Graphs of triangulations and perfect matchings M. E. Houle1 , F. Hurtado2 , M. Noy2 , E. Rivera-Campo3
Dedicated to Professor V´ıctor Neumann-Lara on the occasion of his 70th birthday Abstract. Given a set P of points in general position in the plane, the graph of triangulations T (P ) of P has a vertex for every triangulation of P , and two of them are adjacent if they differ by a single edge exchange. We prove that the subgraph TM (P ) of T (P ), consisting of all triangulations of P that admit a perfect matching, is connected. A main tool in our proof is a result of independent interest, namely that the graph M(P ) that has as vertices the non-crossing perfect matchings of P and two of them are adjacent if their symmetric difference is a single non-crossing cycle, is also connected. Key words. Triangulation. Perfect matching. Non-crossing.
1. Introduction A set of points in the plane are in general position if no three of them lie in a common straight line. A triangulation of a setP of points in general position in the plane is a partition of the convex hull of P into triangles whose vertices are all the points in P . Given a set P of points in general position in the plane, the graph of triangulations T (P ) has a vertex for every triangulation of P , and two of them are adjacent if they differ by a single edge exchange. Graphs of triangulations have been widely studied; see for example [3,6–8]. In particular, it is well-known that T (P ) is a connected graph. In this paper we study the subgraph TM (P ) of T (P ), consisting of all triangulations of P that admit a perfect matching. Not every triangulation contains a perfect matching, so in general TM (P ) is a proper subgraph of T (P ). Our main result is that the graph TM (P ) is connected for any set P in general position. In other words, we show that any two triangulations of P containing a perfect matching can be connected through a sequence of edge exchanges, always resulting in triangulations containing a perfect matching. 1
National Institute of Informatics, Tokyo, Japan,
[email protected]. Departament de Matem`atica Aplicada II, Universitat Polit`ecnica de Catalunya, Espa˜ na,
[email protected],
[email protected]. Partially supported by Projects DGES-SEUID PB98-0933, MCYT-BFM2001-2340, MCYT-FEDER-BFM2002-0557 and Gen. Cat 2001SGR00224. 3 Departamento de Matem´aticas, Universidad Aut´onoma Metropolitana - Iztapalapa, M´exico,
[email protected]. Part of the research was done while this author was on sabbatical leave visiting the Universitat Polit`ecnica de Catalunya with grants by MECD-Espa˜ na and CONACYT-M´exico. 2
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In order to prove our main result, we first prove another result of independent interest, which we now describe. Given a set P in the plane of even cardinality, a perfect matching in P is said to be non-crossing if no two of its edges intersect. The graph M(P ) has as vertices the non-crossing perfect matchings of P , and two of them are adjacent if their symmetric difference is a single non-crossing cycle. The case where P is in convex position was studied in [5]. We show that the graph M(P ) is connected for any set P in general position; this is the key ingredient for proving that TM (P ) is a connected graph. Our graph theory terminology follows that of [2]. The rest of the paper is organized as follows. Section 2 contains the results on graphs of perfect matchings, and Section 3 on graphs of triangulations containing perfect matchings. 2. Graphs of perfect matchings Let P be a set of 2m points in general position in the plane. The symmetric difference of two non-crossing perfect matchings in P is a set of alternating cycles; some of these cycles may have crossings, see Figure 1. We say that two perfect matchings M and N differ in a single alternating non-crossing cycle exchange if their symmetric difference is a single non-crossing cycle; for brevity we say that N is obtained from M by performing a flip.
Fig. 1. Two matchings M and N ; their symmetric difference (right) is the union of two alternating cycles C and D, but only D is non-crossing.
The graph of non-crossing perfect matchings M(P ) of P is the graph with one vertex for each non-crossing perfect matching of P , in which two matchings are adjacent if and only if one can be obtained from the other by a flip. The requirement that the cycle involved in the exchange is non-crossing is not only a natural one, but it is critical when applying Theorem 1 in the next section. Theorem 1. For any set P of 2m points in general position in the plane, the graph M(P ) is a connected graph. Proof. Let τ (P ) be a non-crossing perfect matching of P obtained as follows: let a 0 and b0 be two consecutive points of P lying on the boundary of the convex hull of P0 = P . For i = 0, 1, . . . , m − 1, let Pi+1 = Pi \ {ai , bi } and let ai+1 and bi+1 be two consecutive points of P lying on the boundary of the convex hull of Pi+1 . We call τ (P ) = {a0 b0 , a1 b1 , . . . , am−1 bm−1 } the target matching. In order to prove that M(P ) is connected we show that for any given matching M there is a sequence of flips transforming M into τ (P ). In fact it is enough to prove that we can perform a flip in M in such a way that the resulting matching contains the edge a0 b0 ; the existence of the rest of the sequence then follows due to the inductive definition of τ (P ). Let a = a0 and b = b0 . We assume ab is not an edge in M , as otherwise there is nothing to prove. Without loss of generality we may also assume that a precedes b counterclockwise
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around the boundary of the convex hull CH(P ) of P . For any point p in P , let p 0 denote the point matched with p in M . The basic idea is to perform a piecewise radial sweep searching for a non-crossing alternating path from b0 to a0 . We start by rotating clockwise an open segment b0 t0 , with t0 initially placed at b and moving towards a0 along the edge ba and (possibly) the edge aa0 , until either t0 reaches a0 (Figure 2, left) or the segment b0 t0 encounters the point a0 (Figure 2, center) or a point c1 6= a0 of P in the interior of CH(P ) (Figure 2, right).
Fig. 2. Radial sweep from b0 .
In the first two cases b0 a0 is a visible segment with respect to M , in the sense that it does not intersect any edge of M and therefore we can perform a flip using the non-crossing alternating 4-cycle (bb0 , b0 a0 , a0 a, ab) to bring the edge ab into the matching. In the third case b0 c1 is a visible segment with respect to M . We call c1 and T1 = 0 (b c1 , c1 c01 ), the first contact point and the first alternating path, respectively, and continue the sweep with c01 as the new center of rotation. We iterate the process: when a contact point ci 6= a0 is encountered, the edges c0i−1 ci and ci c0i are added to the current alternating path, the center of rotation is moved to c 0i and the sweep is continued by rotating clockwise an open segment c0i ti , with ti initially placed at ci and moving along the boundary of the area already swept and (possibly) the edges ba and aa0 , until either ti reaches a0 , or the segment c0i ti encounters the point a0 or a point ci+1 6= a0 in the interior of CH (P ) (Figure 3, where i = 1).
Fig. 3. Radial sweep from c01 .
For each contact point cj , the swept area has a reflex angle formed by the edges and cj c0j . Since for any sweeping open segment c0i ti , the point ti slides along the boundary of the swept area and (possibly) the edges ba and aa0 , no contact point cj can be encountered again in the sweep. Nevertheless a sweeping segment c0i ti may encounter, as a new contact point ci+1 , a point c0j previously used as center of rotation (an example is shown in Figure 4). In this case we update the current alternating path back to Tj = (b0 c1 , c1 c01 , c01 c2 , c2 c02 , . . . , c0j−1 cj , cj c0j ) and move the center of rotation back to c0j . To avoid cycling, we continue the sweep by rotating clockwise a new open segment c0j tj , with tj placed, this time, initially at the point x = ti for which the open segment c0i ti encountered the point ci+1 = c0j in the preceding stage (Figure 4, where j = 1 and i = 3). Notice that this choice of starting c0j−1 cj
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Fig. 4. b0 t0 encounters c1 , c01 t1 encounters c2 , c02 t2 encounters c3 , and c03 t3 encounters c01 .
point for tj guarantees that the swept area has a reflex angle at c0j and therefore c0j will not be encountered again in a subsequent stage of the sweep. 0 0 0 We must eventually reach the point a by a visible edge cka in which case, the current alternating path Tk = b0 c1 , c1 c01 , c01 c2 , c2 c02 , . . . , c0k−1 ck , ck c0k , together with the edges c0k a0 , a0 a, ab and bb0 form a non-crossing alternating cycle of length 2k + 4 which we can use to perform a flip that brings the edge ab into the matching. As a remark, notice that the method of the above proof transforms any non-crossing perfect matching of P into the target one τ (P ) using at most m − 1 flips; therefore there is a flip sequence of length at most 2m − 2 connecting any two given matchings in M(P ). 3. Graphs of triangulations Let P be a set of points in general position in the plane. The graph of triangulations T (P ) is the graph with one vertex for each triangulation of P , in which two triangulations T 1 and T2 are adjacent if and only there are edges e ∈ E (T1 ) \ E (T2 ) and f ∈ E (T2 ) \ E (T1 ) such that T2 = (T1 \ {e}) ∪ {f }. In other words, T2 is obtained from T1 by replacing the diagonal of a convex quadrilateral Q by the other diagonal of Q. For any non-crossing set E of line segments with endpoints in P , let T (P, E) be the subgraph of T (P ) induced by the set of triangulations of P whose edge sets contains the set E. The following lemma can be obtained from the proof of Proposition 1 and Observation 3 in [3], for the sake of completeness we include a direct proof here. Lemma 1. Let P be a set of points in general position in the plane, E be a non-crossing set of line segments with endpoints in P and e ∈ / E be a line segment, also with endpoints in P , and such that E ∪ {e} is a non-crossing set. For each triangulation T of P whose edge set E(T ) contains E there is a triangulation S of P , with E ∪ {e} ⊂ E(S), which is connected to T in T (P, E). Proof. Let n be the number of edges of T which are intersected by e. If n = 1 let f = xy be the only edge of T intersected by e. In this case (T \ {f }) ∪ {e} is a triangulation of P which is adjacent to T in T (P, E). We proceed by induction assuming n ≥ 2, and that the result holds for any set E 0 of line segments and any line segment e0 ∈ / E 0 , all with 0 0 0 endpoints in P , such that E ∪ {e } is a non-crossing set, and such that e intersects fewer than n edges of T . Let I be the set of edges of T intersected by e and denote by u and v the endpoints of e. Since e is not an edge of T , there is a triangle avb of T such that ab ∈ I. Rotate a closed segment ut, with t initially placed at v and moving along the edge va, until a point
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x of P is reached; since a ∈ P , such a point x must exist. Do the same operation, now with t moving along the edge vb, until a point y ∈ P is reached (Figure 5). Clearly the quadrilateral uxvy contains no point of P in its interior; we claim that at least one of the segments ux, xv, vy and yx is not an edge of T , otherwise xy would be the only edge of T intersected by e = uv.
Fig. 5. Finding the points x and y in the proof of Lemma 3.1.
Without loss of generality we assume e0 = ux ∈ / E(T ). Since there are no vertices of P in the interior of the quadrilateral uxvy, every edge of T intersected by e 0 is also intersected by e, and since there is at least one edge of T , incident in x, which is intersected by e, the segment e0 intersects less than n edges of T . Let E 0 = E (T ) \I. By induction, there is a triangulation S 0 of P containing E 0 ∪ {e0 } which is connected to T in T (P, E 0 ). Now let E 00 = E 0 ∪ {e0 } and I 0 be the set of edges of S 0 intersected by e. Since e does not intersect any edge in E 00 , we have I 0 ⊂ E (S 0 ) \E 00 and therefore |I 0 | ≤ |E (S 0 )| − |E 00 | = |E (T )| − (|E 0 | + 1) = |E (T ) \E 0 | − 1 = |I| − 1 = n − 1 Again by induction, there is a triangulation S of P containing E 00 ∪ {e} which is connected to S 0 in T (P, E 00 ). Since E ⊂ E 0 ⊂ E 00 , T and S 0 are connected in T (P, E) and also S 0 and S are connected in T (P, E). This implies that T and S are connected in T (P, E). Theorem 2. T (P, E) is a connected graph for any set P of points in general position in the plane, and any non-crossing set E of line segments with endpoints in P . Proof. Let T and S be triangulations of P both of whose edge sets contain the set E. Let E0 = E (T )∩E (S) and {e1 , e2 , . . . , ek } be the set of edges of S not in T . For i = 1, 2, . . . , k let Ei = E0 ∪ {e1 , e2 , . . . , ei }. By Lemma 1 there are triangulations T1 , T2 , . . . , Tk of P such that, for i = 1, 2, . . . , k, Ti contains every edge in Ei , and such that Ti−1 and Ti are connected in T (P, Ei−1 ), where T0 = T . Since E ⊂ Ei for 0, 1, . . . , k, T and Tk are connected in T (P, E); but clearly Tk = S. For any set P of 2m points in general position in the plane, let TM (P ) be the subgraph of T (P ) induced by the set of triangulations of P that admit a perfect matching. Not every triangulation admits a perfect matching (Figure 6), so in general TM (P ) is a proper subgraph of T (P ). Theorem 3. The graph TM (P ) is a connected graph for any set P of 2m points in general position in the plane.
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Fig. 6. The triangulation on the left contains a perfect matching (solid lines) while the one on the right does not, because the 8 independent white vertices are adjacent only to the 6 black vertices.
Proof. Let S and T be triangulations in TM (P ) and MS and MT be perfect matchings of S and T , respectively. By Theorem 1, the graph M (P ) is connected and therefore contains a path MS = M0 , M1 , . . . , Mk = MT joining MS and MT . Since for i = 1, 2, . . . , k, Mi is obtained from Mi−1 by exchanging the edges of an alternating non-crossing cycle, Mi−1 ∪ Mi is a non-crossing set of segments and therefore it can be extended to a triangulation Si of P . Clearly Si is a vertex of TM (P ) for i = 1, 2, . . . , k. Let S0 = S and Sk+1 = T . For i = 0, 1, . . . , k, Si and Si+1 are triangulations of P that contain Mi . By Theorem 2, the graph T (P, Mi ) contains a path joining Si and Si+1 . Since T (P, Mi ) is a subgraph of TM (P ) for each i = 0, 1, . . . , k, the graph TM (P ) contains a path joining S and T . 4. Conclusions Our definition of adjacency of the graph of non-crossing perfect matchings M(P) of P via a single alternating non-crossing cycle exchange contains no constraint on the number of line segments in the cycle. Nevertheless, as pointed out in [4], for the purposes of optimization, enumeration, and random generation, it is desirable that the transformation making a class connected is as local as possible, which amounts to using only exchanges of bounded size. Therefore it is natural to consider a graph of matchings M0 (P ) in which only exchanges in cycles with at most ` segments are considered. It is an open problem to decide whether such graph is connected for some constant value of `. For ` = 4 we have been able to prove that the corresponding graph contains no isolated vertices; yet even this modest fact required a surprisingly long proof. Finally, there are other sets of triangulations of P for which it would be interesting to know whether they induce connected subgraphs of T (P ). For instance, the set of 3connected triangulations of P (see [1] for a related problem), or the set of triangulations of P with minimum degree at least k. References 1. D. Avis, Generating rooted triangulations without repetitions, Algorithmica 16 (1996), 618– 632. 2. G. Chartrand and L. Lesniak, Graphs and digraphs, 3d edition, Chapman and Hall (1996). 3. J. Galtier, F. Hurtado, M. Noy, S. Prenes and J. Urrutia, Simultaneous edge flipping in triangulations, Internat. J. Comput. Geom. Appl. 13 (2) (2003), 113–133.
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4. C. Hernando, M. Houle and F. Hurtado, On local transformation of polygons with visibility properties, Theoretical Computer Science 289 (2) (2002), 919–937. 5. C. Hernando, F. Hurtado and M. Noy, Graphs of non-crossing perfect matchings, Graphs and Combinatorics 18 (2002), 517–532. 6. F. Hurtado and M. Noy, Graph of triangulations of a convex polygon and tree of triangulations, Computational Geometry: Theory and Applications 13 (1999), 179–188. 7. F. Hurtado, M. Noy and J. Urrutia, Flipping edges in triangulations, Discrete and Computational Geometry 22 (1999) 333–346. 8. D.D. Sleator, R.E. Tarjan and W.P. Thurston, Rotation distance, triangulations and hyperbolic geometry, J. Amer. Math. Soc. 1 (1988), 647–682.
