Vertex Unfoldings of Orthogonal Polyhedra: Positive, Negative, and Inconclusive Results

Transcription

1 CCCG 2018, Winnipeg, Canada, August 8 10, 2018 Vertex Unfoldings of Orthogonal Polyhedra: Positive, Negative, and Inonlusive Results Luis A. Garia Andres Gutierrrez Isaa Ruiz Andrew Winslow Abstrat We obtain results for three questions regarding vertex unfoldings of orthogonal polyhedra. The (positive) first result is a simple proof that all genus-0 and genus-1 orthogonal polyhedra have grid vertex unfoldings. The (negative) seond result is an orthogonal polyhedron that is not vertex-unfoldable. The (inonlusive) third result is a vertex unfolding of an orthogonal polyhedron that annot be arranged orthogonally, evidene that deiding whether an orthogonal polyhedron has a vertex unfolding may not lie in NP. 1 Introdution The study of unfolding polyhedral surfaes an be traed to as early as the 1500s when Albreht Dürer onsidered unfoldings of onvex polyhedra via uts only along edges (alled edge unfoldings). To explore unfoldings of non-onvex polyhedra, Biedl et al. [3] onsidered orthogonal polyhedra, in whih all faes are perpendiular to one of the three axes, providing examples of edge ununfoldable orthogonal polyhedra as well as methods for unfolding some lasses of orthogonal polyhedra. Subsequent work has further explored the boundary between unfoldable and ununfoldable orthogonal polyhedra by varying both the types of unfoldings permitted and lasses of orthogonal polyhedra under onsideration. For instane, one line of work has onsidered broadening permitted unfoldings via refinement: adding a regular grid of (potential ut) edges to eah fae. Damian, Flatland, and O Rourke [8] proved that exponential refinement was suffiient to unfold any orthogonal polyhedron; this refinement was later redued to quadrati [7], and then linear [6]. To unfold the edge ununfoldable examples of Biedl et al. [3], it is suffiient to add grid edges formed by the intersetion of orthogonal planes interseting eah vertex of the polyhedron. Suh grid edge unfoldings have been found for several lasses of orthogonal polyhedra [3, 5, 10, 13]. Department of Computer Siene, University of Texas Rio Grande Valley, Edinburg, TX. luis.a.garia01@utrgv.edu, andres.a.gutierrez01@utrgv.edu, isaa.ruiz02@utrgv.edu, andrew.winslow@utrgv.edu Extending grid unfoldings to allow uts meeting at (but exluding) a vertex yields grid vertex unfoldings. Vertex unfoldings were first introdued for general polyhedra [11], and grid vertex unfoldings have been shown to exist for some lasses of orthogonal polyhedra [12], inluding all genus-0 orthogonal polyhedra [9]. Here we obtain three new results on grid vertex unfoldings of orthogonal polyhedra: Every genus-0 and genus-1 orthogonal polyhedron has a grid vertex unfolding (extending a previous result of Damian, Flatland, and O Rourke [9] to inlude genus 1 and providing an alternative, simpler proof). There exists an orthogonal polyhedron with faes homeomorphi to disks that does not have a vertex unfolding (omplementing a vertex-ununfoldable topologially onvex polyhedron of Abel, Demaine, and Demaine [2] and vertex-ununfoldable orthogonal polyhedra with faes not homeomorphi to disks by Biedl et al. [3]). There exists a maximally ut vertex unfolding of an orthogonal polyhedron that annot be made orthogonal, raising the question of whether deiding if an orthogonal polyhedron has a vertex unfolding is in NP (in ontrast with the trivial ontainment in NP of deiding whether an orthogonal polyhedron has an edge unfolding [1]). 2 Definitions This work onsiders orthogonal polyhedra, i.e. polyhedra where all edges are parallel to the x-, y-, or z-axis. A gridded polyhedron has edges everywhere that an xy-, xz-, or yz-plane intersets a vertex of the polyhedron, resulting in faes that are edge-adjaent retangles. A polyube is a speial ase of gridded orthogonal polyhedra whose faes are unit squares. An unfolding is a onneted planar arrangement of the surfae of a polyhedron by the addition of uts. If the uts are restrited to the polyhedron s edges, then the resulting unfolding is an edge unfolding if the surfae remains strongly onneted, and a vertex unfolding otherwise. A surfae that permits no additional edges or verties to be ut without disonneting the surfae is maximally ut.

2 30 th Canadian Conferene on Computational Geometry, 2018 Unfoldings that are (possibly weakly onneted) orthogonal polygons are also alled orthogonal. In the ase of vertex unfoldings, point onnetivity on the surfae yields a hinge or joint allowing portions of the surfae to rotate relative to eah other. Thus vertex unfoldings are weakly onneted polygons that may either be orthogonal or not. The flexibility around verties allows for faes in a vertex unfolding to (potentially) not appear in the same lokwise order around a ommon vertex as they do on the original surfae; here we allow suh rearrangement of vertex-adjaent faes (see [11] for further disussion). 3 A Simple Proof that Genus- 1 Orthogonal Polyhedra have Grid Vertex Unfoldings Here we prove the existene of vertex unfoldings for low-genus polyubes using an approah reminisent of a proof of a similar result [11] for polyhedra with (possibly interseting) triangular faes. This implies a orollary (Corollary 2) for grid vertex unfoldings that extends the previous result by Damian, Flatland, and O Rourke [9] on grid vertex unfoldings to inlude genus-1 polyhedra. Theorem 1 Every genus-0 and genus-1 polyube has a vertex unfolding. Proof. First, we prove that the fae dual graph (obtained by reating a vertex for every fae of the polyube and onneting pairs of edge-adjaent faes) of every suh polyube has a Hamiltonian path. Afterwards, we show how to use a Hamiltonian path through the faes to obtain a vertex unfolding. Bodini and Lefran [4] prove that polyube fae dual graphs are 4-onneted. Intuitively, this is due to the four diretions that must be traversed to obtain a disonneting ut of the surfae of a polyube (i.e. nonontratable yle on the surfae). As they observe, Tutte [15] proves that all genus-0 4-onneted graphs are Hamiltonian, implying that all fae dual graphs of genus-0 polyubes are Hamiltonian. Reently, Thomas, Yu, and Zang [14] proved that all genus-1 4-onneted graphs have a Hamiltonian path. The vertex unfolding onsists of a path of (vertex) onneted faes appearing in the same order as on the Hamiltonian path previously proved to exist. These faes are arranged left to right, and remain either edgeonneted (if the previous fae is left of the next fae) or ut to beome vertex-onneted and rotated by ±90 (if the previous fae is above or below the next fae). These three ases are seen in Figure 1. Only these three ases need be onsidered due to maintaining the invariant that before the edge onneting the previous (gray) fae to the next (green) fae is ut, the next fae is above, below, or to the right of the previous fae. Figure 1: The three ases for vertex unfolding a sequene of onseutive faes along a Hamiltonian path of the fae dual graph. The invariant maintained is that eah subsequent fae (in green) begins attahed to the above, below, or right of the previous fae (indiated as arrows). In the right two ases, the fae is rotated ±90 to maintain the invariant. Observe that the resulting unfolding is also orthgonal. Moreover, beause the resulting unfolding an be partitioned into vertial strips eah ontaining one fae of the polyube, a similar result holds if the faes are edge-adjaent retangles, as they are in all gridded orthogonal polyhedra: Corollary 2 Every genus-0 or genus-1 orthogonal polyhedron has an orthogonal grid vertex unfolding. 4 A Vertex-Ununfoldable Orthogonal Polyhedron Theorem 3 There is an orthogonal polyhedron with simple faes that annot be vertex-unfolded. Proof. The vertex-ununfoldable polyhedron onsists of a box with a thin ridge through two adjaent faes of the box (see Figure 2). Figure 2: An orthogonal polyhedron that annot be vertex unfolded. We prove that the polyhedron is ununfoldable by onsidering only a portion of the surfae onsisting of two large base faes ontaining the ridge (green in Figure 2), and the ell, putt, and bungie faes on the ridge (blue, pink, and yellow, respetively, in Figure 2). In the unfolding, at least one of the two (symmetri) putt faes must be onneted to a base fae via a sequene of faes that exludes the other putt fae. For

3 CCCG 2018, Winnipeg, Canada, August 8 10, 2018 the remainder of the proof, we onsider only this sequene of faes, proving that any suh sequene annot be arranged without overlap. Sine the boundary of the putt shares no boundary with the base fae on the surfae, the sequene must ontain either a bungie or ell fae. Regardless of the sequene of faes onneting the putt to the base fae, the end of the fae sequene is either: 1. ell base, or 2. putt bungie base, or 3. ell bungie base. Case 1: ell base. The first ase implies overlap between the ell fae and base fae (see Figure 3), as these two faes are onneted by either the unique edge they share on the surfae or a vertex of this edge. the entrane of the noth. In the latter ase, the minimum distane between the bungie faes onnetion to the base fae and putt s boundary is at least 8, exeeding the maximum distane that an be spanned by the bungie faes. Figure 5: The two potential putt plaements for Case 2. The red boundary portions denote where bungie faes must onnet (to both the putt and base faes). Case 3: ell bungies base fae. This ase is proved similarly to Case 2. Sine ell faes are 2 units wide, any optimal non-overlapping arrangement of the ell and base faes forms a right triangle onsisting of a portion of the ell fae, with the 90 vertex on the ell and two remaining verties at the entrane of the noth (see Figure 6). 8 Figure 3: Any attahed ell and base fae must overlap. For the other two ases, we simplify the analysis by onsidering the bungie faes as a zero-width urve of length between 0 and the maximum distane between two loations on a sequene of onneted bungie faes (see Figure 4) of 7.30 < 8. That is, we suppose they behave as an elasti bungie ord Figure 4: The maximum length of a onneted sequene of bungie faes is Case 2: putt bungies base fae. In the unfolding, a bungie fae onnets to the putt at boundary loation(s) limited to those drawn in red in Figure 5. Any non-overlapping arrangement of the putt and base faes either has the entire noth filled by the putt (leaving no available loation for any bungie fae in the unfolding) or has no portion of the boundary of the putt to whih the bungie faes an attah more than distane 1 from 1/2 Figure 6: Arranging an ell fae as deep in the noth as possible. By Thales s theorem, the 90 vertex lies on a irle whose diameter is the noth entrane. Then sine this irle has radius 1/2, the 90 vertex (and all other loations on the ell) has distane at most 1/2 from the entrane of the noth. So as in Case 2, the minimum distane between the bungie faes onnetion to the base fae and putt s boundary exeeds the maximum distane that an be spanned by the bungie faes. 5 Evidene that Vertex Unfolding Orthogonal Polyhedra is not in NP Finally, we onsider the omplexity of deiding whether a polyube has a vertex unfolding. Speifially, whether the problem lies in NP. As Abel and Demaine [1] observe, the same problem limited to edge unfoldings is easily seen to be in NP. One proof uses the following simple algorithm: nondeterministially selet a set of maximal set of polyube

4 30 th Canadian Conferene on Computational Geometry, 2018 faes to ut that leaves the fae dual graph onneted (i.e. a set of ut edges that yields a tree-shaped fae dual graph). Then hek whether the resulting surfae is indeed an unfolding, i.e. has no overlaps. This NP algorithm for edge unfolding relies in part on the uniqueness of the indued unfolding. However, in vertex unfoldings, faes may be onneted by a single vertex, allowing infinitely many angles at whih these two faes may be arranged. Thus the same algorithmi approah for vertex unfoldings requires effiiently determining whether a maximally set of ut edges yields a surfae that an be arranged into a vertex unfolding (by areful seletion of adjaent fae angles). One natural approah to resolving this issue is to prove that any maximally ut polyube surfae has a vertex unfolding only if there is suh an unfolding that is orthogonal, i.e. where all adjaent fae angles from the set {0, 90, 180, 270 }. Here, we prove by example that this is not the ase. a a 2 a 3 b 2 a 4 b 3 b 4 b5 b 1 b b 6 Figure 8: A maximally ut vertex unfolding of the polyube in Figure 7 that annot be arranged orthogonally. The solid blak lines represent uts. Figure 7: A polyube with a maximally ut vertex unfolding that annot be arranged orthogonally. Theorem 4 There is a maximally ut vertex unfolding of a polyube with no orthogonal arrangement. Proof. The maximally ut unfolding with no orthogonal arrangement is seen in Figure 8. The four L-shaped regions adjaent to fae (olored fushia, eggshell, light green, and pink in Figure 8) are laws. The two regions attahed by a single vertex to fae onsist of fae sets A = {,..., } and B = {b 1,..., b 5 } For the remainder of the proof, we assume fae is orthogonal. In any orthogonal arrangement of the unfolding, eah of six fae adjaent to lies in one of the eight orthogonal loations adjaent to (see Figure 9). Moreover, faes and b 1 must lie in loations 2, 3, or 4. 1 Claw arrangements up to symmetry. Observe that the laws ome in symmetri pairs (olored fushia/pink and eggshell/green in Fig. 8) and eah pair is also symmetri. Without loss of generality, assume that the 1 Reall that we allow vertex-adjaent faes to appear in a different lokwise ordering around the vertex than they appear on the surfae; the proof holds even when suh unfolding are permitted Figure 9: The eight possible loations for faes adjaent to in orthogonal unfolding. laws appear in the relative order around seen in Figure 8 (i.e. in lokwise order, fushia, pink, light green, eggshell). Then the fushia and pink laws must have faes in loations 1 and 8, respetively, sine otherwise overlap ours between, b 1, and two fushia faes sharing a ommon vertex (see Figure 10). By symmetry, the eggshell and light green laws must lie in loations 5 and 6. Next, onsider the arrangement of the fushia law faes. Figure 11 enumerates the five possible arrangements. The remainder of the proof is dediated to proving that eah arrangement leads to overlap. Arrangements 4 and 5. Both arrangements ause overlap due to more than four faes sharing a ommon vertex loation. For arrangement 4, these faes onsist of two fushia faes,,, and b 1. For arrangement 5, these faes onsist of four fushia faes, one pink fae, and.

5 CCCG 2018, Winnipeg, Canada, August 8 10, 2018 Figure 13: Attempting to unfold A and B by plaing in loation 2. b 2 b 1 Figure 10: Attemping to plae the first fae of the fushia law into loation 2 auses overlap. b 3 a 2 b4 b a : 2 : 3 : 4 : 5 : 6 : b 2 b 1 b 2 ab 1 a 5 4 ba 3 5 b 5 b 3 a 3 ab 4 a 3 a 2 b 4 b 5 a 2 Figure 11: The five possible arrangements of the fushia law faes. Figure 14: Top: required arrangement of faes b 1, b 2, b 3,, a 2. Bottom: the two options for arranging b 4, b 5, and a 3 that further avoid overlap (in both ases, overlap involving a 4 and still ours). a 2 b 2 b 1 b 1 b 2 b 3 b 4 b 5 b 6 a 4 Figure 12: Two possible onfigurations if rossings are allowed. a 3 a 2 We onsider the ases in order. In the ase that is in loation 2, A overlaps with the fushia law, sine due to the uts, a 2 is left of or overlaps the fushia law, and likewise a 3 is either left or above a 2. Next, onsider the ase that b 1 is plaed in loation 2. In this ase, there are unique plaements of faes b 2, b 3,, and a 2 that avoid overlap (the top portion of Figure 14) and only two plaements of b 4, b 5 and a 3 that also avoid overlap (the bottom portion of Figure 14). In both ases, a 4 and overlap with other faes due to a vertex inident to more than four faes. Arrangement 3. The faes of A or B may be arranged so that the first fae ( or b 1 ) lies in loation 2 as seen in Figure 12. As shown the figure, either option auses overlap. Arrangements 1 and 2. Here we onsider plaing A and B. Either or b 1 must be plaed in loation 2 or 4. By symmetry (and ignoring the existene of b 6 ), it suffies to onsider two ases: 1. is plaed in loation b 1 is plaed in loation 2. 6 Open Problems Eah of our results leads diretly to a natural open problem in the same diretion. Sine all genus-0 and genus-1 orthogonal polyhedra have grid vertex unfoldings, what about genus-2? Open Problem 1 Does every genus-2 orthogonal polyhedron have a grid vertex unfolding? Sine there is an orthogonal polyhedron with simple faes and no vertex unfolding, does the same hold for simple faes that are also edge-inident retangles?

6 30 th Canadian Conferene on Computational Geometry, 2018 Open Problem 2 Does every orthogonal polyhedron (of any genus) have a grid vertex unfolding? Finally, we provided an example of a maximally ut polyube with an unfolding, but no orthogonal unfolding, 2 demonstrating that the orthogonal unfoldings of maximally ut grided orthogonal polyhedra do not haraterize the (unrestrited) unfoldings of maximally ut gridded orthogonal polyhedra (eliminating one partiularly simple proof that deiding whether an orthogonal polyhedron has a vertex unfolding is in NP). Thus the following two related problems remain open: Open Problem 3 Is deiding if an orthgonal polyhedron is grid vertex-unfoldable in NP? Is the problem NP-hard? Additionally, the relationship between orthogonal grid vertex-unfoldings and (unrestrited) grid vertexunfoldings also remain open: Open Problem 4 Does there exist a grid vertexunfoldable orthogonal polyheron with no orthogonal grid vertex-unfolding? To our knowledge, the same question also remains open for non-grid vertex-unfoldings: Open Problem 5 Does there exist a vertex-unfoldable orthogonal polyhedron no orthogonal vertex-unfolding? Aknowledgments We would like to thank Erik Demaine for disussion of known results related to vertex unfoldings, and anonymous reviewers whose suggestions improved the readability and orretness of the paper. Referenes [1] Z. Abel and E. D. Demaine. Edge-unfolding orthogonal polyhedra is strongly NP-omplete. In Proeedings of the 23rd Canadian Conferene on Computational Geometry (CCCG), [2] Z. Abel, E. D. Demaine, and M. L. Demaine. A topologially onvex vertex-ununfoldable polyhedron. In Proeedings of the 23rd Canadian Conferene on Computational Geometry (CCCG), [3] T. Biedl, E. Demaine, M. Demaine, A. Lubiw, M. Overmars, J. O Rourke, S. Robbins, and S. Whitesides. Unfolding some lasses of orthogonal polyhedra. In Proeedings of the 10th Canadian Conferene on Computational Geometry (CCCG), pages 70 71, Note that the polyube from whih the example is obtained is indeed vertex (and also edge) unfoldable, and so is not a potential ounterexample to Open Problem 2. [4] O. Bodini and S. Lefran. How to tile by dominoes the boundary of a polyube. In Proeedings of the 13th International Conferene on Disrete Geometry for Computer Imagery (DGCI), volume 4245 of LNCS, pages Springer, [5] E. W. Chambers, K. A. Sykes, and C. M. Traub. Unfolding retangle-faed orthostaks. In Proeedings of the 24th Canadian Conferene on Computational Geometry (CCCG), pages 23 28, [6] Y.-J. Chang and H.-C. Yen. Improved algorithms for grid-unfolding orthogonal polyhedra. International Journal of Computational Geometry & Appliations, 27(1):33 56, [7] M. Damian, E. D. Demaine, and R. Flatland. Unfolding orthogonal polyhedra with quadrati refinement: The delta-unfolding algorithm. Graphs and Combinatoris, 30(1):25 40, [8] M. Damian, R. Flatland, and J. O Rourke. Epsilonunfolding orthogonal polyhedra. Graphs and Combinatoris, 23: , [9] M. Damian, R. Flatland, and J. O Rourke. Grid vertex-unfolding orthogonal polyhedra. Disrete & Computational Geometry, 39(1 3), [10] M. Damian and H. Meijer. Grid edge-unfolding orthostaks with orthogonally onvex slabs. In Proeedings of the 14th Annual Fall Workshop on Computational Geometry, pages 20 21, [11] E. D. Demaine, D. Eppstein, J. Erikson, G.W. Hart, and J. O Rourke. Vertex-unfoldings of simpliial manifolds. In Proeedings of the 18th Symposium on Computational Geometry (SoCG), pages ACM Press, [12] E. D. Demaine, J. Iaono, and S. Langerman. Grid vertex-unfolding orthostaks. International Journal of Computational Geometry & Appliations, 20(3): , [13] J. O Rourke. Unfolding orthogonal terrains [14] R. Thomas, X. Yu, and W. Zang. Hamilton paths in toroidal graphs. Journal of Combinatorial Theory, Series B, 94: , [15] W. T. Tutte. A theorem on planar graphs. Transations of the Amerian Mathematial Soiety, 82:99 116, 1956.