Hamiltonicity in Semi-Regular Tessellation Dual Graphs
Divya Gopinath, Rohan Kodialam, Kevin Lu, Jayson Lynch, Santiago Ospina
HH AMILTONICITY IN S EMI -R EGULAR T ESSELLATION D UAL G RAPHS
Divya Gopinath
Rohan Kodialam
Kevin Lu
Jayson Lynch
Santiago Ospina
October 1, 2019 A BSTRACT
This paper shows NP-completeness for finding Hamiltonian cycles in induced subgraphs of the dualgraphs of semi-regular tessilations. It also shows NP-hardness for a new, wide class of graphscalled augmented square grids. This work follows up on prior studies of the complexity of findingHamiltonian cycles in regular and semi-regular grid graphs [IPS82, AFI +
09, HL18].
In this paper, we consider the problem of determining whether Hamiltonicity (finding a Hamiltonian cycle) is NP-complete in various grid graphs. The problem of finding a Hamiltonian cycle in grid graphs is a heavily-studied prob-lem with a variety of applications in wireless network routing [WGB12] and the lawnmower problem [MAPFM70].We draw on work dating back to 1982, when Itai showed that the Hamiltonian path problem is NP-complete insquare induced grid graphs [IPS82]. More recent research has also shown that the Hamiltonian path problem is alsoNP-complete in other grids [AFI + duals of all eight semi-regular tessellations, in which we create a vertex for each face of the tessellation andan edge between vertices corresponding to adjacent faces. We further consider the problem of Hamiltonicity inaugmented square grids, in which each square cell can optionally contain one or two edges connecting the diagonalsof the square. This class generalizes a number of prior results and includes important grids such as the king’s move,the box-pleat, and the right triangle tessellation grids. A tessellation is a non-overlapping tiling of the plane with polygons. we consider two classes of tessellations: regulartessellations , in which a single kind of regular polygon is used to tile the plane; and semiregular tessellations formedby two or more regular polygons such that the same polygons in the same order surround each of the vertices formedby the tessellation. There are eight such tessellations, as shown in Figure 1 [Wil79]. They are denoted by picking anarbitrary start vertex in the primal graph, and listing the size of the adjacent shapes while walking around a vertex in acounterclockwise fashion. For example, a vertex that is connected to a square and two octagons would be denoted as4.8.8.In this paper, we focus on the duals of these semiregular tessellations. Thus, for each tessellation T we candefine a graph G T as follows: we let each polygon in T correspond to a vertex in G T , and add an edge between twovertices in G T if their corresponding polygons share an edge.We define a Grid Graph G T to be any induced sub-graph created by selecting an arbitrary subset of verticesof G T and edges connecting vertices in this subset. We consider the problem of finding Hamiltonian Cycles in these a r X i v : . [ c s . CC ] S e p .12.123.6.3.63.3.3.3.63.4.6.43.3.4.3.4 3.3.3.4.4 4.8.8 4.6.12 Figure 1: The eight semiregular tessellations
Dual 4.8.8Dual 3.3.4.3.4 Dual 3.3.3.4.4 Dual 4.6.12Dual 3.3.3.3.6Dual 3.4.6.4 Dual 3.6.3.6 Dual 3.12.12
Figure 2: The duals of the eight semiregular tessellations.grid graphs, which are cycles that pass through each vertex of G T exactly once. Specifically, we consider the decisionversion of the Hamiltonicity problem which just determines whether or not the graph can admit such a cycle.A thin grid graph is one in which every vertex of the sub-graph G T its corresponding vertex in the parent graph G T has a neighbor which was not included in G T . This definition differs from that of [AFI +
09] which defines thinwith respect to holes and pixels in the graph. However, this definition is functionally equivalent for the graphs underinspection, is easier to state, and avoids problems of trying to define pixels for augmented square grid graphs.
Generally, we use two main techniques in our reductions. First, we attempt to reduce Hamiltonicity in one grid graphto Hamiltonicity of another grid graph, usually a square or hexagonal one. In this paradigm, we seek to find either asubset of vertices whose induced subgraph is a simple grid, or to create vertex and edge gadgets that form an effectivesimple grid. These are in Sections 3.1 and 3.2.The second type of reduction we employ uses the Tree-Residue Vertex Breaking technique recently developedby Demaine and Rudoy [DR17]. These can be seen in Section 3.3. In this approach, we construct vertex gadgets thathave open and closed configurations. These two techniques suffice for showing NP-hardness results for all of theduals of the semiregular tessellations. We show that Hamiltonicity in the grid graphs corresponding to the duals of all eight semiregular tessellations is NP-Hard. We divide our results into three sections. Section 3.1 examines simple reductions that follow directly from dualgraphs. Section 3.2 focuses on the creation of vertex and edge gadgets to prove NP-Hardness. Finally, Section 3.3gives reductions from Tree-Residue Vertex Breaking. 2 .1 Simple reductions
In this section, we consider graphs where the induced grid graphs can emulate another grid graph by simply picking asubset of vertices whose induced subgraph is a regular tiling of the plane.
Dual of 3.6.3.6
The dual of 3.6.3.6 is known as the rhombile tiling, and contains the hexagonal grid as an induced subgraph. In Figure3, the graph induced by the red vertices is a hexagonal grid graph. Since it is known that Hamiltonicity in hexagonalgrid graphs is NP-hard, Hamiltonicity is also NP-hard in the rhombile tiling.Figure 3: Dual of 3.6.3.6, also known as the rhombiletiling. Figure 4: Red edges are an induced subgraph of therhombile tiling and are a hex grid.
Dual of 4.8.8
The dual of 4.8.8 is known as the Tetrakis tiling, and contains the square grid as an induced subgraph. In Figure5, the graph induced by the red vertices is a square grid graph. Hamiltonicity in square grid graphs is NP-hard, soHamiltonicity is also NP-hard in the Tetrakis tiling.Figure 5: Dual of 4.8.8, also known as the Tetrakisgrid. Figure 6: The red edges are an induced subgraph ofthe Tetrakis grid. They form a square grid.
Dual of 3.12.12
The dual of 3.12.12 is known as the Triakis tiling, and contains the triangular grid as an induced subgraph. In Figure7, the graph induced by the red vertices is a triangular grid graph. Hamiltonicity in triangular grid graphs is NP-hard,so Hamiltonicity is also NP-hard in the Triakis tiling. 3igure 7: Structure of the dual of 3.12.12. Figure 8: The red edges are an induced subgraph ofthe dual of 3.12.12, and form a triangular grid.
In this section, we emulate hexagonal grid graphs via gadgets. For each dual, we construct two kinds of vertexgadgets, which we will call even and odd vertex gadgets respectively. We will also construct edge gadgets that linkeven vertex gadgets to odd vertex gadgets. We note that if we can construct gadgets such that every odd vertexgadget can be connected to three even vertex gadgets and vice versa, we will have a three-regular bipartite graph —this is identical to a hexagonal grid graph, in which Hamiltonicity is known to the NP-Complete. Furthermore, byconstruction Hamiltonian path through our constructed graph will be possible if and only if a Hamiltonian path existsin the corresponding hexagonal grid.In order to create gadgets that allow for the creation of a hexagonal grid in which Hamiltonicity is preserved,we impose the following constraints. First, edge gadgets will have the following properties:1. Edge gadgets contain at least one vertex such that the edge will become disconnected if the vertex is removed.This prevents the edge from being used more than once, since a Hamiltonian Cycle using the edge gadgetwill necessarily pass through this vertex.2. Edge gadgets connect to odd vertex gadgets at one vertex and even vertex gadgets at two adjacent vertices.3. Edge gadgets have at least two Hamiltonian paths:(a) The “traverse” path goes from the even side to the odd side.(b) The “return” path goes from one vertex of the even side to the other vertex on the even side.Vertex gadgets will have the following properties:1. As we are emulating hexagonal grids, all vertex gadgets will have 3 entrances/exits.2. There is a Hamiltonian path through the vertex gadget starting at any entrance and leaving at any otherentrance.3. For even vertex gadgets, the Hamiltonian path must also pass through each of the two adjacent verticesconnecting it to an edge gadget in succession. This ensures that edge gadgets that correspond to unusededges in the emulated hexagonal gadgets can get “picked up” by even vertex gadgets.The gadgets we present in the following section will obey these properties.
Dual of 4.6.12
In Figure 12, green edges are odd vertex gadgets, blue edges are even vertex gadgets, and the red edges are edgegadgets. Figures 9 and 10 demonstrates paths through even and odd vertices, and 11 shows traverse and return pathsthrough the edge gadgets. We show that our gadgets satisfy all necessary properties.First, consider the edge gadget. As shown in Figure 11 there are both transverse and return paths for this gad-get. Furthermore, there is a single vertex that can be removed that will disconnect the odd (green) and even (blue)4ertex gadgets that are connected by the edge gadget. Finally, the edge gadget joins to the odd (green) vertex gadgetat a single vertex, and to the even (blue) vertex gadget at two vertices.Next, consider the vertex gadgets. As in Figures 9 and 10, it is clear that for both vertex gadgets there arethree ways in which an edge gadget can be attached. Furthermore, as depicted for a single possible path, the vertexgadget can only be traversed once by a Hamiltonian path. Lastly, for the even (blue) vertex gadget, as exemplifiedin Figure 10, any Hamiltonian path must visit both of the two adjacent vertices connecting it to an edge gadget insuccession.Figure 9: An odd vertex and a path through it, in-duced by the dual of 4.6.12. Figure 10: An even vertex and a path through it, in-duced by the dual of 4.6.12.Figure 11: Left: an edge gadget for the dual of 4.6.12, connected to an even vertex on the left and an odd vertex on theright. Middle: A traverse path through the edge gadget. Right: A return path through the edge gadget.Figure 12: Left: an example of an induced subgraph of the dual of 4.6.12 that emulates a hexagonal grid graph. Right:A Hamiltonian path through said subgraph. The grey lines represent the path through the corresponding hexagonalgrid.
Dual of 3.3.3.3.6
Here we present the edge and vertex gadgets for the dual of the 3.3.3.3.6 gadget. Figures 13 and 14 shows the evenand odd vertex gadgets, and example paths through them, while Figure 15 shows traverse and return paths through theedge gadgets. These gadgets also hold the desired properties to show that Hamiltonicity is NP-Complete.5igure 13: The odd vertex gadget of the dual of 3.3.3.3.6, and two example paths through it.Figure 14: The even vertex gadget of the dual of 3.3.3.3.6, and two example paths through it. Notice that while theshape of the gadget is identical to the odd vertex, the starting and ending points of the paths are different.
The tree-residue vertex breaking problem asks, given a graph with some “breakable” vertices, is it possible to breaksome of them such that the resulting graph is a tree. Demaine and Rudoy [DR17] characterize the complexity of thisproblem in planar and bounded degree settings. In particular, they show the problem is still hard in the case of a planargraph with breakable vertices of maximum degree .Our proofs will use a different type of vertex and edge gadget. Vertex gadgets must be able to satisfy “bro-ken” and “unbroken” states; furthermore, edge gadgets are simply pairs of paths that have no (or at the very least, nopotentially usable) shared edges between them. In each of our proofs below we will focus on the vertex gadgets inbroken and unbroken states, as edge gadgets can be any reasonable pairs of paths. We only need to check that theedge gadgets near the vertex do not interfere with each other.Figure 15: Traverse and return paths through the edge gadgets of the dual of 3.3.3.3.6 are shown in green. We alsodemonstrate how the edge gadget is attached to even and odd vertices; red edges represent an even vertex and blueedges represent an odd vertex. 6igure 16: In the dual of 3.4.6.4, three edge gadgets can be joined to emulate an unbreakable degree 3 vertex. Theblack lines show edges in the graph being emulated.Figure 17: Vertex gadget for the dual of 3.4.6.4. Figure 18: Example edge gadget for the dualof 3.4.6.4. Dual of 3.4.6.4
For the dual of 3.4.6.4, we use the 4-regular breakable vertex gadget shown in Figure17. The broken and unbrokenversions are shown in Figure19. Edge gadgets can be, for example, pairs of parallel paths extending in the four cardinaldirections. The existence of these gadgets proves that Hamiltonicity in the dual of 3.3.4.3.4 is NP-hard.A note: it was previously believed that a single degree breakable vertex gadget was not sufficient to prove NP-hardness, because there is no planar graph with only degree vertices. However, edge gadgets can modified to createarbitrary degree unbreakable vertex gadgets, so our two gadgets are sufficient. As an example, in figure 16, three edgegadgets in the dual of 3.4.6.4 combine for emulate an unbreakable degree 3 vertex. The existence of this gadget showsthat the Hamiltonian path problem in this grid is NP-hard.7igure 19: Broken and unbroken states for a vertex gadget for the dual of 3.4.6.4Figure 20: Vertex gadget for the dual of 3.3.4.3.4 Dual of 3.3.4.3.4
For the dual of 3.3.4.3.4, we use the 4-regular breakable vertex gadget shown in Figure 20. The breakable andunbreakable versions are shown in Figure 21. Edge gadgets can be, for example, pairs of parallel paths extendingin the four cardinal directions. The existence of these gadgets proves that Hamiltonicity in the dual of 3.3.4.3.4 isNP-hard. Figure 21: Broken and unbroken states for a 3.3.4.3.4 dual vertex gadget8igure 22: Left: vertex gadget for the dual of 3.3.3.4.4. Middle and right: broken and unbroken states for said gadget.
Dual of 3.3.3.4.4
For the dual of 3.3.3.4.4, we use the 4-regular breakable vertex gadget shown in Figure 22, with the correspondingbroken and unbroken versions shown in the red edges. The full gadget is simply the union of the red and blue edges.The existence of this vertex gadget proves that Hamiltonicity in the dual of 3.3.3.4.4 is NP-hard. With this, we havethat the Hamiltonian path problem is NP-hard in the duals of all eight semi-regular tessellations.
In this section we show that Hamiltonicity in augmented square grids is NP-complete. Further, our reduction worksfor thin graphs, giving alternate proofs and sometimes stronger results for a number of well known grid graphs.A full augmented square grid graph is an infinite graph formed by taking the infinite square grid graph and addingeither or both diagonals of each square pixel may exist as edges in the graph. An augmented square grid graph is aninduced subgraph of a full augmented square grid graph. We will show that the Hamiltonian Cycle Problem for anyaugmented square grid is NP-hard. As before, we will use regular degree-4 tree-residue vertex breaking.Many interesting families of graphs are augmented square grids. These include: the King’s graph, the box-pleat grid(important in origami design), the triangular grid graph, the 3.3.4.3.4 and 3.3.3.4.4 semi-regular tessellations, andother combinations of unit right triangles and squares. It is also interesting to note that this includes aperiodic tilingsof the plane, for example by including the diagonals in every pixel whose bottom-leftmost point has coordinates thatsum to powers of 10.Edge gadgets are simple, we can just use arbitrary parallel paths that are distance apart. All vertex gadgetswill be contained in a by square, and will contain 4 “side” pieces and 4 “corner” pieces. Each side piecewill connected to exactly 2 corner pieces and each corner piece will be connected to exactly 2 side pieces; the finalconstruction will look somewhat like a square. We will focus on a × region that amounts to one eighth of thefinal vertex gadget, which is focused on the connection between a side and corner piece; we will demonstrate how tocombine the regions into a full vertex gadget after all of our casework. Our diagrams will concentrate on the right halfof the top side of the vertex gadget.Consider Figure 23. All the black edges will be guaranteed to be part of our gadget. We now have threecases. If the blue edge exists, we use the connection in Figure 24. If the blue edge does not exist but both red ones do,we use the connection in Figure 25. Finally, if either red edge does not exist, we use the connection in Figure 26.In each of the three cases we described, we have only drawn the minimum number of edges in the induced subgraph(there may exist other induced diagonal edges in the square pixels. However, notice that in each connection thereexists a vertex such that, if that vertex is removed, the gadget becomes separated. This is essential, because it meansthe connection can be traversed at most one time.Furthermore, we need to verify that for each side or corner piece, it is possible to create a Hamiltonian path througheach edge/corner piece, with all combinations of “half pieces,” that starts at the edge gadget and ends at the connectionpoint. The case for side and corner pieces is shown in figures 27 and 28.9igure 23: Edges to consider for gadget creation inaugmented grid graphs. The green edges denote apossible edge gadget. Figure 24: Case 1: connection used when the blueedge exists. The red dashed line may or may not beinduced; the gadget works either way.Figure 25: Case 2: connection used when the solidblue edge doesn’t exist but both red edges do. Thedotted blue edge may or may not exist, the gadgetworks either way. Figure 26: Case 3: When at least one red edge doesnot exist. The gadget still works if only one of themexists.Figure 27: Demonstration of a Hamiltonian path through a sidepiece. Combine one half side piece on the left with one reflectedhalf side piece on the right for a path through a full side gadget.See Figure 29 for a few examples. Figure 28: Demonstration of a Hamiltonianpath through a corner piece; all corner piecesare one of these three cases.10igure 29: Top: A full vertex gadget combining multiple types of connections. Bottom: Broken and unbroken statesfor this particular gadget.As an example of what a complete gadget, see Figure 29. This demonstrates combining eight different gadgets tocreate a breakable degree-4 vertex gadget. Green lines represent edge gadgets, which are relatively unconstrained. Itis straightforward to verify that any Hamiltonian path or cycle involving this vertex gadget forces it into either thebroken or unbroken states. The existence of this gadget proves that the Hamiltonian path problem is NP-hard in allaugmented square grid graphs. In this paper, we have shown that the Hamiltonian path problem in many tessellations is NP-Hard. Using reductionsto Hamiltonicity in other grid graphs, we have shown that Hamiltonicity in the duals of the 3.6.3.6, 4.8.8, 3.12.12,4.6.12, and 3.3.3.3.6 tessellations is NP-hard. Furthermore, using reductions to tree-residue vertex breaking, we haveshown Hamiltonicity in the duals of the 3.4.6.4, 3.3.4.3.4, and 3.3.3.4.4 tessellations, as well as augmented squaregrid graphs, is NP-hard.A wide variety of tessellations can be categorized as certain augmented square grid graph, so those tessellations mustalso be NP-hard. This leads to alternative proofs for Hamiltonicity in e.g. the triangular, hexagonal, 3.3.3.4.4, and11.3.4.3.4 grids. Overall, the frameworks we have set up for constructing gadgets to convert Hamiltonicity in variousgrid graphs to Hamiltonicity in a simple regular tiling or to TRVB will be powerful ways to simplify other openproblems in this area.There are still some questions that remain to be answered. A natural extension to augmented square grids areaugmented hexagonal grids. The substantial blow-up in cases for the hexagonal grid makes this more difficult totackle. Perhaps restricting the augmentation to single edges in each pixel will still provide interesting results butremain more tractable.Although we can show Hamiltonicity in some aperidic tilings is hard, it would be nice to see this shown for a naturalone such as the Penrose tiling, Truchet tiling, or Conway’s pinwheel tiling. These tessellations may require some newtechniques, as it seems somewhat problematic to tackle with techniques we have explored here.Finally, one could examine the thin, solid, and polygonal cases of the semi-regular tessellations and their duals.
Acknowledgements
This paper came out of the MIT course 6.892 Algorithmic Lower Bounds: Fun with Hardness Proofs taught byProfessor Erik Demaine. In addition to Professor Demaine, we would like to thank the students and teaching staff fortheir support and ideas. We would also like to thank Kaiying Hou for some discussion and ideas related to this paper.
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