y, y, y ] = null then 4: Are-Cheking (,,) 5: end i 6: i i = root && AC i [ x, x, x, y, y, y ] = true then 7: i re w h then 8: re=wh 9: end i 40: end i 41: end or 42: end or 4: end or

22 198 Mondl et l. Minimum-Are Drwings o Plne -Trees Algorithm 4 Are-Cheking(,,) 1: {,, re outer verties o G nd is the reresenttive vertex} 2: i (mx{ x, x, x } min{ x, x, x } = 0) (mx{ y, y, y } : min{ y, y, y } = 0) then 4: AC [ x, x, x, y, y, y ] = Flse 5: X [ x, x, x, y, y, y ] = Flse 6: Y [ x, x, x, y, y, y ] = Flse 7: else i (mx{ x, x, x } min{ x, x, x } > 1) & (mx{ y, y, y } 8: min{ y, y, y } > 1) & ( is n internl node) then 9: or ll integer oints ( x, y ) inside the tringle with verties,, do 10: i Are-Cheking(,,)& Are-Cheking(,,)& 11: Are-Cheking(,,) then 12: AC [ x, x, x, y, y, y ] = True 1: X [ x, x, x, y, y, y ] = x 14: Y [ x, x, x, y, y, y ] = y 15: rek 16: end i 17: end or 18: else 19: Comute AC [ x, x, x, y, y, y ], X [ x, x, x, y, y, y ] 20: nd Y [ x, x, x, y, y, y ] y Theorem : end i Lemm 12 Let G t e nested tringles grh with n verties nd t = (n/) nested tringles. Then there exists lne -tree G n with n verties suh tht G n ontins n/ nested tringles. Proo. The se t = 1 is trivil sine G 1 is tringle whih is the lne -tree G. So suose tht t > 1 nd the lemm holds or ll nested tringles grhs hving less thn t nested tringles. We delete the three outer verties o G t to get G t 1. By indution hyothesis, there exists lne -tree G n with (n/) 1 nested tringles. Let the outerverties o G n e d, e nd. We ut G n inside tringle {,,} nd dd the edges (,e), (,d), (,), (,), (,), (,e) s shown in Figure 11(). The resulting grh is the required G n i it is lne -tree nd ontins n/ nested tringles. Sine G n is lne -tree, we n delete its interior verties reursively in suh wy tht the resulting grh remins tringulted t eh ste. We n then delete the verties d, e nd one ter nother to otin the tringle {,,}. As illustrted in Figures 11() (d), the deletion o d, e, nd one ter nother kees the resulting grh tringulted t eh ste. Thus we n lwys delete ninnervertexog n insuhwytht tehstethe resultinggrhremins lne -tree; nd hene, G n is lne -tree. Moreover, sine the numer o nested tringles in G n is (n )/, the numer o nested tringles in G n is n/ in totl.

23 JGAA, 15(2) (2011) 199 d e G (n/ * ) 1 G n * () e () () (d) Figure 11: Illustrtion or the roo o Lemm 12. Ft 1 [20] Let Γ e ny lnr drwing o grh G, nd let t 2 nd t 1 e two disjoint -yles o G suh tht t 2 > t 1 in Γ. The height (width) o t 2 in Γ is t lest two units igger thn the height (width) o t 1. We denote y G 6, G 7 nd G 8 the three lne -trees deited y Figures 12(), () nd (), resetively. G 6 G 7 G 8 () () () Figure 12: Drwings o G 6, G 7 nd G 8. Ft 14 The minimum-re stright-line grid drwings or G 6 requires 2 6 or 4 re, G 7 requires 6 re nd G 8 requires 8 or 4 6 re. Proo. We n rove the t y se study or y Algorithm Minimum-Are resented in Setion 5. We now hve the ollowing theorem or the lower ound on re o lne grhs. The roo o the theorem uses G 6, G 7 nd G 8 s the uilding loks or the grhs ttining the lower ound with n 6 verties s illustrted in Figure 12. Note tht when n is multile o three, this ound is the sme s the one y Frti nd Ptrignni [20]. In t, the grh they used s the uilding lok is G 6. Theorem 6.1 For eh n 6, there is n-vertex lne grh G suh tht the re required tootin stright-line grid drwing o G is t lest 2n 1 2 n.

24 200 Mondl et l. Minimum-Are Drwings o Plne -Trees Proo. As n existentil roo, we onstrut lne -trees or whih the lower ound holds. We orm those grhs y enlosing G 6, G 7 nd G 8 with re 4, 6 nd 4 6 in the innermost tringle o G n 6, G n 7 nd G n 8 where n = m, m+1 nd m+2 or m 2, resetively. We enlose the drwings o Figure 12() nd () with re 4 nd 4 6 sine drwings enlosing the lterntive drwings o G 6 nd G 8 will tke the sme or more re. Thereore the new loweround or the rew H ollowsromft 1 14ndLemm 12. ( 2n 5 ) ( 2n+4 ) i n = m+1 nd m 2; W H = ( 2n 4 ) ( 2n+2 ) i n = m+2 nd m 2; ( 2n ) ( 2n ) i n = m nd m 2. It n e esily shown tht or ll n 6, the lower ound or the re o n-vertex lne grh is 2n 1 2 n. We onlude this setion with the onjeture tht or n > 6, the 2n 1 2 n lower ound on the re requirement o lne grhs lso hold or the lss o lne -trees shown in Figure Figure 1: A lss o lne -trees. 7 Conlusion We hve shown tht or ixed lnr emedding o lne -tree G, minimum-re drwing n e otined in olynomil time. Sine lne -tree G hs only liner numer o lnr emeddings, we n omute the re requirements o ll the emeddings o G nd determine the lnr emedding whih gives the est re ound; nd thus we n otin minimum-re drwing o G in olynomil time when the emedding o G is not ixed. Sine the re minimiztion rolem or lne -trees n e solved in olynomil time, it remins oen to investigte whether ny other omuttionlly hrd rolem in the re o grh drwing n e solved in olynomil time or lne -trees. Mny suh rolems yet to e nlyzed n e ound in [, 6, 24]. It is hllenge to ind simler lgorithm or otining minimum-re drwings o lne -trees nd to exlore urther roerties o this sulss o lnr grhs. It is lso let s uture work to ind other lsses o lnr grhs or whih the re minimiztion rolem n e solved in olynomil time. It iswellknowntht ideisionrolemongrhsosmll treewidth n e deined in mondi seond-order logi, there is liner-time lgorithm or testing the rolem [11]. Sine the treewidth o lne -trees is ounded y

25 JGAA, 15(2) (2011) 201 three, it would e interesting to study whether the re minimiztion rolem is deinle in mondi seond-order logi or not. Aknowledgement This work is done under the rojet Minimum-Are Drwings o Plne Grhs, suorted y CASR, BUET, in Grh Drwing & Inormtion Visuliztion Lortory o the Dertment o CSE, BUET estlished under the rojet Fility Ugrdtion or Sustinle Reserh on Grh Drwing & Inormtion Visuliztion suorted y the Ministry o Siene nd Inormtion & Communition Tehnology, Government o Bngldesh. We thnk the nonymous reerees or their useul omments nd suggestions.

26 202 Mondl et l. Minimum-Are Drwings o Plne -Trees Reerenes [1] M. J. Alm, M. A. H. Smee, M. M. Ri, nd M. S. Rhmn. Minimumlyer uwrd drwings o trees. Journl o Grh Algorithms nd Alitions, 14(2): , [2] S. Arnorg nd A. Proskurowski. Cnonil reresenttions o rtil 2- nd -trees. Behviour & Inormtion Tehnology, 2(2): , [] G. Di Bttist, P. Edes, R. Tmssi, nd I. G. Tollis. Grh Drwing: Algorithms or the Visuliztion o Grhs. Prentie Hll, [4] T. Biedl nd L. E. R. Velsquez. Drwing lnr -trees with given e-res. In the 17th Interntionl Symosium on Grh Drwing (GD 2009), volume 5849 o Leture Notes in Comuter Siene, ges Sringer, [5] H. L. Bodlender nd T. Kloks. Eiient nd onstrutive lgorithms or the thwidth nd treewidth o grhs. Journl o Algorithms, 21(2):58 402, [6] F. Brndenurg, D. Estein, M. T. Goodrih, S. Koourov, G. Liott, nd P. Mutzel. Seleted oen rolems in grh drwing. In the 11th Interntionl Symosium on Grh Drwing (GD 200), volume 2912 o Leture Notes in Comuter Siene, ges Sringer, [7] F. J. Brndenurg. Drwing lnr grhs on 8 9 n2 re. In the Interntionl Conerene on Toologil nd Geometri Grh Theory, volume 1 o Eletroni Notes in Disrete Mthemtis, ges Elsevier, [8] M. Chrok nd S. Nkno. Minimum width grid drwings o lne grhs. In the DIMACS Interntionl Worksho on Grh Drwing, volume 894 o Leture Notes in Comuter Siene, ges Sringer, [9] T. H. Cormen, C. E. Leiserson, R. L. Rivest, nd C. Stein. Introdution to Algorithms. The MIT Press, [10] S. Cornelsen, T. Shnk, nd D. Wgner. Drwing grhs on two nd three lines. In the 10th Interntionl Symosium on Grh Drwing(GD 2002), volume 2528 o Leture Notes In Comuter Siene, ges Sringer, [11] B. Courelle. The mondi seond-order logi o grhs. I. Reognizle sets o inite grhs. Inormtion nd Comuttion, 85(1):12 75, [12] H. de Frysseix, J. Ph, nd R. Pollk. How to drw lnr grh on grid. Comintori, 10:41 51, [1] D. Dolev, T. Leighton, nd H. Trikey. Plnr emedding o lnr grhs. Advnes in Comuting Reserh, 2: , 1984.