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Open Problems from CCCG 2006 CCCG 2007, Ottawa, Ontario, August 20–22, 2007 Open Problems from CCCG 2006 Erik D. Demaine∗ Joseph O’Rourke† The following is a list of the problems presented on between any two unit-length 3D trees of the same August 14, 2006 at the open-problem session of the 18th number of links using edge reconnections? Canadian Conference on Computational Geometry held in Kingston, Ontario, Canada. Update: At the conference, a related move was Edge Reconnections in Unit Chains defined: a tetrahedral swap alters consecutive ver- Anna Lubiw tices (a, b, c, d) of a chain to either (a, c, b, d) or University of Waterloo (a, d, b, c), whichever of the two yields a chain. It [email protected] was established (by Erik Demaine, Anna Lubiw, and Joseph O’Rourke) that, if tetrahedral swaps Is it possible to reconfigure between any two con- are permitted without regard to self-intersection, figurations of a unit-length 3D polygonal chain, or then they suffice to unlock any chain. more generally unit-length 3D polygonal trees, by “edge reconnections”? Edge Swaps in Planar Matchings Ferran Hurtado (via Henk Meijer) a Univ. Polit´ecnicade Catalunya [email protected] b Is the space of (noncrossing) plane matchings on (a) c a set of n points in general position connected by “two-edge swaps”? A two-edge swap replaces two edges ab and cd of a plane matching M with a dif- ferent pair of edges on the same vertices, say ac and bd, to form another plane matching M 0. (The swap is valid only if the resulting matching is non- crossing.) Notice that ac and bd together with their a c a=c c replacement edges form a length-4 alternating cycle that does not cross any segments in M. Is it pos- b sible to reach any plane matching from any other (b) b b 2 (c) 1 on the same point set by a sequence of such two- edge swaps? In other words, if we form the graph Figure 1: An edge reconnection. where vertices correspond to plane matchings and edges correspond to two-edge swaps, is this graph An edge reconnection is an operation on a path connected? of three consecutive vertices (a, b, c) of a chain or This is a reposing of one question from a series of tree linkage; refer to Figure 1. If it is possible to related questions posed at CCCG 2003 [DO04]; see fold the linkage, preserving edge lengths and avoid- also [MR04]. It is known that k-edge swaps suffice ing self-intersection, to bring the two incident edges to connect the space of plane matchings, but only ab and bc into coincidence, then the edge reconnec- if no bound is imposed on k [HHNR05]. The same tion breaks the connection between the edges at b, work also proved that a two-edge swap always ex- splits b into two vertices b and b , and fuses a=c 1 2 ists, i.e., the two-edge-swap graph on plane match- together. This operation changes a chain into a ings cannot have an isolated node. (The proof, tree. If we apply this operation to a tree, there will however, was not included in the journal version of generally be attachments at b; allow them to at- the paper.) For n = 2m points in convex position, tach to either b or b . Is it possible to reconfigure 1 2 the two-edge-swap graph is well understood: its ∗MIT Computer Science and Artificial Intelligence Laboratory, vertex connectivity is m − 1, it contains no Hamil- 32 Vassar Street, Cambridge, MA 02139, USA, [email protected] tonian path for m odd and greater than 3, and it † Department of Computer Science, Smith College, Northamp- contains a Hamiltonian cycle for even values of m ton, MA 01063, USA. [email protected] greater than 2 [HHN02]. 19th Canadian Conference on Computational Geometry, 2007 References 18th Canad. Conf. Comput. Geom., pages 59–62, 2006. [DO04] Erik D. Demaine and Joseph O’Rourke. Open problems from CCCG 2003. [NG94] S. Ntafos and L. Gewali. External In Proc. 16th Canad. Conf. Comput. watchman routes. Visual Comput., Geom., pages 209–211, 2004. 10:474–483, 1994. [HHN02] Carmen Hernando, Ferran Hurtado and [PD90] H. Plantinga and C. R. Dyer. Visibility, Marc Noy. Graphs of non-crossing per- occlusion, and the aspect graph. Inter- fect matchings. Graphs and Combina- nat. J. Comput. Vision, 5(2):137–160, torics 18(3):517–532, 2002. 1990. [HHNR05] Michael H. Houle, Ferran Hurtado, The SS-Divide Marc Noy, Eduardo Rivera-Campo. Joseph O’Rourke Graphs of triangulations and perfect Smith College matchings. Graphs and Combinatorics, [email protected] 21(3):325–331, 2005. [MR04] Henk Meijer and David Rappaport. Si- Fix a nonvertex point p on the surface of a convex multaneous edge flips for convex sub- polyhedron P of n vertices. Draw shortest paths to divisions. In Proc. 16th Canad. Conf. all vertices, breaking ties arbitrarily. Connect the Comput. Geom., pages 57–59, 2004. vertices by geodesics in the angular order of their shortest paths from p. See Figure 2. This geodesic Shortest Aspect Tour 7 Joseph O’Rourke 6 7 6 Smith College [email protected] 3 3 2 How quickly can one compute an exterior tour of 2 a polyhedron that “views it from all different an- 5 gles”? More precisely, the aspect graph of a poly- hedron P represents the cells of the arrangement of 0 planes containing the faces of P : from each cell, one 1 4 0 1 5 sees a different “aspect” of P . The aspect graph has Θ(n6) nodes [PD90]. The goal is to find a closed Figure 2: The ss-divide for a cube with respect to a point path π in 3 of minimum possible length that visits R 3 up the middle of the front face. every aspect, i.e., π meets every cell of the arrange- 4 ment, and does not penetrate the interior of the polyhedron P . polygon is called the ss-divide in [DO07], because it divides the source unfolding from the star unfolding The same problem may be posed in the plane, with respect to p. It partitions the surface into two and with or without the constraint that the path halves, neither of which contains any vertices, and be a closed tour. both of which unfold without overlap. 1. Under what conditions are both unfolded Update: The problem is related to the external polygons convex? watchman route problem, where the goal is to find a tour exterior to the polyhedron such that every 2. Under what conditions are both unfolded point on the polyhedron’s surface is visible from polygons congruent? (They are in the exam- some point on the tour. This problem was first ple in Figure 2.) posed and solved in [NG94]; details even for planar 3. It is established in [AAOS97] that there are convex polygons remain unresolved [AW06]. An Θ(n4) distinct permutations of the vertices re- aspect tour is a more stringent requirement, so in alized by ss-divides for different points p. Can general is longer than the optimal external watch- the shortest ss-divide for a fixed polyhedron man route. be computed in o(n4) time? References 4. Given a set of permutations of vertices cor- responding to all the ss-divides for some un- [AW06] Rafa Absar and Sue Whitesides. On known polyhedron, how difficult is it to con- computing shortest external watchman struct some polyhedron P that realizes those routes for convex polygons. In Proc. permutations? CCCG 2007, Ottawa, Ontario, August 20–22, 2007 References On the other hand, the Sprague-Grundy theory of impartial games gives a polynomial-time algo- [AAOS97] Pankaj K. Agarwal, Boris Aronov, rithm for determining the winner from a position Joseph O’Rourke, and Catherine A. of Divide-and-Conquer. Specifically, a board posi- Schevon. Star unfolding of a polytope tion fitting in a x × y × z box has at most x2y2z2 with applications. SIAM J. Comput., future board positions; each can be associated with 26:1689–1713, 1997. a “nimber” describing the game-theoretic value of [DO07] Erik D. Demaine and Joseph O’Rourke. the position; and the nimber of a position can be Geometric Folding Algorithms: Link- computed from the nimber of all its possible moves ages, Origami, Polyhedra. Cambridge using the “mex rule”. See [Dem01]. Is there a sim- University Press, July 2007. http:// pler characterization of the winner? www.gfalop.org. References Divide-and-Conquer: The Game Joseph O’Rourke [BCH03] Elwyn R. Berlekamp, John H. Conway, Smith College and Richard K. Guy. Winning Ways for [email protected] Your Mathematical Plays. 2nd edition, A K Peters, 2003. Chomp is described in What is the outcome of the following two-player volume 3 on page 632. combinatorial game? The board is the 3D integer lattice, and a board position is a set S of unit- [Dem01] Erik D. Demaine. Playing games with cube lattice cells marked as solid. For example, algorithms: Algorithmic combinatorial these cells might form an orthogonal polyhedron. game theory. In Proc. 26th Sym- The two-player game is impartial: both players can pos. Math. Found. in Comput. Sci., make the same moves from the same board posi- LNCS 2136, August 2001, pages 18–32. tion. In a move, a player chooses an integral co- arXiv:cs.CC/0106009 ordinate plane that has at least one cube of S to Blokus either side, and removes from S all the cubes in one Erik Demaine half-space bounded by the plane.
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