Constrained tri-connected planar straight line graphs

Transcription

1 Constrained tri-connected lanar straight line grahs Marwan Al-Jbeh Gill Bareet Mashhood Ishae Diane L. Soaine Csaba D. Tóth Andrew Winslow Abstract It is known that for any set V of n 4 oints in the lane, not in conex osition, there is a 3-connected lanar straight line grah G = (V, E) with at most 2n 2 edges, and this bond is the best ossible. We show that the er bond E 2n contines to hold if G is constrained to contain a gien grah G 0 = (V, E 0 ), which is either a 1-factor (i.e., disjoint line segments) or a 2-factor (i.e., a collection of simle olygons), bt no edge in E 0 is a roer diagonal of the conex hll of V. Since there are 1- and 2-factors with n ertices for which any 3-connected agmentation has at least 2n 2 edges, or bond is are nearly tight in these cases. We also examine ossible conditions nder which this bond may be imroed, sch as when G 0 is a collection of interior disjoint conex olygons in a trianglar container. 1 Introdction A grah is k-connected if it remains connected on deleting any k 1 ertices along with all incident edges. Connectiity agmentation roblems are an imortant area in otimization and network design. The k-connectiity agmentation roblem asks for the minimm nmber of edges needed to agment an int grah G 0 = (V, E 0 ) to a k-connected grah G = (V, E), E 0 E. In abstract grahs, the connectiity agmentation roblem can be soled in O( V + E ) time for k = 2 [4, 7, 8], and in olynomial time for any fixed k [9]. Researchers hae considered the connectiity agmentation roblems oer lanar grahs where both the int G 0 and the ott G hae to be lanar (that is, they hae no K 5 or K 3,3 minors). Kant and Bodlaender [10] roed that already the 2-connectiity agmentation oer lanar grahs is NP-hard, and they deised a 2-aroximation algorithm that rns in O(n log n) time. We consider 3-connectiity agmentation oer lanar geometric grahs, where the gien straight line embedding of the int grah has to be resered. A lanar straight-line grah (for short, slg) is a grah with a straight-line embedding in the lane That is, the ertices are distinct oints in the lane and the edges are straight-line segments between the incident endoints (that do not ass throgh any other ertices). The k-connectiity agmentation for slgs asks for the minimm nmber of edges needed to agment an int slg This material is based on work sorted by the National Science Fondation nder Grant No Research by Tóth was also sorted by NSERC grant RGPIN Preliminary reslts hae been resented at the 26th Eroean Worksho on Comtational Geometry (2010, Dortmnd) and at the 20th Annal Fall Worksho on Comtational Geometry (2010, Stony Brook, NY). Deartment of Comter Science, Tfts Uniersity, Medford, MA, USA. {maljb01, bareet, mishae, dls, cdtoth, awinsl02}@cs.tfts.ed Deartment of Comter Science, Technion, Haifa, Israel. bareet@cs.technion.ac.il Deartment of Mathematics and Statistics, Uniersity of Calgary, Calgary, AB, Canada. cdtoth@calgary.ca 1

2 G 0 = (V, E 0 ) to a k-connected slg G = (V, E), E 0 E. Rtter and Wolff [13] showed that this roblem is NP-hard for any 2 k 5. Note that the roblem is infeasible for k 6, since eery lanar grah has a ertex of degree at most 5. There are two ossible aroaches to get arond the NP-hardness of the agmentation roblem: (i) aroximation algorithms, as was done for lanarity-resering 2-connectiity agmentation; and (emhii) roing extremal bonds for the minimm nmber of edges sfficient for the agmentation in terms of the nmber of ertices, which we do here. It is easy to see that for eery n 4, there is a 3-connected lanar grah with n ertices and 3n/2 edges, where all bt at most one ertex hae degree 3. On a set V of n 4 oints in the lane, howeer, a 3-connected slg may reire many more edges. García et al. [5] roed that if 3 h < n oints lie on the conex hll of V, then it admits a 3-connected slg G = (V, E) with at most max( 3n/2, n + h 1) 2n 2 edges, and this bond is best ossible. If the oints in V are in conex osition (that is, h = n), then V does not admit any 3-connected slg. Tóth and Valtr [14] characterized the 3-agmentable lanar straight-line grahs, which can be agmented to 3-connected slgs. Secifically, a slg G 0 = (V, E) is 3-agmentable if and only if E 0 does not contain any edge that is a roer diagonal of the conex hll of V. Eery 3-agmentable slg on n ertices can be agmented to a 3-connected trianglation, which has to 3n 6 edges, bt in some cases significantly fewer edges are sfficient. As mentioned aboe, the 3-connectiity agmentation roblem for slgs is NP-hard, and no aroximation is known for this roblem. It is also not known how many new edges are sfficient for agmenting any 3-agmentable slg with n ertices. Sch a worst case bond is known only for edge-connectiity: Al-Jbeh et al. [3] roed recently that eery 3-edge-agmentable slg with n ertices can be agmented to a 3-edge-connected slg by adding at most 2n 2 new edges. Or reslts. In the 3-connectiity agmentation roblem for slgs, we are gien a slg G 0 = (V, E 0 ), and asked to agment it to a 3-connected slg G = (V, E), E 0 E. Intitiely, the edges in E 0 are either sefl for constrcting a 3-connected grah or they are obstacles that reent the addition of new edges which wold cross them. In this note, we exlore which 3-agmentable slgs with n 4 ertices can be agmented to a 3-connected slgs which hae at most 2n edges. Recall that 2n 2 edges may be necessary een for a comletely nobstrcted int G 0 = (V, ). We roe that if G 0 is 1-reglar (that is, a crossing-free erfect matching) or 2-reglar (a collection of airwise noncrossing simle olygons), then it can be agmented to a 3-connected slg which has at most 2n 2 or 2n edges, resectiely. Theorem 1. Eery 1-reglar 3-agmentable slg G 0 = (V, E 0 ) with n 4 ertices can be agmented to a 3-connected slg G = (V, E), E 0 E, with E 2n 2 edges. Theorem 2. Eery 2-reglar 3-agmentable slg G 0 = (V, E) with n 4 ertices can be agmented to a 3-connected slg G = (V, E), E 0 E, with E 2n edges. Figres 1(a)-1(b) deict 1- and 2-reglar slgs, resectiely, where all bt one of the ertices are on the bondary of the conex hll. Clearly, the only 3-connected agmentation is the wheel grah, which has 2n 2 edges. We conjectre that Theorems 1 and 2 can be generalized to slgs with maximm degree at most 2. Conjectre 1.1. Eery 3-agmentable slg G 0 = (V, E) with n 4 ertices and maximm degree at most 2 can be agmented to a 3-connected slg G = (V, E), E 0 E, with E 2n 2 edges. 2

3 (a) (b) (c) Figre 1: (a-b) 1- and 2-reglar slgs whose only 3-connected agmentation is the wheel grah. (c). Nested coies of K 4, for which eery 3-connected agmentation has at least 9 4 n 3 edges. It is not ossible to extend Theorem 1 and 2 to 3-reglar slgs. For examle, if G 0 = (V, E 0 ) is a collection of nested 4-clies as in Fig. 1(c), then eery 3-connected agmentation reires 3( n 4 1) new edges, which gies a total of 9 4 n 3 edges. As mentioned aboe, eery set of n 4 oints in the lane, h n of which lie on the bondary of the conex hll, admits a 3-connected slg with at most max( 3n/2, n + h 1) 2n 2 edges [5]. We cold not strengthen or Theorems 1 and 2 to be sensitie to the nmber of hll ertices. Some imroement may be ossible for 1-reglar slgs with fewer than n 1 ertices on the conex hll; the best lower bond constrction we fond with a trianglar conex hll reires only 7 4 (n 2) edges in total (Fig. 2(a)). For 2-reglar slgs, howeer, one cannot exect significant imroement een if h = 3. If G 0 = (V, E 0 ) consists of n 3 nested triangles (Fig. 2(b)), then any agmentation to a 3-connected slg has at least 2n 3 edges. (a) (b) (c) Figre 2: (a) A 1-reglar slg on n ertices with a trianglar conex hll whose 3-connected agmentations hae at least 7 4 (n 2) edges. (b) A 2-reglar slg on n ertices with a trianglar conex hll sch that eery 3-connected agmentation has at least 3n 3 edges. (c) Interior disjoint simle olygons in a trianglar container, for which eery 3-connected agmentation has 3n 3 edges. Obstacles in a container. We hae considered whether Theorem 2 can be imroed for collections of simle olygons, where the conex hll is a triangle, and there is no nesting among the remaining olygons. We model sch 2-reglar slgs as a collection of interior disjoint simle 3

4 olygons in a trianglar container. Figre 2(c) shows a constrction where eery 3-connected agmentation still reires 3n 3 edges. In this examle the olygons are nonconex, and they are nested in the sense that each olygon is isible from at most one larger olygon. In Section 5, we derie lower bonds for the 3-connectiity agmentation of 2-reglar slgs G 0, where G 0 is a collection of interior disjoint conex olygons (called obstacles) lying in a trianglar container. All or lower bonds in this section are below 2n 2, which sggests that Theorem 2 may be imroed in this secial case. Organization. In Section 2, we introdce a general framework for 3-connectiity agmentation, and roe that eery nonconex simle olygon with n ertices can be agmented to a 3-connected slg which has at most 2n 2 edges. We roe Theorems 1 and 2 in Section 3 and 4, resectiely. Lower bonds for the model of disjoint conex obstacles in a trianglar container are resented in Section 5. We conclde with oen roblems in Section 6. 2 Preliminaries In this section, we roe two reliminary reslts abot abstract grahs, which are directly alicable to the 3-connectiity agmentation of simle olygons. In an (abstract) grah G = (V, E), a sbset U V is called 3-linked if G contains at least three disjoint aths between any two ertices of U. (Two aths between the same two ertices are called disjoint if they do not share any edges or ertices aart from their endoints.) By Menger s theorem, a grah G = (V, E) is 3-connected if and only if V is 3-linked in G. The following lemma gies a criterion for incrementing a 3-linked set of ertices with one new ertex. Lemma 2.1. Let G = (V, E) be a grah sch that U V is 3-linked. If G contains three disjoint aths from V \ U to three distinct ertices in U, then U {} is also 3-connected. Proof. Assme that G contains three disjoint aths from V \U to distinct ertices 1, 2, 3 U. It is enogh to show that for eery U, there are three ertex disjoint aths between and. By Menger s theorem, it is enogh to show that if we delete any two ertices w 1, w 2 V \ {, }, the remaining grah G \ {w 1, w 2 } still contains a ath between and. Since there are three disjoint ath from to 1, 2, and 3, the grah G \ {w 1, w 2 } contains a ath from to i for some i {1, 2, 3}. If i =, then we are done. Otherwise, G \ {w 1, w 2 } contains a ath from i to, since U is 3-connected. The nion of these two aths (from to i and from i to ) contains a ath from to. Lemma 2.2. Let G A = (V, A) be a connected grah with minimm degree 2, and let G C = (V, C) be a 3-connected grah with A C. Let U A V be the set of ertices that hae degree 3 or higher in G A, and assme that U A is 3-linked in G A. Then G A = (V, A) can be agmented to a 3-connected grah G B = (V, B) with A B C, by adding at most V \ U A new edges. Frthermore, if U A =, then V 2 new edges are enogh for the agmentation. Proof. We describe an algorithm that agments G A = (V, A) to a 3-connected grah G B = (V, B) incrementally. We maintain a grah G i = (V, E i ) with A E i C. Initially, we start with i = 0 and E 0 = A. We agment G i incrementally by adding new edges from C ntil G i becomes 3-connected, and then ott G B = G i. We also increment a set U i V of ertices that hae degree 3 or higher in G i, and maintain the roerty that U i is 3-linked in G i. In each ste, we will 4

5 increment G i with one new edge sch that U i increases by at least one new ertex. If U = or U = 2, then we can add one new edge sch that U increases by two new ertices. Or algorithm terminates with U i = V, and the aboe roerties garantee that altogether at most V \ U 0 new edges are added, and if U 0 =, then at most V 2 new edges are added. (a) (b) (c) (d) Figre 3: Illstration for the roof of Lemma 2.2. Edges of G A are black, additional edges of G C are gray, ertices in U i are marked with large dots. (a) G A is a Hamiltonian cycle. (b) G A has two ertices of degree 3. (c) Both and lie in the interior of some aths between ertices of U i. (d) Vertex is in U i. It remains to describe one ste of the agmentation, in which we increment E i with one new edge from C. We distingish between three cases. Case 1: U i =. Since G i is connected and has minimm degree 2, it is a Hamiltonian cycle (Fig. 3(a)). Pick an arbitrary edge C \ E 0, and set E i+1 = E i {}. Let U i+1 = {, } be the set of the two ertices of degree 3. Note that U i+1 is indeed 3-linked, as reired. Case 2: U i = 2. Denote the ertices in U i by and. Eery edge in E i is art of a ath between and (Fig. 3(b)). Let P i denote the set of all (at least three) aths of G i between and. Note that eery ertex in V \ U i lies in the interior of a ath in P i. Since G i is a simle grah, at least two aths in P i hae interior ertices. Let P P i be a ath with at least one interior ertex. Grah G C contains some edge C between an interior ertex of P and a ertex otside of P, otherwise the deletion of and wold disconnect G C. Set E i+1 = E i {} and U i+1 = U i {, }. Note that G i+1 now contains three disjoint aths between any two ertices of U i+1 = {,,, }. Case 3: U i 3. In this case, eery edge in E i is art of a ath between two ertices in U i. Let P i denote the set of all aths of G i between ertices in U i. Note that eery ertex in V \ U i lies in the interior of a ath in P i. Pick two ertices, U i connected by a ath in P i, and let V be the set of interior ertices of all aths in P i between and. Let P P i be a ath with at least one interior ertex, and denote its endoints by, P (Fig. 3(c) 3(d)). Grah G C contains some edge C between a ertex V and a ertex otside V {, }, otherwise the deletion of and wold disconnect G C. Set E i+1 = E i {}. Now G i+1 contains three disjoint aths from to three ertices of U i : disjoint aths to and along a ath in P i, and a third ath starting with edge and, if U i, then contining along a ath containing to a third ertex in U i. Similarly, if U i, then G now contains there disjoint aths from to three ertices of U i {}. By Lemma 2.1, U i {, } is 3-linked in G i+1. So we can set U i+1 = U i {, }. Let H be a simle olygon in the lane with n ertices, that is, a straight-line embedding of a Hamiltonian cycle. We show that it can be agmented to a 3-connected slg with at most 2n 2 edges. 5

6 Corollary 2.3. Eery simle olygon with n 4 ertices, not all in conex osition, can be agmented to a 3-connected slg which has at most 2n 2 edges. Proof. The edges and ertices of a simle olygon form a Hamiltonian cycle G 0 = (V, E 0 ). By the reslts of Valtr and Tóth [14], if the olygon is nonconex, then it is 3-agmentable, so there is a 3-connected slg G 2 = (V, E 2 ), E 0 E 2. Lemma 2.2 comletes the roof. 3 Disjoint Line Segments In this section, we roe Theorem 1. Let G A = (V, A) be a straight-line embedding of a erfect matching with n 4 ertices, not all in conex osition. We show that if no edge in M is a roer chord of the conex hll of the ertices, then G A can be agmented to a 3-connected slg which has at most 2n 2 edges. We se the reslt by Hoffmann and Tóth [6] that G A can be agmented to a Hamiltonian slg G H. If G A is 3-agmentable, then G H is also 3-agmentable and can be agmented to a 3-connected Hamiltonian slg G C. In the following lemma, we se sch a 3-connected Hamiltonian sergrah, bt we no longer rely on the straight-line embedding of the matching G A. Lemma 3.1. Let G A = (V, A) be a erfect matching with n 4 ertices, and let G C = (V, C) be a 3-connected Hamiltonian lane grah with A C. Then G A = (V, A) can be agmented to a 3-connected grah G B = (V, B) with A B C, sch that B 2n 2. Proof. Let (V, H) be an arbitrary Hamiltonian cycle in G C. If A H, then the reslt follows from Lemma 2.2. Sose that A H. We constrct a 3-connected grah G B, A B C, incrementally. We maintain a 2-connected grah G i = (V, E i ) with E i C. We also maintain a set U i V of secial ertices called hbs, and a set P i of aths in G i between hbs. We maintain the following roerties for G i. (i) U i V is 3-linked in G i, (ii) E i contains eery edge of A sanned by ertices of U i, (iii) E i (n 2) + U i, (i) for eery edge e E i there is a ath P P i sch that either e P or e joins an endoint of P to an interior oint of P. () no ath in P i is dangeros (defined below). In each ste, we will modify G i sch that the set of hbs strictly increases, and the nmber of edges is bonded by E i (n 2) + U i. The algorithm terminates when U i = V. At that time, G i is a 3-connected sbgrah of G C, all edges of A are contained in E i, and E i 2n 2, so we can ott G B = G i. We note here that the set of hbs U i monotonically increases, bt the set of edges E i does not always increase. Sometimes we may delete edges from E i. In or algorithm, we will maintain the that P i contains no dangeros ath (roerty ()). Let P be a ath in P i or a roer sbath of some ath in P i. We say that P is dangeros if (1) each endoint, of P is connected to some interior oint of P by an edge in A \ E i, and (2) for eery edge st in C between an interior ertex s of P and a ertex t otside of P, there is an edge in A \ E i between s and an endoint of P (see Fig. 4). 6

7 In order to aoid dangeros aths, we also need the following definition. An interior ertex of a ath P P i with endoints and is dangeros if the sbath of P between and or between and is dangeros. P s t Figre 4: A dangeros ath P between and, and a dangeros ertex. Solid edges are in E i, dashed edges are in A \ E i, and gray edges are in C \ E i, resectiely. Initialization. Recall that G H = (V, H) is a Hamiltonian cycle in G C, with n edges, sch that A H. Let A be an arbitrary chord of H. Vertices and decomose the Hamiltonian cycle H into two aths, each of which has some interior ertices. Since G C is 3-connected, it contains an edge st C between two interior ertices of two distinct aths. Let G 0 = (V, E 1 ) with E 0 = H {, st}. Let U 0 = {,, s, t}, which is 3-linked in G 0. The matching A contains and ossibly st, so E 0 contains all edges sanned by U 0. We hae G 0 = n + 2, U 0 = 4, and so E 0 (n 2) + U 0 holds. All edges in E 0 lie along aths between hbs, which we denote by P 0. Note also that eery ath in P 0 with interior ertices is incident to or, which are incident to the nie edge A of the matching, so no ath in P 0 is dangeros. General Ste i. We are gien a grah G i = (V, E i ), a set of hbs U i, and a set of aths P i with roerties (i) (). We distingish three cases. In all three cases, we agment G i with an edge where is an interior ertex of a ath P P i and is otside of ath P. We will add ertex to U i. If haens to be an interior ertex of another ath P P i, then we add to U i as well, and we also agment G i with any ossible edge of A \ E i that joins to another ertex of P. This ensres that een if is a dangeros ertex of P, the two sbaths of P in P i+1 will not be dangeros. Since ertex is treated the same way in all three cases, we will not discss these ossibilities below we assme for simlicity that is in U i. The three cases differ only on the handling of ertex and ath P P i. Case 1. There is an edge A sch that is an interior ertex of a ath P P i and is otside of ath P. (Fig. 5(a).) Let E i+1 = E i {} and U i+1 = U i {}. By Lemma 2.1, U i+1 is 3-linked in G i+1. Vertex decomoses P into two sbaths, which are not dangeros. We constrct P i+1 by relacing P with its two sbaths. Case 2. Eery edge in A \ E i connects ertices within the same ath of P i. There is an edge in C \ (E i A) sch that is an interior ertex of a ath P P i, is otside of ath P, is not a dangeros ertex, and there is no edge in A \ E i between and an endoint of ath P. (See Fig. 5(b).) Let E i+1 = E i {} U i+1 = U i {}. By Lemma 2.1, U i+1 is 3-linked in G i+1. Vertex decomoses ath P into two sbaths, which are not dangeros. We constrct P i+1 by relacing P with its two sbaths. 7

8 P P P G i G i G i (a) (b) (c) Figre 5: Cases 1-3a. Vertex is in the interior of a ath P and is otside of ath P. (a) Case 1: A \ E i. (b) Case 2: C \ A bt there is no edge in A \ E i between and an endoint of P. (c) Case 3: C \ A and there is an edge in A \ E i between and an endoint of P. Case 3. Eery edge in A \ G i connects ertices within the same ath in P i. For eery edge C between an interior ertex of a ath P P i and a ertex otside of that P, either is dangeros or there is an edge in A \ E i between and an endoint of P. We consider two sbcases. Sbcase 3a: There is an edge C sch that is an interior ertex of a ath P P i, ertex is otside of P, and edge m A \ G i connects to an endoint of P. (Fig. 5(c).) Denote the two endoints of P by and, and assme withot loss of generality that m =. We wold like to add to U i, bt then we hae to agment G i with both and (to maintain roerty (ii)). We will agment U i with three interior ertices of P. Vertex decomoses ath P into two aths: let P 1 P be the sbath between and, and P 2 between and. Note that P 1 has at least one interior ertex since A \ E i, bt P 2 may be a single edge. Since C is 3-connected, there is some edge st between an interior ertex s of P 1 and some ertex t otside of P 1. Obsere that s cannot be a dangeros ertex, and there is no edge in A \ E i between s and an endoint of P, otherwise P wold be a dangeros ath. Therefore t mst be a ertex of P, that is, either t is an interior ertex of P 2 or we hae t =. We examine both ossibilities. Sbcase 3a(i): There is an edge st C sch that s is an interior ertex of P 1 and t is an interior ertex of P 2. Let E i+1 = E i {,, st} and U i+1 = U i {, s, t}. Sbcase 3a(ii): For eery edge st C sch that s is an interior ertex of P 1 and t is otside of P 1, we hae t =. We show that P 2 has no interior ertices. Sose, to the contrary, that P 2 has interior ertices. Since T is 3-connected, there is an edge s t between an interior ertex s of P 2 and a ertex t otside of P 2. Note that there is no edge in A \ E i between s and an endoint of P, otherwise P wold be dangeros, and s is not dangeros, since s C. Hence t mst be a ertex of ath P. We hae assmed that t is not an interior ertex of P 1, and t becase T is lanar. Hence t cannot be otside of P 2, which is a contradiction. We conclde that P 2 is a single edge P 2 = {}. Let E i+1 = (E i \ {}) {,, s} and U i+1 = U i {, s}. Sbcase 3b. Eery edge in A \ E i connects ertices within the same ath in P i. For eery edge C between an interior ertex of a ath P P i and a ertex otside of that P, ertex is dangeros. Denote the two endoints of P by and. Vertex decomoses ath P into two aths: let P 1 P be the sbath between and, and P 2 between and. 8

9 Assme withot loss of generality that P 1 P is a dangeros ath. Let and be the interior ertices of P 1 sch that, A \ E i. Since G C is 3-connected, C has an edge between an interior ertex of P 1 and a ertex otside of P 1. Howeer, P 1 is a dangeros ath, so only or may be connected to a ertex otside of P 1. Note that and are not dangeros ertices of P. Therefore, they can only be connected to some ertex in P. If there is an edge t between and an interior ertex of P 2, then let E i+1 = E i {,, t} and U i+1 = U i {,, t}. Similarly, if there is an edge t between and an interior ertex of P 2, then let E i+1 = E i {,, t} and U i+1 = U i {,, t}. Now assme that neither nor is adjacent to any interior ertex of P 2. Then at least one of them is adjacent to. Similarly to case 3a, we can show that P 2 is a single edge P 2 = {}. Sose, to the contrary, that P 2 has interior ertices. Since G C is 3-connected, there is an edge s t between an interior ertex s of P 2 and a ertex t otside of P 2. Note that there is no edge in A \ E i between s and an endoint of P, otherwise P wold be dangeros, and s is not dangeros, since C contains an edge between and one of,. Hence t mst be a ertex of ath P. We hae assmed that t is not an interior ertex of P 1, and t becase C is lanar. Hence t cannot be otside of P 2, which is a contradiction. We conclde that P 2 is a single edge P 2 = {}. We distingish two sbcases deending on the order of ertices and along ath P 1. P 1 P 1 P 2 G i (a) P 1 P 2 G i (b) s P 2 t P 3 (c) G i Figre 6: Cases 3b. Solid edges are art of grah G i, dashed edges are in A \ E i. Vertex is dangeros in the interior of a ath P. (a) Case 3b: There is an edge between and an interior ertex of P 2. (b) Case 3b(i): Vertices,,, aear in this order along P 1. (c) Case 3b(ii): Vertices,,, aear in this order along P 1. Sbcase 3b(i): The ertices,,, aear in this order along P 1. If T, then let E i+1 = (E i \ {}) {,, } and U i+1 = U i {, }. If T, then let E i+1 = (E i \ {}) {,, } and U i+1 = U i {, }. Sbcase 3b(ii): The ertices,,, aear in this order along P 1. Denote by P 3 the sbath of P between and. Path P 3 has an interior ertex becase M \ G i. Since T is 3-connected, there is an edge s t in T sch that s is an interior ertex of P 3, and t is otside of P 3. By or assmtions, t mst be a ertex of P 1 (ossibly t = ). Let U i+1 = U i {,, s, t }. If t U i, then let E i+1 = E i {,, s t }; otherwise agment E i with {,, s t } and any edge in A \ E i incident to t. 9

10 Corollary 3.2. Eery 3-agmentable lanar straight-line matching with n 4 ertices can be agmented to a 3-connected slg which has at most 2n 2 edges. Proof. Let G A = (V, A) be a 3-agmentable lanar straight-line matching with n 4 ertices. By the reslts of Hoffmann and Tóth [6], there is a slg Hamiltonian cycle H on the ertex set V that does not cross any edge in A. Since the Hamiltonian cycle H is crossing-free, none of its edges is a chord of the conex hll of ertices (otherwise the remoal of this edge wold disconnect H). Hence both (V, H) and (V, A H) are 3-agmentable [14]. That is, there is a 3-connected slg G C = (V, C) sch that A H C. Lemma 3.1 comletes the roof. 4 A Collection of Simle Polygons In this section, we roe Theorem 2. We are gien a 2-reglar slg G A = (V, A) with n 4 ertices and n edges. If G A is 3-agmentable, then it is contained in some 3-connected slg G C = (V, C), say a trianglation of G A, which may hae to 3n 6 edges. Note that the oter face of G C is a simle olygon. We will constrct an agmentation G B = (V, B), A B C, with B 2n edges. Lemma 4.1. Let G A = (V, A) be a 2-reglar grah with n 4 ertices, and let G C = (V, C) be a 3-connected lane grah with A C sch that all bonded faces are triangles. Then G A = (V, A) can be agmented to a 3-connected grah G B = (V, B) with A B C, sch that B 2n. Proof. Consider a straight-line embedding of G C. Since Q C is 3-connected, its oter face is a simle olygon, which we denote by Q C. We constrct a 3-connected grah G B, A B C, incrementally. We maintain a 2-connected grah G i = (V i, E i ) with V i V and E i C. We also maintain a set U i V of ertices, called hbs, which is the set of all ertices in V i with degree 3 or higher in G i. The hbs natrally decomose G i into a set P i of aths in G i between hbs. We maintain the following roerties for G i. (i) Q C E i C, (ii) U i V i is 3-linked in G i, (iii) eery bonded face of G i is incident to at least three ertices in U i, (i) no edge of C \ E i joins nonconsectie ertices of any ath in P i. Initially G 0 will hae 4 ertices, and we incrementally agment it with new edges and ertices, ntil we hae U i = V. The ertex sets V i, U i, and the edge set E i will monotonically increase dring this algorithm, and we gradally add all edges of A to E i. When or algorithm terminates and U i = V, the grah G i is a 3-connected sbgrah of G C, which contains all edges of A. Wheneer we add an edge e C \ A to E i, we charge e to one of the endoints of e, sch that eery ertex is charged at most once. This charging scheme garantees that we add at most n edges from C \ A, in addition to the n edges of A. Initialization. We constrct the initial grah G 0 with U 0 = 4 hbs. Consider the 3-connected slg G C with where Q C is the bondary of the oter face. Let V be a ertex in the interior of Q C, and let and w be its two neighbors in the 2-reglar grah G A. Constrct an axiliary grah G C = (V {a, b}, C ), with C C, as follows. The edges of are all edges in C, edges a, a, and aw, and edges connecting the the axiliary ertex b to G C 10

11 all ertices of the oter face Q C. By Lemma 2.1, G 2 is 3-connected (albeit not necessarily lanar). Hence G C contains three disjoint ath between a and b. Fix three disjoint aths of minimal total length. The minimality imlies that no two nonconsectie ertices in any ath are joined by an edge of C. Relace the edges a and aw with and w, resectiely, to obtain three disjoint aths in C from V to three distinct ertices of the oter face Q C, sch that two of these aths leae along edges of A. Denote by P 1, P 2, P 3 the three aths, with endoints 1, 2, 3 along Q C, resectiely. b 1 a 2 w w 3 (a) (b) (c) Figre 7: (a) A 2-reglar slg G A (black) in a 3-connected trianglation G C (gray). (b) Grah G C with two axiliary ertices, a and b, is also 3-connected. (c) Three disjoint aths from to three bondary oints 1, 2, and 3. Let or initial grah G 0 = (V 0, E 0 ) consists of all edges and ertices of Q C P 1 P 2 P 3. There are exactly for ertices of degree 3, namely U 0 = {, 1, 2, 3 }, which are 3-linked in G 0. Each of the three bonded faces G 0 is incident to 3 hbs. So G 0 has roerties (i) (iii). For roerty (i), note also that no edge in C joins nonadjacent ertices of Q C, otherwise G C wold not be 3-connected. Let s estimate how many edges of G 0 are from C \ A. Orient Q C conterclockwise, and charge eery edge e C \ A along Q C to its origin. Clearly, eery ertex if Q C is charged at most once. Direct the aths P 1, P 2, and P 3 from to 1, 2, and 3 ; and charge each edge e C \ A along the aths to its origin. Since two aths leae along edges of A, ertex is charged exactly once. All interior ertices of the three aths are charged at most once, becase the aths are disjoint. Phase 1. In the first hase of or algorithm we agment G i = (V i, E i ) ntil V i = V, bt at the end of this hase some edges of A may still not be contained in E i. We agment G i = (V i, E i ) with new edges and ertices incrementally. It is enogh to describe a general ste of this hase. Pick an arbitrary ertex V \ V i. We will agment G i to inclde (and ossibly other ertices). Or argment is similar to the initialization. Let Q denote the bondary of the face of G i that contains. Let G be the sbgrah of C that contains all edges and ertices of G C in the closed olygonal domain bonded by Q. Let and w be the neighbors of in the 2-reglar grah G A ; note that both and mst be ertices of G. Constrct an axiliary grah G as follows. The ertices of G are the ertices of G and two axiliary ertices, a and b. The edges of G are the edges of G ; the edges a, a, and aw; and edges between b and eery hb ertex along Q. We claim that grah G is 3-connected. Indeed, it is easy to erify that the deletion of any two ertices cannot disconnect G. Therefore, there are three disjoint aths in G between a and b. Fix three disjoint aths with a minimm total 11

12 nmber of edges lying in the interior of Q. The minimality imlies that each ath goes from a to a ertex along Q, then follows Q to a hb on Q, and then contines to b along a single edge. In articlar, no two nonconsectie ertices of any of the three aths between a and Q are joined by an edge of G (i.e., no shortcts). Relace the edges a and aw with edges and w, resectiely, to obtain three disjoint aths from to three distinct hbs along Q, sch that two of these aths leae along edges of A. Denote by P 1, P 2, and P 3 the initial ortions of the aths between a and Q ; and let 1, 2, and 3 be their endoints on Q (these endoints are not necessarily hbs of G i ). We constrct G i+1 by agmenting G i with all ertices and edges of the aths P 1, P 2, and P 3. The new ertices of degree 3 are and, if they were not hbs already, 1, 2, and 3. In G i+1, three disjoint aths connects to three hbs in U i, so U i {} is 3-linked in G i+1. Similarly, 1, 2, and 3 are each connected to three hbs in U i {} along three edge disjoint aths. We conclde that U i+1 := U i {, 1, 2, 3 } is 3-linked in G i+1. We constrct P i from P i by adding the three new aths P 1, P 2, and P 3 ; and slitting the aths containing 1, 2, and 3 into two ieces if necessary. Paths P 1, P 2, and P 3 decomose a face of G i into three faces, each of which is incident to at least three hbs of U i+1. So roerties (i) (i) hold for G i+1. Direct the aths P 1 P 2 P 3 from to 1, 2, and 3 ; and charge any new edge e C \ A to its origin. Each new ertex of V i+1 is charged at most once: is charged at most once becase two incident new edges are contained in A; and any other new ertices are charged at most once becase the aths P 1, P 2, and P 2 are disjoint. Phase 2. In the second hase, we agment G i = (V, E i ) with edges of A \ E i sccessiely ntil A E i. We can add all edges of A at no charge, we only need to check that that roerties (i) (i) are maintained. We describe a single ste of the agmentation. Consider an edge A \ E i. Let G i+1 = (V, E i+1 ) with E i+1 = E i {} and U i+1 = U i {, }. By Lemma 2.1, U i+1 is 3-linked in G i+1. The edge sbdiides a bonded face of G i into two faces of G I+1. Since does not join two ertices of the same ath in P i, both new faces are incident to at least three hbs in U i+1 (inclding and. The aths in P i+1 are obtained from P i by adding the 1-edge ath, and ossibly decomosing the aths containing and into two. Since P i has roerty (i), no edge in C joins two nonconsectie ertices of any ath in P i+1, either. So roerties (i) (i) hold for G i+1. Phase 3. We hae a grah G i = (V, E i ) with A E C, where the set U i of ertices of degree 3 or higher is 3-linked in G i. This imlies that eery ertex V \ U i has degree 2 in G i. Since G A is 2-reglar, and A E i, no edges of C \ A hae been charged to. Let x denote the nmber of ertices of G i of degree 2. Aly Lemma 2.2 to agment G i = (V, E i ) to a 3-connected grah G B with x additional edges. The int grah G A is 2-reglar, and has n edges. In all three hases, we agmented G i with at most n edges from C \A. So when or algorithm terminates, G B = G i has at most 2n edges. 5 Obstacles in a Container In this section, we consider agmenting a slgs G 0 = (V, E) with n 6 ertices that consists of a set of interior disjoint conex olygons (obstacles) in the interior of a trianglar container. Since no edge is a roer chord of the conex hll, eery sch slg is 3-agmentable [14], and in fact is 12

13 it not difficlt to see that any trianglation of G 0 is a 3-connected grah with 3n 6 edges. We beliee, howeer, that significantly fewer edges are sfficient for 3-connectiity agmentation. The best lower bonds we were able to constrct reire fewer than 2n 2 edges. When there is only one conex obstacle, three edges are obiosly reired for connecting it to the container. Howeer, for k N conex obstacles at least 3k 1 edges are necessary in the worst case. Or lower bond constrction is deicted in Figre 8(a). It incldes one large conex obstacle which hides one small obstacle behind each side (excet the base), sch that each small obstacle can see only three different ertices (the to ertex of the container and two adjacent ertices of the large obstacle). Ths, we need three edges for each small obstacle and only two edges for the larger obstacle, connecting its two bottom ertices to the two endoints of the base of the container. (a) (b) Figre 8: (a) 3-connectiity agmentation for k interior disjoint conex obstacles in a trianglar container reires 3k 1 new edges. (b) For k interior disjoint trianglar obstacles in a trianglar container, we need (5k + 1)/2 new edges. The large obstacle in the aboe constrction is a conex k-gon, and so the lower bond 3k 1 does not hold if eery obstacle has at most s sides, for some fixed 3 s < k. In that case we se a similar constrction, in which a big s-sided obstacle hides s 1 smaller obstacles behind all its sides excet one, and the constrction is reeated recrsiely. This constrction corresonds to a comlete tree with branching factor s 1, in which the smaller obstacles are the children of a larger obstacle. For a fixed ale of s, we set h as the height of the comlete (s 1)-ary tree. Ths, the nmber of obstacles, k = (s 1)h 1, s 2 can be as high as we desire. The nmber of leaes in the tree is (s 1) h 1. A simle manilation of this eation shows that this nmber eals k k 1 s 1. Hence, the nmber of internal nodes in the tree is k 1 s 1. For the 3-connectiity agmentation, each leaf obstacle needs at least s new edges and each nonleaf obstacle needs at least two new edges. The total nmber of edges reired is at least ( s k k 1 ) ( ) k = sk s 2 ( ) s 2 n 3 (k 1) = (n 3) s 1 s 1 s 1 s 1 1, s which ranges from 5 6 n 5 2 to n O( n) for 3 s k. Figre 8(b) deicts this lower bond constrction for s = 3. 13

14 6 Discssion We hae shown that a 1- or 2-reglar slg with n ertices, where no edge is a chord of the conex hll, can be agmented to a 3-connected slg which has at most 2n 2 edges (Theorems 1 and 2). We conjectre that or reslt generalizes to slgs with maximm degree at most 2 (Conjectre 1.1). The bond of 2n 2 for the nmber of edges is the best ossible in general, bt it may be imroed if few ertices lie on the conex hll, and the comonents of the int grah are interior disjoint conex obstacles, ossibly with a container. It remains an oen roblem to derie tight extremal bonds for 3-connectiity agmentation for (i) 1-reglar slgs with n ertices, h of which lie on the conex hll; and (ii) 2-reglar slgs formed by n s interior-disjoint conex olygons, each with s ertices for s 3. The 3-connectiity agmentation roblem (finding the minimm nmber of new edges for a gien slg) is known to be NP-hard [13]. Howeer, the hardness roof does not aly to 1- or 2-reglar olygons. It is an oen roblem whether the connectiity agmentation remains NP-hard restricted to these cases. We hae comared the nmber of edges in the reslting 3-connected slgs with the benchmark 2n 2, which is the best ossible bond for 0-, 1-, and 2-reglar slgs. More generally, for a 3-agmentable slg G 0 = (V, E 0 ) with n 4 ertices, let f(g 0 ) = E 1 be the minimm nmber of edges in a 3-connected agmentation (V, E 1 ) of the emty slg (V, ); and let g(g 0 ) = E 2 be the minimm nmber of edges in a 3-connected agmentation (V, E 2 ), E 0 E 2, of the slg G 0. It is clear that f(g 0 ) g(g 0 ). With this notation, we can characterize the slgs G 0 where all edges in E 0 are sefl for 3-connectiity: these are the slgs for which f(g 0 ) = g(g 0 ) is ossible. In general, it wold be interesting to stdy the behaior of the difference g(g 0 ) f(g 0 ). References [1] R. K. Ahja, T. L. Magnanti, and J. B. Orlin, Network Flows: Theory, Algorithms, and Alications, Prentice Hall, NJ, [2] O. Aichholzer, S. Bereg, A. Dmitresc, A. García, C. Hemer, F. Hrtado, M. Kano, A. Márez, D. Raaort, S. Smorodinsky, D. L. Soaine, J. Urrtia, and D. R. Wood, Comatible geometric matchings, Comtational Geometry: Theory and Alications 42 (2009), [3] M. Al-Jbeh, M. Ishae, K. Rédei, D. L. Soaine, and C. D. Tóth, Tri-edgeconnectiity agmentation for lanar straight line grahs, Proc. Int. Sym. on Algorithms and Comtation, ol of Lectre Notes in Comter Science, Sringer, 2009, [4] K. P. Eswaran and R. E. Tarjan, Agmentation roblems, SIAM J. on Comting 5 (1976), [5] A. García, F. Hrtado, C. Hemer, J. Tejel, and P. Valtr, On triconnected and cbic lane grahs on gien oint sets, Comtational Geometry: Theory and Alications 42 (2009), [6] M. Hoffmann and C. D. Tóth, Segment endoint isibility grahs are Hamiltonian, Comtational Geometry: Theory and Alications 26 (2003),

15 [7] T.-S. Hs and V. Ramachandran, On finding a minimm agmentation to biconnect a grah, SIAM J. on Comting 22 (5) (1993), [8] T.-S. Hs, Simler and faster biconnectiity agmentation, J. of Algorithms 45 (1) (2002), [9] B. Jackson and T. Jordán, Indeendence free grahs and ertex connectiity agmentation, J. of Combinatorial Theory, Ser. B 94 (2005), [10] G. Kant and H. L. Bodlaender, Planar grah agmentation roblems, Proc. 2nd Worksho on Algorithms and Data Strctres, ol. 519 of Lectre Notes in Comter Science, Sringer, 1991, [11] D. L. Millman, M. O Meara, J. Snoeyink, and V. Verma, Maximm geodesic roting in the lane with obstacles, Proc. 22nd Canadian Conf. on Comtational Geometry, Winnieg, MB, 2010, [12] J. Plesník, Minimm block containing a gien grah, Archi der Mathematik 21 (1976), [13] I. Rtter and A. Wolff, Agmenting the connectiity of lanar and geometric grahs, P roc. Int. Conf. on Toological and Geometric Grah Theory, Paris, France, Electronic Notes in Discrete Mathematics, 31 (2008), [14] C. D. Tóth and P. Valtr, Agmenting the edge connectiity of lanar straight line grahs to three, in: 13th Sanish Meeting on Comtational Geometry, Zaragoza, Sain,