Proportional Contact Representations of 4-Connected Planar Graphs

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1 Proortionl Contct Reresenttions of -Connected Plnr Grhs Md. Jwherul Alm nd Stehen G. Koourov Dertment of Comuter Science, University of Arizon, Tucson, AZ, USA Astrct. In contct reresenttion of lnr grh, vertices re reresented y interior-disjoint olygons nd two olygons shre non-emty common oundry when the corresonding vertices re djcent. In the weighted version, weight is ssigned to ech vertex nd contct reresenttion is clled roortionl if ech olygon relizes n re roortionl to the vertex weight. In this er we study roortionl contct reresenttions of -connected internlly tringulted lnr grhs. The est known lower nd uer ounds on the olygonl comlexity for such grhs re nd 8, resectively. We nrrow the g etween them y roving the existence of reresenttion with comlexity 6. We then disrove 0-yer old conjecture on the existence of Hmiltonin cnonicl cycle in -connected mximl lnr grh, which lso imlies tht reviously suggested method for constructing roortionl contct reresenttions of comlexity 6 for these grhs will not work. Finlly we rove tht it is NP-hrd to decide whether -connected lnr grh dmits roortionl contct reresenttion using only rectngles. Introduction Contct grh reresenttions for lnr grhs re well-studied lterntive to the trditionl node-link digrm. In most contct grh reresenttions, vertices re reresented y geometric ojects such s circles, tringles nd rectngles, while edges corresond to two ojects touching in some secified fshion. Here we consider contct reresenttions of lnr grhs, with vertices reresented y simle interior-disjoint olygons nd edges reresented y non-emty shred oundries etween corresonding olygons. In the weighted version, the inut is lnr grh G =(V,E) nd weight function w : V R +.Aroortionl contct reresenttion of G is one where ech vertex v is reresented y olygon with w(v) re. When the olygons re mde of only xis-ligned sides, the unweighted nd weighted reresenttions re clled rectiliner dulsnd rectiliner crtogrms, resectively. Contct reresenttions hve rcticl lictions in crtogrhy [5], geogrhy [7], sociology [] nd floor-lnning for VLSI lyout [9]. Other lictions re in visuliztion of reltionl dt, where using the djcency of regions to reresent edges in grh cn led to more comelling visuliztion thn just drwing line segment etween two oints [3]. In this context it is often desirle, for esthetic, Reserch suorted in rt y NSF grnt CCF-597 nd grnt from the Humoldt Foundtion. W. Didimo nd M. Ptrignni (Eds.): GD 0, LNCS 770,. 3, 03. c Sringer-Verlg Berlin Heidelerg 03

2 M.J. Alm nd S.G. Koourov rcticl nd cognitive resons, to limit the olygonl comlexity of the reresenttion, mesured y the mximum numer of sides in olygon. Similrly, it is lso desirle to minimize the unused re, lso known s holes in floor-lnning nd VLSI lyouts. With these considertions in mind, we study the rolem of constructing hole-free roortionl contct reresenttions with miniml olygonl comlexity.. Relted Work Koee s theorem [] is n erly exmle of contct reresenttion, showing tht ny lnr grh cn e reresented y touching circles. Any lnr grh lso hs reresenttion with tringles [7] nd with cues in 3D [6]. However ll these results yield reresenttions tht contin oint-contcts etween djcent olygons. When djcent olygons must shre non-trivil common oundries (lso known s side-contct reresenttion) it hs een shown tht convex hexgons re sometimes necessry nd lwys sufficient []. The rectiliner vrint of the side-contct reresenttion rolem ws first studied in grh theoretic context, nd then with renewed interest in VLSI lyouts nd floor lnning. It is known tht 8 sides re sometimes necessry nd lwys sufficient for rectiliner duls of mximl lnr grhs [0,, 9]. Chrcteriztions for lnr grhs with rectiliner duls of comlexity nd 6 re lso known [3, 6, 8]. While ll the ove results del with the unweighted version of the rolem, the weighted version ws first studied ck in 93 when Risz descried rectngulr crtogrms, i.e., rectiliner crtogrms tht use only rectngles [5]. In the generl rectiliner setting it hs een recently shown tht 8 sides re lwys sufficient for rectiliner crtogrm of mximl lnr grhs [] while it hs lwys een known tht 8 sides re sometimes necessry. It hs een shown tht 7 sides re sometimes necessry nd lwys sufficient if we dro the rectiliner restriction while still reuiring roortionl side-contct reresenttions []. Note tht the reresenttion with 7 sides comes t the exense of mny holes, when comred to the reresenttion with 8 sides. In this er we study roortionl contct reresenttions for -connected internllytringulted lnr grhs. There re severl erlier results on contct reresenttion for this lrge grh clss nd for some suclsses thereof. It is known tht the clss of grhs tht hve rectiliner duls with rectngles nd without ny degree- oints in the reresenttions re exctly the clss of -connected lnr grhs with tringulr internl fces nd non-tringulr outerfce [3, 8]. However, the sme cnnot e sid out roortionl contct reresenttions; there re instnces of -connected lnr grhs with tringulr internl fces nd non-tringulr outerfce tht hve no roortionl contct reresenttions with rectngles for some weight functions. Recently, Estein et l. [5] chrcterized the clss of re-universl rectngulr duls, i.e., rectngulr duls tht cn relize ny secified re for the rectngles. In summry, the est known lower nd uer ounds for hole-free roortionl contct reresenttions of -connected lnr grhs re nd 8, resectively [, ].. Our Contriutions We re interested in nrrowing the g etween the known lower nd uer ounds on the olygonl comlexity in roortionl contct reresenttions for -connected lnr grhs in vrious settings (rectiliner or not, hole-free or with holes). We lso

3 Proortionl Contct Reresenttions of -Connected Plnr Grhs 3 resent some comuttionl comlexity results out nturl recognition rolem, nd disrove 0-yer old conjecture. To summrize: () We descrie n lgorithm for constructing hole-free roortionl contct reresenttion of -connected internlly tringulted lnr grh using 6-sided olygons with ritrrily smll crtogrhic error. We then rove the existence of reresenttion without crtogrhic error. Note tht this result lso imroves the known uer ound on the olygonl comlexity of reresenttions where holes re lso llowed. () We disrove conjecture, osed indeendently y two sets of uthors [, 8], out the existence of Hmiltonin cnonicl order ( cnonicl order tht induces Hmiltonin cycle) in -connected mximl lnr grh. In rticulr, this shows tht method suggested in [] for constructing 6-sided rectiliner roortionl contct reresenttions for -connected grhs will not work. (3) We show tht it is NP-hrd to decide whether -connected lnr grh hs roortionl contct reresenttion with rectngles. Proortionl Contct Reresenttion of Comlexity 6 In this section we give n lgorithm for constructing 6-sided roortionl contct reresenttion of n internlly tringulted -connected lnr grh with ritrrily smll crtogrhic error. With the hel of this lgorithm, we then rove the existence of reresenttion with comlexity 6 nd no crtogrhic error.. Reresenttions with Crtogrhic Error We rove the following min theorem in the rest of this section. Theorem. Let G =(V,E) e n internlly tringulted -connected lne grh nd let w : V R + e weight function on the vertices of G. Then for ny ɛ>0, G hs contct reresenttion where ech vertex v of G is reresented y 6-sided olygon with re in the rnge [w(v),w(v) +ɛ]. Assume first tht the outer fce of G is not tringle. Then G dmits rectngulr dul Γ ( rectiliner dul tht uses only rectngles) due to [3, 8]. We need the following definitions for rectngulr dul Γ of G. The grh G is the dul of Γ. Since ech olygon in Γ is rectngle, the djcency etween two rectngles in Γ reresenting n edge of G cn occur in one of two wys: (i) through shred horizontl segment (horizontl djcency), or (ii) through shred verticl segment (verticl djcency). We sy Γ is toologiclly euivlent to nother rectngulr lyout Γ when oth Γ nd Γ hve the sme dul grh G nd ech edge of G is reresented y the sme tye of djcency (horizontl or verticl) in oth Γ nd Γ.Aline-segment in Γ is the union of inner edges of Γ forming consecutive rt of stright-line. A line-segment not contined in ny other line-segment is mximl. A mximl line-segment s is clled one-sided if it forms full side of t lest one rectngulr fce, or in other words, if the Crtogrhic error in reresenttion of grh G is the mximum over the vertices v in G of the vlue A(v) w(v),wherea(v) is the re for v nd w(v) is its weight.

4 M.J. Alm nd S.G. Koourov Fig.. Illustrtion for the roof of Lemm erendiculr line segments tht ttch to its interior re ll on one side of s.otherwise, s is two-sided. Estein et l. roved tht if ll the mximl segments in rectngulr dul Γ re one-sided then Γ is re-universl, which mens tht ny distriution of res to the rectngles in Γ cn e relized with toologiclly euivlent lyout [5]. Unfortuntely, not every internlly tringulted -connected lne grh hs rectngulr dul tht is lso re-universl. With the next lemm we cn rove slightly weker sttement which cn hel us reduce the olygonl comlexity. Secificlly, we cn show tht for ny such grh with non-tringle outerfce, there exists rectngulr dul where ll two-sided mximl segments re horizontl. Lemm. Let G e n internlly tringulted -connected lne grh with nontringle outerfce. Then G hs rectngulr dul with no verticl two-sided segment. Proof Sketch. We strt y comuting (ossily two-sided) rectngulr dul Γ of G [3]. If G is one-sided or hs only horizontl -sided mximl segments we re done. Let s e verticl mximl segment in Γ. Cll every degree-3 oint on s junction on s. Ifs is not one-sided, then going from ottom to to long s there will e t lest one of the two configurtions in Fig.. In oth cses, we modify the lyout loclly, s illustrted in Fig.. If we reetedly ly this oertion for ech verticl two-sided segment in Γ, there will e no more verticl two-sided segment. Once we hve rectngulr dul of lnr grh G with no two-sided verticl mximl segments, we modify the reresenttion into contct reresenttion with 6-sided olygons to relize ny set of weights on the vertices of G, t the exense of ɛ-crtogrhic error, ɛ>0. Lemm. Let G =(V,E) e n internlly tringulted lne grh nd let Γ e rectngulr dul of G such tht Γ contins no verticl two-sided mximl segment. Then G dmits contct reresenttion Λ such tht ech vertex v of G is reresented y olygon of comlexity t most 6 with re in the rnge [w(v),w(v) +ɛ], where w : V R + is n ritrry weight function nd ɛ>0. Proof Sketch. If Γ contins no two-sided mximl segment, then it cn relize ny weight function [5] nd we re done. Otherwise, for ech horizontl two-sided mximl segment s, we relce s y rectngle with smll height (< the minimum feture size of Γ ) nd with horizontl sn sme s s; see Fig. () (). It is esy to see tht These oertions re well-known s flis in much of the relted work; see [9] for exmle.

5 Proortionl Contct Reresenttions of -Connected Plnr Grhs 5 s l l l l l l () () (c) (d) Fig.. Illustrtion for the roof of Lemm () () (c) (d) (e) (f) Fig. 3. Illustrtion of the construction with 6-sided olygons nd ɛ-crtogrhic error this modifiction mkes ll mximl segments one-sided, which mkes the resulting reresenttion Γ re-universl. Let Γ e rectngulr lyout, where the re of ech newly formed rectngle is ɛ nd the res for ll other rectngles relize the weight function w; see Fig. (c). Suose s ws horizontl two-sided segment in Γ nd let l nd l e the to nd ottom side of the corresonding rectngle in Γ (lso in Γ ). We select some oints on l corresonding to ll the junctions on l so tht the order of ll these junctions defined y s is resected. We then dd n edge from ech junction on l to its corresonding oint on l. In this wy the re of the rectngle defined y l nd l which hd re ɛ is distriuted mong the rectngles ove l. For ech rectngle t most two dditionl corners re thus dded mking it olygon with t most 6 sides. We re now redy to rove Theorem. Proof of Theorem. If the outerfce of G is not tringle, then y Lemm nd Lemm, G dmits desired reresenttion. We thus ssume tht the outerfce of G is tringle c.thengdmits no rectngulr dul. However, G dmits rectiliner dul where one of the outer vertices, sy, is reresented y 6-sided L-shed rectiliner olygon nd ll other vertices re reresented y rectngles [6]. Furthermore, the reresenttion is contined inside rectngle nd so is the union of the rectngles reresenting ll vertices of G = G {}. We then otin contct reresenttion of G with 6-sided olygons y Lemm nd Lemm. Since the oundry of the rectngulr dul of G is not chnged, we cn still dd the L-shed olygon for round it with the desired re nd correct djcencies.

6 6 M.J. Alm nd S.G. Koourov Figure 3 illustrtes the construction of contct reresenttion of -connected lne grh G with 6-sided olygons using the ove rocedure.. Reresenttions without Crtogrhic Error Here we rove tht n internlly tringulted -connected lne grh hs roortionl contct reresenttion of comlexity 6 with no crtogrhic error for ny weight function. We egin with reresenttion tht hs ɛ-crtogrhic error nd rgue tht it cn e modified so s to remove the errors, while reserving the toology of the lyout. Let G =(V,E) e grh with weight function w : V R +. Consider vertex order v,v,...,v n of the vertices of G. LetΛ = λ,λ,...,λ n e list of n nonnegtive rel numers.then the Λ-vicinity of w is the set of ll weight functions w : V R + such tht w(v i ) w (v i ) λ i for ech vertex v i of G. Ifλ = λ =... = λ n = λ, then the Λ-vicinity of w is lso clled the λ-vicinity of w. Wehvethe following lemm, whose roof is ommitted in this extended strct. Lemm 3. Let G =(V,E) e n internlly tringulted lne grh nd let Γ e rectngulr crtogrm of G for weight function w : V R + where Γ contins no verticl two-sided segment. Then there exists sufficiently smll λ>0 such tht for ny weight function w : V R + in the λ-vicinity of w, there is rectngulr crtogrm Γ (w ) of G with resect to w,whereγ (w ) is toologiclly euivlent to Γ. Using the ove lemm, we cn rove the existence of crtogrm for internlly tringulted -connected grh without ny crtogrhic error. Theorem. Let G =(V,E) e n internlly tringulted -connected lne grh nd let w : V R + e weight function on the vertices of G. ThenG hs roortionl contct reresenttion of comlexity 6 with resect to w. Proof. Here we use the lgorithm from Theorem. We ssume tht G hs non-tringle outerfce since the cse with tringle outerfce cn e hndled in the sme wy s in Theorem. Let Γ ereresenttionofg otined y this lgorithm. Ech olyline etween two horizontl segments in Γ consists of two segments: verticl segment followed y segment with n ritrry sloe. Cll ech such olyline verticl -line nd cll the common oint etween the two segments of verticl -line ivot oint. For exmle, in Fig. (d), there re three such verticl -lines. Ech ivot oint cn e moved u or down, resulting in the increse or decrese of the res of its djcent olygons. We use this flexiility to find reresenttion without crtogrhic error. Let Γ 0 denote the rectngulr lyout when we set ɛ =0, where ech ivot oint hs the sme y-coordinte s the ottommost oint of its verticl -line. Note tht Γ 0 my no longer reresent G ecuse the djcencies etween rectngles on oosite sides of horizontl segment my chnge. We now modify the weights for the rectngles in Γ 0 so tht the totl weight for ll rectngles remins the sme nd we cn move the ivot oints to correct the error creted y this chnge. We choose the new weight function w to e in the λ-vicinity of w,forsomeλ>0, y lying Lemm 3, so tht we cn find rectngulr crtogrm Γ toologiclly euivlent to Γ 0. We set the width nd height of

7 Proortionl Contct Reresenttions of -Connected Plnr Grhs 7 LL L C R RR xl xr xl 3x + xr x +3x x () () Fig.. Illustrtion for Theorem l l r r this lyout to e B nd H,whereB H = v V w(v).thenb min =(w min + λ)/h nd H min =(w min + λ)/b re the minimum width nd height of rectngle in Γ, resectively, where w min = min v V w(v). We now define the weights for ll the rectngles ove ech two-sided horizontl mximl segment s s follows. Let us consider the set S of ll the verticl -lines incident to the to-side of rticulr rectngle R. Fig. illustrtes set of such verticl -lines where for convenience of visuliztion the ivot oints nd the ottommost oints re lced t different y-coordintes. Let x l nd x r e the x-coordintes of the left nd right side of the rectngle to the ottom of the horizontl segment. In the finl reresenttion with 6-sided olygons, we wnt the ottommost oint of ech of these verticl -lines to e lced etween one-fourth nd three-fourth of its horizontl sn; i.e., etween (3x l + x r )/ nd (x l +3x r )/. In this resect we rtition the verticl -lines in S into five clsses s follows, deending on the x-coordintes of the ivot oints; see Fig. () (ivot oints re highlighted y lck dots). (i) LL-lines: ivot oints re to the left of x l (ii) L-lines: ivot oints re etween x l nd (3x l + x r )/ (iii) C-lines: ivot oints re etween (3x l + x r )/ nd (x l +3x r )/ (iv) R-lines: ivot oints re etween (x l +3x r )/ nd x r (v) RR-lines: ivot oints re to the right of x r By left-to-right scn, we set the weight for ll the rectngles to the left of LL-line nd L-line in S. For ech LL-line, we set the weight of the rectngle to its left such tht the totl weight for ll the rectngles to its left is decresed y n mount ɛ>0 where ɛ<λnd ɛ<(b min H min )/8. For ech L-line, we set the weight of the rectngle to its left so tht the weight of ll rectngles to its left is decresed y ɛ>0 where ɛ<λnd ɛ<(b min H min )/(8 S ). Similrly y right-to-left scn, we set the weight for ech the rectngle to the left of RR-line (res. R-line) so tht the weight of ll rectngles to the left of the olyline is incresed y ɛ>0 where ɛ<λnd ɛ<(b min H min )/8 (res. ɛ<(b min H min )/(8 S )). For ech C-line, we set the weight of the rectngle to its left so tht the totl weight of ll the rectngles to its left remins the sme. Once we comute ɛ for ll the verticl -lines, we cn clculte the exct weights to e ssigned to ech rectngle in S. By Lemm 3, we then find rectngulr crtogrm Γ with the new weight function such tht Γ is toologiclly

8 8 M.J. Alm nd S.G. Koourov euivlent to Γ. By shifting the ivot oints u s needed, we relize exct weights for ech rectngle. By the weight distriution, it is never reuired to shift ny ivot oint more thn H min distnce, the minimum verticl distnce etween two horizontl segments. By choosing ɛ smll enough, we mke sure tht the segments etween the ivot oints nd ottommost oints of verticl -lines do not cross. 3 Hmiltonin Cnonicl Cycles Let G =(V,E) e mximl lne grh with outer vertices u, v, w in clockwise order. A cnonicl order of the vertices v = u, v = v, v 3,..., v n = w of G, is one tht meets the following criteri for every i n, whereg i is the sugrh of G induced y the vertices v, v,..., v i : G i G is iconnected, nd its outer-cycle C i contins the edge (u, v). vertex v i is in the outerfce of G i, nd its neighors in G i form n (t lest -element) suintervl of the th C i (u, v). A Hmiltonin Cnonicl Cycle in mximl lnr grh G is cnonicl order v, v,..., v n of the vertices of G such tht v v...v n is lso Hmiltonin cycle of G. Whether every -connected mximl lnr grh hs Hmiltonin cnonicl cycle is uestion sked t lest two times y two sets of uthors in different contexts [, 8]. In fct, n lgorithm in [] roduces 6-sided rectiliner roortionl contct reresenttion of ny mximl lnr grh tht hs Hmiltonin cnonicl cycle. If the ove conjecture were true, tht would suffice to lower the current est known uer ound on the olygonl comlexity for rectiliner crtogrms of -connected mximl lnr grhs from 8 sides to 6 sides. Unfortuntely, we show tht the conjecture is not true y constructing -connected mximl lnr grh with no Hmiltonin cnonicl cycle. Theorem 3. There exist -connected mximl lnr grhs tht do not hve ny Hmiltonin cnonicl cycle in ny emedding. To rove this clim we construct -connected grh G where there re two internl fces of length nd the remining fces (including the outerfce) re tringles. We ut one isomorhic coy of the grh K in Fig. 5 inside ech of the fces of length in G (so tht the four vertices on the fces re suerimosed with,, c nd d). In ny emedding of this grh, t lest one coy of K will retin its emedding. Thus in order to rove Theorem 3, it suffices to rove tht there is no Hmiltonin cnonicl cycle of ny lne grh tht contins n isomorhic coy of K with the fixed emedding. Lemm. Let G e mximl lne grh with n isomorhic coy of the grh K in Fig. 5 s n emedded sugrh. Then G hs no Hmiltonin cnonicl cycle. The grh K is very symmetric nd it contins four isomorhic coies of the grh L in Fig. 6. Lemm 5 shows tht ny Hmiltonin cnonicl cycle of K must follow restricted th inside L. This cn e used to rove Lemm. Let C =v, v,..., v n e Hmiltonin cnonicl cycle of mximl lnr grh G. SinceC induces cnonicl order, we cn consider it s directed cycle where the edge (v i,v i+ ) is directed towrds v i+ for i n nd the edge (v n,v ) is directed towrds v. This lso induces direction for ny suth of C.

9 Proortionl Contct Reresenttions of -Connected Plnr Grhs 9 d L 3 h r g L s o c L e f L Fig. 5. The grh K used in Theorem 3 nd Lemm Lemm 5. Let G e mximl lne grh tht contins n isomorhic coy of grh L of Fig. 6 s n emedded sugrh. Then there exists no Hmiltonin cnonicl cycle C of G such tht it contins suth P where: (i) the first two vertices on P re from the set {A, B}, (ii) the third vertex on P is lso from the sugrh L, nd (iii) either the lst vertex of P is Z nd F is fter Z in C or the lst vertex of P is F. Proof Sketch. Assume for contrdiction tht there exists Hmiltonin cnonicl cycle C of G with suth P such tht the conditions (i) (iii) hold. Denote the vertex set of L y S. First note tht the vertex set T = {A, B, Z, F } defines serting cycle of G.LetL e the grh induced y S T.ThenP cn enter nd exit L only once. The first two vertices on P re A nd B. Without loss of generlity, is the third vertex. Since C is the only vertex for P to go etween vertices on different sides of the olyline BCF, C must er on P fter ll the vertices from the left of BCF hve ered. Then since C is Hmiltonin cycle s well s cnonicl order, creful oservtion shows tht the initil suth of P is the one drwn y the thick lck olyline, followed y the one drwn y the dotted grey olyline in Fig. 6. On the other hnd, the suth of P ending t C must e the one drwn y the thick gry solid olyline. However there is no edge to go etween these two suths, contrdiction. We re now redy to rove Lemm. We re only giving the roof sketch here. Proof Sketch of Lemm. Assume for contrdiction tht G hs Hmiltonin cnonicl cycle C. For ny emedded sugrh H of G ounded y serting cycle C H, cll the sugrh induced y the vertices of H C H the inside of H. SinceC is cnonicl order, withour loss of generlity the first vertex on C inside K is e. Then oth nd er efore e nd C cn enter nd exit the inside of K only once nd either c or d is the only exit vertex. Cll the suth of C etween the entry nd the exit vertex P. Assume due to symmetry tht P enters in the inside of L fter e. Since either c or d is the exit vertex, P must enter nd exit the inside of L through s nd h, resectively

10 0 M.J. Alm nd S.G. Koourov F C B A Z Fig. 6. The grh L used in Lemms nd 5 nd o must immeditely receed s on P. Looking ck t L, in cse P goes from either or f to o, it must go from h to the inside of L 3 nd eventully exit the inside of K through c. However, this is not ossile y Lemm 5. Thus P must visit ll the vertices of L nd exit through f nd go vi the inside of L,toc. Since,, c hve ll een lredy visited y P, it must then exit though d from the inside of K; which is lso not ossile due to Lemm 5. The roof of Theorem 3 follows from Lemm since we could construct grh tht contins n isomorhoc coy of the grh K s n emedded grh. NP-Hrdness for Rectngulr Reresenttions Here we consider the following rolem. Given -connected lne grh G =(V,E) with tringle nd udrngle internl fces nd non-tringle outerfce nd weight function w : V R +, we wnt to determine whether G hs rectngulr crtogrm with resect to w. Let us cll the rolem RectngleCrtogrm (RC). We now show tht this rolem is NP-hrd y reduction from the well-known NP-hrd rolem Prtition defined s follows. Given set of integers S = {x,..., x n } with n i= x n = A for some integer A, we wnt to find suset I of S such tht x x i I i = A. Given n instnce of Prtition, we construct n instnce of RC s follows. For ech integer x i of S, we hve sugrh with eight vertices: X i, i, i+, i, i+, i, i, c i. We highlight such sugrh for x i in Fig. 7. The constructed grh is then -connected with nontringle outerfce where ech internl fce is tringle or udrngle. We define the weight function s follows. Define m = min n i= x n. For ech vertex X i,wedefinew(x i )=x i. We give very smll weight, sy δ = m/0 to ech vertex i, i, c i, i, i for i n nd to ech vertex L, R, T nd B. Wegiveverylrge weight W (M) to the vertex M such tht W (M)+A +(n +)δ W (M) < m. Finlly we give weights to the vertices t nd t such tht w(t ) : w(t ) = W (M)+A +(n +)δ W (M) : W (M). We now hve the following lemm nd we only sketch its roof here.

11 Proortionl Contct Reresenttions of -Connected Plnr Grhs T t t L X X 3 i Xi i i i+ n Xn n n n+ M c c 3 i c i i+ n c n n+ B R Fig. 7. The grh constructed from n instnce of Prtition rolem Lemm 6. There exists suset I of S such tht x i I x i = A if nd only if there is rectngulr crtogrm of G with resect to the weight function w. Proof Sketch. Suose first tht there exists crtogrm Γ of G with resect to w.then the rectngles for L, T, R, B, t nd t define rectngulr frme F inside which lie the remining rectngles. The left nd ottom of the rectngle for M define two lines l v nd l h nd ech sugrh for x i hs drwing in one of two configurtions; see Fig. 8. Then the vertices corresonding to the rectngles X i lying to the left of l v form the suset I of S. Conversely if we re given suset I of S such tht x i I x i = A,then M M X i l h l h l v l v X i () () Fig. 8. Illustrtion of the roof of Lemm 6 we first drw the rectngles for L, T, R, B, t nd t such tht they enclose sure frme F of size W (M) +A +(n +)δ. We then drw the sugrhs for x i s in Fig. 8() if x i I; otherwise we drw this sugrh s in Fig. 8(). We cn thus reduce n instnce S of Prolem Prtition to n instnce (G, w) of Prolem RectngleCrtogrm such tht S hs solution if nd only if G hs rectiliner crtogrmwith resect to w. This yields the following theorem. Theorem. Prolem RectngleCrtogrm is NP-hrd.

12 M.J. Alm nd S.G. Koourov 5 Conclusions nd Oen Prolems We ddressed the rolem of roortionl contct reresenttion of -connected internlly tringulted lnr grhs nd showed tht non-rectiliner olygons with comlexity 6 re sufficient. Nrrowing the g etween this uer ound nd the currently est known lower ound of remins oen. With the dditionl restriction of using rectiliner olygons, there re instnces where comlexity 6 is reuired nd here we showed tht it is NP-hrd to decide whether -connected lnr grh with tringle nd udrngle internl fces dmits reresenttion with rectngles. It will e interesting to investigte the comlexity of the sme rolem for internlly tringulted grhs. On the other hnd, the currently est known uer nd lower ounds for rectiliner crtogrms of -connected internlly tringulted lnr grhs re 8 nd 6, resectively. Thus to find out whether the true ound in this cse is 6 or 8 is oen. Acknowledgments. We thnk Torsten Ueckerdt nd Michel Kufmnn for discussions out the rolem. References. Alm, M.J., Biedl, T., Felsner, S., Kufmnn, M., Koourov, S.G.: Proortionl Contct Reresenttions of Plnr Grhs. In: vn Kreveld, M., Seckmnn, B. (eds.) GD 0. LNCS, vol. 703, Sringer, Heidelerg (0). Alm, M.J., Biedl, T.C., Felsner, S., Kufmnn, M., Koourov, S.G., Ueckerdt, T.: Comuting crtogrms with otiml comlexity. In: Symosium on Comuttionl Geometry, SoCG 0,. 30 (0) 3. Buchsum, A.L., Gnsner, E.R., Procoiuc, C.M., Venktsurmnin, S.: Rectngulr lyouts nd contct grhs. ACM Trnsctions on Algorithms () (008). Duncn, C.A., Gnsner, E.R., Hu, Y.F., Kufmnn, M., Koourov, S.G.: Otiml olygonl reresenttion of lnr grhs. Algorithmic 63(3), (0) 5. Estein, D., Mumford, E., Seckmnn, B., Vereek, K.: Are-universl nd constrined rectngulr lyouts. SIAM Journl on Comuting (3), (0) 6. Felsner, S., Frncis, M.C.: Contct reresenttions of lnr grhs with cues. In: Symosium on Comuttionl Geometry, SoCG 0, (0) 7. de Frysseix, H., de Mendez, P.O., Rosenstiehl, P.: On tringle contct grhs. Comintorics, Proility nd Comuting 3, 33 6 (99) 8. de Frysseix, H., de Mendez, P.O.: On toologicl sects of orienttions. Discrete Mthemtics 9(-3), 57 7 (00) 9. Fusy, É.: Trnsversl structures on tringultions: A comintoril study nd stright-line drwings. Discrete Mthemtics 309(7), (009) 0. He, X.: On floor-ln of lne grhs. SIAM Journl on Comuting 8(6), (999). House, D.H., Kocmoud, C.J.: Continuous crtogrm construction. In: IEEE Visuliztion, (998). Koee, P.: Kontktroleme der konformen Aildung. Berichte üer die Verhndlungen der Sächsischen Akdemie der Wissenschften zu Leizig. Mth.-Phys. Klsse 88, 6 (936) 3. Koźmiński, K., Kinnen, E.: Rectngulr duls of lnr grhs. Networks 5, 5 57 (985)

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