On Alliance Partitions and Bisection Width for Planar Graphs

Transcription

1 Joural of Graph Algorithms ad Applicatios vol. 17, o. 6, pp (013) DOI: /jgaa O Alliace Partitios ad Bisectio Width for Plaar Graphs Marti Olse 1 Morte Revsbæk 1 AU Herig Aarhus Uiversity, Demark MADALGO, Departmet of Computer Sciece Aarhus Uiversity, Demark Abstract A alliace i a graph is a set of vertices (allies) such that each vertex i the alliace has at least as may allies (coutig the vertex itself) as o-allies i its eighborhood of the graph. We show how to costruct ifiitely may o-trivial examples of graphs that ca ot be partitioed ito alliaces ad we show that ay plaar graph with miimum degree at least 4 ca be split ito two alliaces i polyomial time. We base this o a proof of a upper boud of o the bisectio width for 4-coected plaar graphs with a odd umber of vertices. This improves a recetly published + 1 upper boud o the bisectio width of plaar graphs without separatig triagles ad supports the folklore cojecture that a geeral upper boud of exists for the bisectio width of plaar graphs. Submitted: May 013 Reviewed: September 013 Article type: Regular paper Revised: October 013 Published: November 013 Accepted: October 013 Commuicated by: S. K. Ghosh Fial: October 013 addresses: martio@hih.au.dk (Marti Olse) mrevs@madalgo.au.dk (Morte Revsbæk) Ceter for Massive Data Algorithmics, a ceter of the Daish Natioal Research Foudatio.

2 600 Olse ad Revsbæk O Alliace Partitios ad Bisectio Width for Plaar Graphs 1 Itroductio A alliace is a set of vertices (allies) such that ay vertex i the alliace has at least as may allies (icludig the vertex itself) as o-allies i its eighborhood of the graph. The alliace is said to be strog if this holds eve without icludig the vertex itself amog the allies. Alliaces of vertices i graphs [13] are used to model amog other thigs alliaces of idividuals or atios ad alliaces appear may places i the literature uder various ames: Flake et al. [9] refer to a strog alliace as a commuity ad base their work o the assumptio that web pages related to each other form commuities i the web graph. Gerber ad Kobler [11] look at what they refer to as the Satisfactory Graph Partitio Problem where the objective is to partitio a graph ito two strog alliaces. A partitio of a graph ito strog alliaces ca also be viewed as a so called Nash stable partitio of a Additive Hedoic Game [15]. As metioed above, alliaces have bee used to model scearios that might be plaar i ature, so i this paper we focus o the problem of partitioig a plaar graph ito two alliaces. I Sectio we show how to compute such a partitio i polyomial time for ay plaar graph with miimum degree at least 4. To prove this, we eed a upper boud of o the bisectio width of 4-coected plaar graphs with a odd umber of vertices. We prove this upper boud i Sectio 3. This tight upper boud is a improvemet over the recetly published [8] + 1 upper boud for plaar graphs without separatig triagles, ad it supports the folklore cojecture [8], that a geeral upper boud of exists for the bisectio width of plaar graphs. 1.1 Prelimiaries Cosider the coected graph G with vertex set V ad edge set E where V = ad E = m. The degree d(v) of a vertex v i G is the umber of edges icidet to v i G. Similarly, for a subset X V we defie the degree d X (v) of a vertex v i the subgraph of G iduced by X {v} as d X (v) = {u X : {v, u} E}. We deote the miimum degree of the vertices i G as δ ad the maximum degree as. A graph G is k-coected whe at least k vertices are required to be removed i order to discoect G. A d-regular graph is a graph where all vertices have degree d. A clique is a fully coected graph ad a maximal plaar graph is a plaar graph with the property that the additio of ay ew edge destroys plaarity. A alliace i G is a o empty set A V such that u A : d A (u) + 1 d V A (u). This defiitio ca be tighteed givig the otio of a strog alliace which is a o empty set A V such that u A : d A (u) d V A (u). A partitio of G is a collectio of o-empty disjoit subsets V 1... V l of V such that l i=1 V i = V. For a partitio of G ito two subsets V 1 ad V we will deote the set of edges crossig this partitio as e(v 1, V ) = {{u, v} E : u V 1 v V }. A bisectio of G is a partitio of G ito V 1 ad V such that V 1 V 1 ad the bisectio width of G is defied as the miimum e(v 1, V ) over all bisectios. Throughout this paper whe cosiderig a plaar graph, we will implicitly cosider a embeddig of

3 JGAA, 17(6) (013) 601 the graph. A separatig triagle i a plaar graph is a triagle where both the iterior ad the exterior are o-empty. 1. Related Work The problem of partitioig a graph ito two strog alliaces is NP-hard if we put o restrictios o the graph []. There are however classes of graphs for which we ca decide whether a partitio ito strog alliaces exists ad compute it i polyomial time. Examples of such classes are graphs with maximum degree at most 4 ad graphs with girth at least 5 ad miimum degree at least 3 [, 3]. Shafique [1] presets amog other thigs results tellig precisely whe lie graphs ca be partitioed ito alliaces. For a geeral graph G, the computatioal complexity of partitioig G ito two alliaces is a ope problem [4]. Fricke et al. [10] show that ay graph G cotais a efficietly computable alliace with o more tha vertices, while the problem of decidig whether a alliace with less tha k members exists i G is NP-complete if k is part of the iput. This eve holds if G is plaar [7]. Fa et al. [8] prove a upper boud of + 1 for the bisectio width for plaar graphs without a separatig triagle ad a upper boud o for the bisectio width for ay triagle-free plaar graph. The latter upper boud has subsequetly bee improved by Li et al. [14] for triagle-free plaar graphs. Diks et al. [6] show how to obtai a partitio V 1, V with mi( V 1, V ) 3 ad o more tha crossig edges i liear time for ay plaar graph. Alliaces i Plaar Graphs Our mai aim of this sectio is to show that a partitio ito two alliaces exists for ay plaar graph with miimum degree at least 4 ad that this partitio ca be computed i polyomial time. Before doig this i Sectio. we preset more geeric results o alliace partitios where we amog other thigs describe o-trivial ifiite families of graphs that ca ot be partitioed ito alliaces..1 Geeric Observatios o Alliace Partitios If all the vertices i a graph have a odd degree the the graph ca be efficietly partitioed ito two alliaces by usig the decompositio techique by Bazga et al. [3]. O the other had, it is easy to see that ay clique with a odd umber of vertices ca ot be partitioed ito alliaces. Ispired by this fact we ow preset a observatio cotaiig a recipe for costructig ifiitely may less trivial examples of graphs that caot be partitioed ito alliaces: Observatio 1 Let G(V, E) be a graph ad let B 1, B,..., B l be a partitio of V for some l. The graph G ca ot be partitioed ito alliaces if the followig coditios hold: B 1 B is odd (1)

4 60 Olse ad Revsbæk O Alliace Partitios ad Bisectio Width for Plaar Graphs u B 1 : d(u) = d B1 B (u) = B 1 B 1 () i u B i : d Bi 1 (u) > d V \Bi 1 (u) + 1 (3) Proof: Now assume that V 1, V is a partitio of V ito alliaces. The coditios (1) ad () imply that B 1 must be fully cotaied i V 1 or i V otherwise there would be at least oe member of B 1 with too few allies sice B 1 B is odd. The coditio (3) implies that all vertices i B i are members of the alliace cotaiig B i 1 for i ad we arrive at a cotradictio sice this leaves us with a empty alliace. Figure (1a) shows a example of a graph satisfyig the coditios of Observatio 1 for l = 3 where the B-sets are idicated by the differet colors. Bazga et al. [4] ote that K 3,3,3 show i Figure (1b) ca ot be partitioed ito alliaces. The followig observatio shows amog other thigs how to costruct a d-regular graph that is ot a clique ad impossible to partitio ito alliaces for ay eve d 6. As a example of such a graph we metio the 8-regular graph with 11 vertices defied by havig a complemet graph cosistig of two triagles ad a 5-cycle. Observatio Let G(V, E) be a regular graph with a odd umber of vertices ad d(u) = 3 for all u V. The graph G ca be partitioed ito two alliaces if ad oly if G cotais a clique with vertices. Checkig whether such a clique exists ca be doe i polyomial time. Proof: If G cotais a clique C V with vertices the this clique is a alliace. The complemet graph Ḡ is -regular so there are = 1 edges i Ḡ with exactly oe edpoit i C ad cosequetly oe edge i Ḡ with both edpoits i V \ C. We therefore coclude that V \ C is a alliace as well. Now let us o the other had assume that G does ot cotai a clique with vertices. I this case it is ot hard to see that we ca ot have a alliace with or fewer vertices. A clique is a idepedet set i the complemet graph ad it is easy to fid the size of the biggest idepedet set i a -regular graph i polyomial time. We ow tur our attetio to graphs for which alliace partitios exist. I Lemma 1 we characterize a group of geeral graph partitios that ca be refied ito a alliace partitio usig a polyomial time algorithm. I Sectio. we will use the lemma to obtai a partitio ito alliaces for plaar graphs with miimum degree 4 i polyomial time. Lemma 1 is a precise formulatio of the well kow priciple [4, 10] that a partitio ito two sets of vertices forms a good startig poit for obtaiig a partitio ito alliaces if the umber of crossig edges is relatively small compared to the cardiality of the smallest set of vertices.

5 JGAA, 17(6) (013) 603 (a) A graph satisfyig the coditios of Observatio 1 for l = 3 where the B-sets are idicated by the differet colors. (b) The graph K 3,3,3. Figure 1: Examples of graphs that ca ot be partitioed ito alliaces. Lemma 1 A graph G ca be partitioed ito two alliaces if there exists a partitio V 1, V of G such that e(v 1, V ) mi( V 1, V ) < δ. (4) The alliaces ca be computed i polyomial time if V 1 ad V ca be obtaied i polyomial time. Proof: Let V 1, V be a partitio of G satisfyig (4). We ow ru the followig simple algorithm: 1. Let A 1 = V 1 ad A = V.. If A 1 ad A both are alliaces or if oe of them is empty we stop. Otherwise we go to step Assume that A 1 is ot a alliace (otherwise we process A similarly). There must be a u A 1 with d A1 (u) + 1 < d A (u). We ow move u from A 1 to A ad go to step. The umber of crossig edges e(a 1, A ) decreases with or more every time step 3 is executed so the algorithm must stop after o more tha m steps. Assume that the algorithm stops because A 1 is empty ad let u be the last vertex to leave A 1. We ow cosider the poit i time where A 1 = {u}: d V (u) = e(a 1, A ) e(v 1, V ) (mi( V 1, V ) 1). We obtai a cotradictio sice (4) implies that the right had side is less tha δ. We coclude that the algorithm caot stop with A 1 or A beig empty. It has to stop with A 1 ad A beig alliaces.

6 604 Olse ad Revsbæk O Alliace Partitios ad Bisectio Width for Plaar Graphs For oe applicatio of this lemma we cosider some sparse graphs: The expected umber of e(v 1, V ) is slightly bigger tha m for a radom bisectio if is big. So m just has to be slightly less tha to esure a partitio ito alliaces computable i polyomial time usig deradomizatio. If a big graph has average degree a little below 4 it ca thus be partitioed ito alliaces. We omit the techical details o this observatio i this paper.. Alliace Partitios of Plaar Graphs We ow prove that all plaar graphs with miimum degree at least 4 allow a partitio of the vertices ito two alliaces ad that this partitio ca be computed i polyomial time. This is trivially also true for plaar graphs with miimum degree 1 (let oe alliace cosist of a sigle vertex with degree 1), while for plaar graphs with miimum degree ad 3 we ca use Observatio 1 ad costruct examples of graphs that ca ot be partitioed ito alliaces. Figure shows two examples of such graphs where the black vertices ad the gray vertices form the sets B 1 ad B from Observatio 1 respectively. It is worth otig that we do ot guaratee the alliace partitio to be a bisectio. (a) (b) Figure : Examples of plaar graphs that ca ot be partitioed ito alliaces. The black vertices ad the gray vertices form the sets B 1 ad B from Observatio 1 respectively. We are ow ready to state ad prove the mai result of this sectio. Theorem 1 Ay plaar graph with δ 4 ca be partitioed ito two alliaces i polyomial time. Proof: We start by expadig the graph by addig edges util it is a maximal plaar graph which ca be doe i polyomial time. We ow cosider two cases: The expaded graph has a separatig triagle: A separatig triagle has vertices both iside ad outside of the triagle. Let V 1 be the vertices o the side of the triagle cotaiig the fewest vertices ad let V = V \V 1. There ca be o more tha oe vertex i V 1 havig edges to all three vertices i the separatig triagle so e(v 1, V ) V This iequality also holds i the origial graph so we ca ow use Lemma 1. The detectio ad processig of the separatig triagle case is easily doe i polyomial time.

7 JGAA, 17(6) (013) 605 The expaded graph does ot have a separatig triagle: I this case the graph is 4-coected sice all maximal plaar graphs without a separatig triagle are 4-coected [5] ad thus cotais a hamiltoia cycle computable i liear time [1]. Fa et al. [8] show how to efficietly compute a bisectio V 1, V of V with e(v 1, V ) + 1 for such a graph. This makes it possible for us to apply Lemma 1 i the case where is eve but for odd a upper boud o for the bisectio width is eeded to make iequality (4) hold. I Sectio 3 we prove Theorem 3 statig the existece of a efficietly computable bisectio V 1, V with e(v 1, V ) for ay 4-coected plaar graph G(V, E) with a odd umber of vertices. We ow use Lemma 1 i the case where is odd. As metioed above, Fricke et al. [10] have show that ay graph cotais a alliace with o more tha members. We ca ow improve this upper boud for plaar graphs with δ 4 sice such graphs ca be partitioed ito two alliaces that caot both have more tha members: Corollary 1 Ay plaar graph with δ 4 cotais a alliace with o more tha members. A plaar graph with < 1 18 ca be partitioed ito two alliaces i polyomial time. Theorem A plaar graph ca be partitioed ito two alliaces i polyomial time if < Proof: For δ = 1 the case is clear. For δ > 1 we use the work of Diks et al. [6] ad obtai a partitio V 1, V of G with e(v 1, V ) ad mi( V 1, V ) 3 i liear time. Fially, we use Lemma 1. A similar result holds eve for strog defesive alliaces. 3 A Upper Boud for the Bisectio Width We ow show that a bisectio V 1, V with e(v 1, V ) ca be computed i polyomial time for ay 4-coected plaar graph with a odd umber of vertices. Some of the techiques used are similar to the techiques used by Fa et al. [8] but we also use other techiques ad the aalysis is cosiderably more complicated compared to the aalysis of Fa et al. Sice the bisectio width ever icreases whe removig edges from a graph, it is sufficiet to oly cosider maximal 4-coected plaar graphs with a odd umber of vertices. Lemma A maximal 4-coected plaar graph with a odd umber of vertices has a vertex u with d(u) 5 such that G u is Hamiltoia. The vertex u ad the hamiltoia cycle of G u ca be foud i polyomial time. Proof: Cosider a maximal 4-coected plaar graph G with a odd umber of vertices. There is at least oe vertex u i G with d(u) 5 sice otherwise

8 606 Olse ad Revsbæk O Alliace Partitios ad Bisectio Width for Plaar Graphs we would have v V d(v) = m = (3 6) 4 that could oly happe if 5 which would cotradict 4-coectedess. The graph G is 4-coected so the graph G u has a Hamiltoia cycle computable i polyomial time as showed by Thomas ad Yu [16]. v W W u L u u u t L (a) (b) (c) (d) Figure 3: Illustratios of a cofiguratio. Figure (a) shows the hamiltoia cycle with its k hamiltoia bisectios (the dotted lies) ad the cycle legth of edge {v, t}. Figure (b) shows a sigle hamiltoia bisectio where the vertices are colored accordig to which side of the bisectio they belog to. Also, it shows L ad W for the cofiguratio. Figure (c) shows a compacted eighbor cofiguratio with L ad W. Figure (d) shows a heavy compacted eighbor cofiguratio where the dotted edges are the ier edges of the cofiguratio ad the dashed edges are the outer edges of the cofiguratio. We poit out that the graph i figure (d) is ot 4-coected but meat as a illustratio. Let G be a maximal 4-coected graph with a odd umber of vertices ad let u be a vertex i G with d(u) 5 ad C a Hamiltoia cycle i G u. We will say that the tuple (G, u, C) represets a cofiguratio of G. For ay such cofiguratio, there are exactly k = differet ways to split C ito two coected ad equally sized parts. From ay such split ito two parts, we costruct a hamiltoia bisectio V 1, V of G by addig u to the part where it has the most eighbors i.e. the part that miimizes e(v 1, V ) (ties are broke arbitrarily). Refer to Figure 3(a) ad 3(b). I the followig we let T (G, u, C) deote the sum of e(v 1, V ) over the k possible hamiltoia bisectios of (G, u, C). We will sometimes omit the argumets if they are clear from the cotext. The cycle legth of a edge {v, t} i G u is the miimum distace betwee v ad t i the graph iduced by the cycle. The cotributio to T (G, u, C) of a edge i G u is precisely the cycle legth of the edge. Refer to Figure 3(a). We let L deote the legth of the logest path alog C startig ad edig at a eighbor from u but visitig o other eighbors of u ad let W deote the legth of the secod logest such path. Refer to Figure 3(b). We refer to the cofiguratio (G, u, C) as a compacted eighbor cofiguratio if the eighbors of u ca be divided ito two subsets N 1 ad N of size d(u) ad respectively such that each subset occupies a coected subpath of d(u) the hamiltoia cycle C. Refer to Figure 3(c). The ier edges are the edges o the same side of C as u. The ier edges that are ot icidet to u are aturally

9 JGAA, 17(6) (013) 607 grouped ito (at most) two groups i a compacted eighbor cofiguratio. A compacted eighbor cofiguratio is called heavy if the edges from both these groups have cycle legths, 3, 4,..., k, k 1, k, k 3,... (for both groups we start the sequece from the left) ad if the set of outer edges has two edges of legth, two edges of legth 3,..., two edges of legth k 1 ad oe edge of legth k. Refer to Figure 3(d). I what follows, we will show that T (G, u, C) < k(+1) for ay cofiguratio (G, u, C) of a maximal 4-coected plaar graph with a odd umber of vertices. Sice T (G, u, C) is the sum of bisectio sizes for the k hamiltoia bisectios this implies that there exists at least oe hamiltoia bisectio V 1, V such that e(v 1, V ) which the gives us the upper boud o the bisectio width. Fa et al. [8] use the same strategy to prove the + 1 upper boud for the bisectio width for plaar graphs without a separatig triagle but Fa et al. cosider a Hamiltoia cycle i G where we cosider a Hamiltoia cycle i G u makig the aalysis cosiderably more complicated. To prove T (G, u, C) < k( + 1) we will first show that the heavy compacted eighbor cofiguratios ca be cosidered as a set of worst case cofiguratios such that for ay cofiguratio (G, u, C) there exists a heavy compacted eighbor cofiguratio (G, u, C ) where T (G, u, C) T (G, u, C ). We the exploit that the heavy compacted eighbor cofiguratios are simple eough that T (G, u, C ) < k( + 1) ca be show for this set of cofiguratios. L u Ŵ u ˆL W Figure 4: I the cofiguratio (G, u, C) to the left, the hamiltoia bisectios where edges icidet to u cotribute with d(u)/ are show with dotted lies. Similarly, i the cofiguratio (Ĝ, u, C) to the right, the hamitoia bisectios where edges icidet to u cotribute with d(u )/ are show with dotted lies. Lemma 3 If (G, u, C) is a cofiguratio the it is possible to costruct a heavy compacted eighbor cofiguratio (G, u, C) where G ad G have the same umber of vertices ad d(u) = d(u ) such that T (G, u, C) T (G, u, C). Proof: Let (G, u, C) represet a arbitrary cofiguratio. We ow remove those edges i G that are ot o C ad ot icidet to u. We the replace u (ad the edges icidet to u) with a vertex u with d(u) = d(u ) such that the resultig cofiguratio (Ĝ, u, C) is a compacted eighbor cofiguratio. Fially, we put i edges to create the graph G such that (G, u, C) is a heavy

10 608 Olse ad Revsbæk O Alliace Partitios ad Bisectio Width for Plaar Graphs L ˆL L ˆL u u u u (a) (b) Figure 5: I (a) we show those hamiltoia bisectios which fully cotai the vertices o the cycle path correspodig to L i (G, u, C) (to the left) ad i (Ĝ, u, C) (to the right). I (b) we show those hamiltoia bisectios which does ot fully cotai the vertices i (G, u, C) (to the left) ad i (Ĝ, u, C) (to the right). compacted eighbor cofiguratio. Below, we first argue that the cotributio to T of edges icidet to u i G is ot higher tha the cotributio to T of edges icidet to u i G. Secodly, we argue that the cotributio to T of edges i G u is ot higher tha the cotributio to T of edges i G u. Edges icidet to u : We separate our aalysis ito a case aalysis based o the value of L i G. The values of L ad W i Ĝ are deoted by ˆL ad Ŵ respectively. Case 1: L k: We cosider the followig subcases: If L + d(u) k we build the compacted eighbor cofiguratio (Ĝ, u, C) such that ˆL Ŵ is miimized (0 or 1). Refer to Figure 4. The cotributio to T of edges icidet to u is k which is d(u) the maximum obtaiable value sice u always chooses to joi the partitio which cotributes the least to T. Refer to Figure 4. The coditio L + d(u) k makes it possible for us to obtai the k d(u) cotributio to T from edges icidet to u ad at the same time obtai ˆL L ad Ŵ W that is importat whe we cosider the cotributio from the other edges. If L + d(u) > k we build the compacted eighbor cofiguratio (Ĝ, u, C) such that L = ˆL ad such that the vertices formig the log paths i G ad Ĝ with legth L alog C with o eighbors of u or u respectively are the same. Refer to Figure 5. For each of the k hamiltoia bisectios i (Ĝ, u, C) we ow show that the umber of crossig edges icidet to u has ot decreased compared to the correspodig (same partitio of C) hamiltoia bisectio i (G, u, C). - We first cosider a bisectio V 1, V where the vertices o the path alog C of legth L = ˆL is fully cotaied withi either V 1 or V say V 1. I this case, u must choose to joi V. The

11 JGAA, 17(6) (013) 609 umber of eighbors of u i V 1 is at least as high as the umber of eighbors of u i V 1 i G so the umber of crossig edges for such a bisectio has ot decreased. Refer to Figure 5(a). - We ow cosider a bisectio where the vertices o the path alog C of legth L = ˆL are ot fully cotaied withi either side of the bisectio. Whe u has chose a side of the bisectio u has oly crossig edges to members of either N 1 or N (the two groups of eighbors of u ). If u has d(u ) crossig edges the case is clear. Otherwise, the umber crossig edges has ot dropped sice every vertex o the other side of the cut ad ot o the log path is a eighbor to u. Refer to Figure 5(b). Case : L > k: We build the compacted eighbor cofiguratio (Ĝ, u, C) with L = ˆL. Cosider a bisectio V 1, V of (Ĝ, u, C). Whe u chooses side of the bisectio u ca ot have crossig edges to both N 1 ad N. If there are o crossig edges the same would be the case for the correspodig bisectio of the origial cofiguratio. Refer to Figure 6(a). The eighbors of u are packed aroud the path with ˆL 1 cosecutive vertices o the cycle that are ot eighbors of u so if there are crossig edges the the umber of eighbors o the other side caot have decreased. Refer to Figure 6(b). Edges ot icidet to u : Sice C is i both G ad G the edges o C obviously cotribute with the same to T. We ow cosider the edges i G ot icidet to u ad ot o C ad the edges of G ot icidet to u ad ot o C. We will refer to these sets of edges as the G-set ad the G -set respectively. We will ow argue that a oe-to-oe correspodece betwee the two sets of edges exists such that ay edge i the G-set is matched with a edge i the G -set with the same cycle legth or a bigger cycle legth. The first thig we do is to cosider the ier edges i the G-set that go betwee L+1 cosecutive vertices o the cycle with oly the first ad last vertex beig a eighbor of u. Fa et al. [8] show how to elimiate ay triagle of such edges ad obtai a ew set of edges with higher cycle legths by replacig some of the edges ad Fa et al. also argue that repeated elimiatio of triagles will produce a heavy cofiguratio we refer to [8] for more details. I this way we are able to match ay edge i the cosidered subset of the G-set with a edge i the G -set with the same or a bigger cycle legth but with a cycle legth ot exceedig L. We ow repeat this argumet where we cosider the ier edges i the G-set that go betwee W + 1 other cosecutive vertices o the cycle with oly the first ad last vertex beig a eighbor of u. By doig this we are also able to match these edges with edges i the G -set with the same or a bigger cycle legth but with a cycle legth ot exceedig W. The remaiig umatched edges i the G -set all have cycle legth at least W which makes it possible to match the remaiig ier edges i the G-set with a edge i the G -set with the same cycle legth or a bigger cycle legth. We repeat the

12 610 Olse ad Revsbæk O Alliace Partitios ad Bisectio Width for Plaar Graphs procedure for the outer edges as well. The cotributio to T of the edges ot icidet to u ca cosequetly ot decrease durig the trasformatio sice all the edges i the G-set have bee matched with a edge i the G -set with the same cycle legth or a bigger cycle legth. u L u ˆL u L u ˆL (a) (b) Figure 6: I (a) we illustrate the case where there are o crossig edges for a hamiltoia bisectio i G ad the correspodig bisectio of Ĝ. I (b) we show the bisectios where there are crossig edges i which case the umber of crossig edges ca ot have decreased i Ĝ. Lemma 4 Let (G, u, C) be a heavy compacted eighbor cofiguratio with d(u) eve. The cotributio to T (G, u, C) of the edges icidet to u is d(u) 4 + W d(u) d(u). Proof: We group the edges icidet to u ito pairs such that a pair of edges cuts C ito two pieces with the same umber of eighbors of u. For a give hamiltoia bisectio the cotributio to T (G, u, C) of a pair is 1 if the edpoits of the edges are separated ad 0 otherwise. The shortest route alog the cycle betwee two vertices i a pair cotais (W 1) + d(u) vertices betwee the two vertices. There are cosequetly W + d(u) 1 bisectios that separate each pair so it is ow easy to compute the cotributio to T (G, u, C) of the edges icidet to u: d(u) d(u) (W + 1). Lemma 5 If (G, u, C) is a heavy compacted eighbor cofiguratio the we have the followig: T (G, u, C) < k( + 1). Proof: We divide the proof ito three cases.

13 JGAA, 17(6) (013) 611 Assume that L k 1 ad that d(u) is eve: We compute T i the followig way: ( ) k 1 k 1 d(u) 3 ( d(u) T = k+ k + i + k + i (W + i) + + W d(u) d(u) ). 4 i= i= The first term is the sum of cycle legths from the edges o the cycle, the secod term is the sum of cycle legths for the outer edges, the third term is the sum of cycle legths for the ier edges ot icidet to u, ad the fourth term is the cotributio from edges icidet to u give by Lemma 4. We ow use ( ) k 1 i= i = (k 1)k 1 ad = k + 1: ( d(u) T k( + 1) = 4 + W d(u) i=1 d(u) ) d(u) 3 (W + i) 4. We ow work o a part of this sum multiplied by 4 i order to exclusively have itegers i the computatio: ( 4 d(u) + W d(u) d(u) ) d(u) 3 (W + i) 4 = (d(u) )d(u) + W d(u) 4(d(u) 3)W (d(u) 3)(d(u) ) = d(u) +8d(u) 1+1W W d(u) = (d(u) 6)(d(u) )+1W W d(u) = d(u) + 8d(u) 1 + 1W W d(u) = (d(u) 6)(d(u) + W ), ad fially we get (d(u) 6)(d(u) + W ) T k( + 1) = 4 < 0, (5) 4 i=1 where we have used that the degree of u is at least 6. i=1 Now assume that L k ad that d(u) is eve: I this case we get T = L W i + i + d(u) 4 i= L(L + 1) = 1 + implyig i= W (W + 1) + W d(u) 1 + d(u) 4 4T 4k(k + ) d(u) + W d(u) + k + k d(u) + k + k = L(L+1) 8+W (W +1)+d(u) +W d(u) d(u)+8k+4k 8 4k(k+) = L(L + 1) + W (W + 1) + d(u)(d(u) + W ) 4k 16.

14 61 Olse ad Revsbæk O Alliace Partitios ad Bisectio Width for Plaar Graphs We ow use W + L + d(u) = k: 4T 4k(k+) = L(L+1)+W (W +1)+(k+ W L)(k+W L) 4k 16 We ow use L W i (6): = 3L + W 4kL + 4W + 4k 16. (6) 4T 4k(k + ) 4L + (4 4k)L + 4k 16 implyig = 4((L 1)(L k + ) ). T k( + 1) (L 1)(L k + ) < 0 for L {1,,..., k }. Now assume that d(u) is odd: We remove the edge of u from the group with edges that is closest to the path alog the cycle correspodig to W. It d(u) is ot hard to see that the cotributio to T of the edges of u is uchaged after the removal of this edge. For d(u) > 5 there is cosequetly a heavy compacted eighbor cofiguratio cosidered above with a higher value of T compared to the origial graph. Now cosider the case d(u) = 5. We ow remove a edge of u as described above implyig o chage i the cotributio to T of the edges of u ad isert a edge with cycle legth W + 1 ad obtai a heavy compacted eighbor cofiguratio with d(u) = 4. We ca ow use (5) with d(u) = 4 ad W replaced by W + 1 to compute T for this cofiguratio. Sice we have iserted a edge with cycle legth W + 1 we have to subtract W + 1 to obtai the value for T for the origial cofiguratio with d(u) = 5: (4 6)(4 + (W + 1)) T k( + 1) = 4 (W + 1) = 3 < 0. 4 We are ow ready to preset the mai theorem of this sectio: Theorem 3 A bisectio V 1, V exists with e(v 1, V ) for ay 4-coected plaar graph G(V, E) with a odd umber of vertices ad such a bisectio ca be obtaied i polyomial time. Proof: Let G(V, E) be a 4-coected plaar graph with a odd umber of vertices. As oted earlier, we ca assume that G is a maximal plaar graph without loss of geerality. Lemma assures that we ca efficietly obtai a cofiguratio (G, u, c). We ow examie all the k hamiltoia bisectios of the cofiguratio. By usig Lemma 3 ad Lemma 5 we kow that at least oe of the hamiltoia bisectios satisfies e(v 1, V ).

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