the topology of bendless orthogonal three-dimensional ...eppstein/0xde/xyz-tucson.pdf · the...
TRANSCRIPT
![Page 1: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/1.jpg)
The Topology of Bendless OrthogonalThree-Dimensional Graph Drawing
David EppsteinComputer Science Dept.Univ. of California, Irvine
![Page 2: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/2.jpg)
Graph drawing: visual display of symbolic information
Vertices and edges in a graphhave some inherent meaning
Must be placed geometricallyin plane or 3d space
Aesthetic criteria(drawing should be pretty)
Usability criteria(drawing should conveythe important informationabout the relations betweenthe objects it depicts)
Flips between triangulations of 3x3 grid(clustered by short diagonal placement)
![Page 3: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/3.jpg)
Topological graph theory: graphs on surfaces
“Tucker’s Genus Two Group,” by DeWitt Godfrey and Duane Martinez(at Technical Museum of Slovenia, photo by DE)
Abstract mathematicaltheory of embeddings
E.g. represent embeddingon oriented surfaceas circular permutationof edges at each vertex
Study properties ofcomplexes of vertices,edges, and faces
Not directly relatedto visualization
![Page 4: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/4.jpg)
What’s in this talk?
Unexpected equivalence between a style of graph drawingand a type of topological embedding
3d grid drawings in which each vertex has three perpendicular edges
2d surface embeddings in which the faces meet nicely and may be 3-colored
...and its algorithmic consequences
![Page 5: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/5.jpg)
Outline
Motivation: Aesthetic criteria leading to xyz drawings
Definitions and examples
Topological equivalence
Algorithms
Computational complexity
![Page 6: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/6.jpg)
Outline
Motivation: Aesthetic criteria leading to xyz drawings
Definitions and examples
Topological equivalence
Algorithms
Computational complexity
![Page 7: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/7.jpg)
Uniform spacing of vertices
Leads to placements on integer lattice points
2d lattice: directly usable as drawing3d lattice: can be projected to 2d drawing
GDEA logo, gdea.informatik.uni-koeln.de
![Page 8: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/8.jpg)
Variations of grid drawing
Only vertices on grid, or edges and vertices both grid-aligned
Edges may have bends, or no bends allowed
Edges must have unit length, or longer edges allowed
Graph distance = grid distance, or distances may differ
Parallel edges at same vertex, or all must have different slopes
![Page 9: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/9.jpg)
Minimizing the number of slopes of edges[e.g., Dujmović, E., Suderman, Wood, CGTA 2007]
Long used to help legibility of subway maps
Tokyo subway system
![Page 10: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/10.jpg)
The fewest slopes of any drawing of any graph?
In d dimensions, need at least d slopeselse drawing would lie in a lower dimensional subspace
If there are exactly d slopes,can choose affine transform to align with coordinate axes
Two dimensions:planar graphs with horizontal and vertical edges
reasonably well understood
Three dimensions:graphs with three axis-aligned slopes?
![Page 11: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/11.jpg)
Angular resolution
[Malitz, STOC 1992; Carlson & E., GD 2006; etc.]
Avoid sharp angles between edges at same vertexas it makes edges difficult to follow
Usual definition of angular resolution:minimum angle between rays through edges
Modified definition:minimum angle between lines through edges
Avoids nearly-straight angles, difficultto distinguish from edges passing near vertex
Optimal resolution for modified definition: 90 degrees
![Page 12: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/12.jpg)
Outline
Motivation: Aesthetic criteria leading to xyz drawings
Definitions and examples
Topological equivalence
Algorithms
Computational complexity
![Page 13: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/13.jpg)
xyz graphs
Let S be a set of points in three dimensionssuch that each axis-aligned line contains zero or two points of S
Draw an edge between any two points on an axis-aligned line
![Page 14: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/14.jpg)
Three xyz graphs within a 3 x 3 x 3 grid
Note that edges are allowed to cross
Crossings differ visually from vertices as vertices never have two parallel edges
![Page 15: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/15.jpg)
The permutohedron
Convex hull of all permutations of (1,2,3,4) in 3-space x+y+z+w=10Forms a truncated octahedron
(4,1,2,3)(4,2,1,3)
(3,2,1,4)
(3,1,2,4)
(2,1,3,4)
(1,2,3,4)
(1,2,4,3)
(1,3,2,4)
(2,1,4,3)
(2,3,1,4)
(3,1,4,2)
(4,1,3,2)
(4,2,3,1)
(3,2,4,1)(2,4,1,3)
(1,4,2,3)
(1,3,4,2)
(2,3,4,1)
(1,4,3,2)
(2,4,3,1)
(3,4,2,1)
(4,3,2,1)
(4,3,1,2)
(3,4,1,2)
![Page 16: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/16.jpg)
Inverting the permutohedron
Move each permutation vertex to its inverse permutationaffine transform so that the edges are axis-aligned
![Page 17: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/17.jpg)
A polyhedron for the inverse permutohedron
Rearrange face planes to form nonconvex topological sphere
![Page 18: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/18.jpg)
A different xyz graph on 4-element permutations
Project (x,y,z,w) to (x,y,z)
![Page 19: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/19.jpg)
xyz graphs with many vertices in a small bounding box
In n x n x n box, place points such that x+y+z = 0 or 1 mod n
n = 4, the Dyck graph
![Page 20: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/20.jpg)
Basic properties of xyz graphs
3-regular (each vertex has exactly three edges)
Triangle-freeand 5-cycle-free
(but may have longer odd cycles)
3-connected(can replace any edge by paths of alternating parallel and perpendicular edges,
with two different choices of perpendicular direction)
Are these (or similar simple properties) sufficient to characterize them?
![Page 21: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/21.jpg)
Puzzle: which of these three graphs is not bipartite?
![Page 22: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/22.jpg)
Puzzle solution
![Page 23: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/23.jpg)
Outline
Motivation: Aesthetic criteria leading to xyz drawings
Definitions and examples
Topological equivalence
Algorithms
Computational complexity
![Page 24: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/24.jpg)
From xyz graphs to surface embeddings
Edges parallel to any coordinate planeform degree-two subgraph (collection of cycles)
Form a face of a surface for each cycle
![Page 25: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/25.jpg)
Basic properties of xyz surfaces
All faces are topological disks (by construction)
If two faces meet, they lie on perpendicular planesthe planes meet in a line
and the faces meet in an edge lieing on that line
The faces may be given three colors(by the direction of the planes they lie in)
and are thus properly 3-colored
![Page 26: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/26.jpg)
From xyz surfaces to xyz graphs
Le G be a 3-regular graph embedded on a surface, so thatfaces are topological disks
any two intersecting faces meet in a single edgethe faces are properly 3-colored
(say, red, blue, and green)
Number the faces of each color
Assign coordinates of a vertex:x = the number of its red facey = the number of its blue facez = the number of its green face
The result is an xyz graph embedding!
![Page 27: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/27.jpg)
Great rhombicuboctahedron
By Robert Webb using Great Stella, http://www.software3d.com/Stella.htmlimage from http://commons.wikimedia.org/wiki/Image:Great_rhombicuboctahedron.png
![Page 28: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/28.jpg)
Great rhombicosidodecahedron
By Robert Webb using Great Stella, http://www.software3d.com/Stella.htmlimage from http://commons.wikimedia.org/wiki/Image:Great_rhombicosidodecahedron.png
![Page 29: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/29.jpg)
Truncated dodecadodecahedron
By Robert Webb using Great Stella, http://www.software3d.com/Stella.htmlimage from http://commons.wikimedia.org/wiki/Image:Truncated_dodecadodecahedron.png
![Page 30: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/30.jpg)
F18
F24
F42
F54
Symmetric graphs on the torus
Start with regular tiling of plane by 3-colored hexagons
Cut out a 60-120 rhombus with matching edge coloringsglue together opposite edges to form a torus
![Page 31: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/31.jpg)
More xyz tori
Leftmost example is order-4 cube-connected cycles networkEmbedding generalizes to any even order CCC
![Page 32: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/32.jpg)
Bipartiteness and orientability
Theorem: Let G be an xyz graph.Then G is bipartite if and only if the corresponding
xyz surface is orientable
Orientable surfaces: sphere, torus, ...Non-orientable surfaces: Möbius strip, projective plane, Klein bottle, ...
Photo by David Benbennick, http://commons.wikimedia.org/wiki/Image:M%C3%B6bius_strip.jpg
![Page 33: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/33.jpg)
Planar xyz graphs
Lemma: If G is a planar xyz graph, its xyz surface must be a topological sphere
Therefore, every planar xyz graph is 3-connected and bipartite
Known:
every 3-connected planar graph is the skeleton of a polyhedron(so faces meet at most in single edges)
every bipartite polyhedron has 3-colorable faces
Therefore: a planar graph has an xyz embeddingif and only if it’s 3-connected and bipartite
![Page 34: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/34.jpg)
Outline
Motivation: Aesthetic criteria leading to xyz drawings
Definitions and examples
Topological equivalence
Algorithms
Computational complexity
![Page 35: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/35.jpg)
Testing if a surface embedding is xyz
Choose arbitrarily two colors for two adjacent faces
Propagate colors:If some face has neighbors of two colors, assign it the third color
Must successfully color all faces of any xyz surface(colors are forced by triples of faces along a path connecting any two faces)
So embedding is xyz iff faces intersect properly and coloring succeeds
Testing if a partition of edges into parallel classes is xyz
Find the xyz surface embedding that would correspond to the partition
Check that faces intersect properly and color it
![Page 36: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/36.jpg)
Testing if a graph has an xyz embedding
Try all partitions of its edges into three matchings
Backtracking algorithm:
order vertices so all but two have both incoming and outgoing edges
assign edges of first vertex to matchings, arbitrarily
for each remaining vertex, in order:try all assignments of its incident edges to matchings
that are consistent with previous choices
Vertex with two incoming edges has only one choiceVertex with two outgoing edges has two choices
So number of search paths ≤ 2n/2-1 and total time = O(2n/2)
![Page 37: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/37.jpg)
Corollary:Any 3-regular graph has O(2n/2) partitions into matchings
Tight for prisms [G. Kuperberg, personal comm.]
But partitions needed for xyz surfaces have additional properties(e.g. in any 4-cycle, opposite edges must be in same partition)
Maybe can be used to reduce number of partitions to test?
By Robert Webb using Great Stella, http://www.software3d.com/Stella.htmlimage from http://commons.wikimedia.org/wiki/Image:Decagonal_prism.png
![Page 38: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/38.jpg)
Implementation
123 lines of Python
http://www.ics.uci.edu/~eppstein/PADS/xyzGraph.py
Successfully run on graphs on up to 54 vertices
Could probably benefit from additional optimization:
— Faster test for each edge partition
— Early backtrack for bad partial partitions
![Page 39: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/39.jpg)
Outline
Motivation: Aesthetic criteria leading to xyz drawings
Definitions and examples
Topological equivalence
Algorithms
Computational complexity
![Page 40: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/40.jpg)
Uniqueness of xyz embeddings
Planar graphs have unique embeddings
But this 32-vertex graph has two (isomorphic torus) embeddings:
Similar “brick wall” patternsgive larger graphs withmultiple nonisomorphic embeddings
a b c d
c d a b
e
f
e
f
a b c d
c d a b
e
f
e
f
![Page 41: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/41.jpg)
Forcing embeddings to be unique
The “connector gadget”
Messy case analysis of surface embedding face colorings shows:left three edges must be mutually perpendicularright three edges must be mutually perpendicular
each left edge is parallel to the opposite right edge
Surface embedding forms tube connecting left to right
![Page 42: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/42.jpg)
Attaching a connector gadget to a surface
Forces the two attachment points to have compatible colorings
![Page 43: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/43.jpg)
Recognizing xyz graphs is NP-complete
Proof idea: reduction from graph 3-coloring
Represent color as orientation of an edge of a connector gadget
Vertex of graph to be colored becomes planar graph in possible xyz graph
Edge in graph to be colored becomes edge gadgetformed from three connectors and two ambiguous tori
a b c d
c d a b
eu v
f
e
f
g h i j
i j g h
ku v
l
k
l
![Page 44: The Topology of Bendless Orthogonal Three-Dimensional ...eppstein/0xDE/xyz-Tucson.pdf · The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer](https://reader031.vdocument.in/reader031/viewer/2022022600/5b3f4ba77f8b9a2f138becb1/html5/thumbnails/44.jpg)
Conclusions
Interesting type of 3d graph drawing
Equivalence with 2d surface embedding leads to some deep theory
It’s NP-complete
but...
that doesn’t prevent us from implementing algorithms and finding drawings