Periodische Graphen und
ein Resultat von Tutte
DMV Jahrestagung 2004, Heidelberg
Olaf Delgado-Friedrichs
Wilhelm-Schickard-Institut für Informatik, Universität Tübingen
Periodische Graphen und ein Resultat von Tutte – p.1/22
Periodic graphs
G = (V, E, s) a finite, directed multigraph.
Periodische Graphen und ein Resultat von Tutte – p.2/22
Periodic graphs
G = (V, E, s) a finite, directed multigraph.
s: E → Zd a “shift”-function.
Periodische Graphen und ein Resultat von Tutte – p.2/22
Periodic graphs
G = (V, E, s) a finite, directed multigraph.
s: E → Zd a “shift”-function.
Define the d-periodic graph G∗ = (V ∗ , E ∗ ) by:
Periodische Graphen und ein Resultat von Tutte – p.2/22
Periodic graphs
G = (V, E, s) a finite, directed multigraph.
s: E → Zd a “shift”-function.
Define the d-periodic graph G∗ = (V ∗ , E ∗ ) by:
V ∗ = { (v, z) | v ∈ V ; z ∈ Zd }.
Periodische Graphen und ein Resultat von Tutte – p.2/22
Periodic graphs
G = (V, E, s) a finite, directed multigraph.
s: E → Zd a “shift”-function.
Define the d-periodic graph G∗ = (V ∗ , E ∗ ) by:
V ∗ = { (v, z) | v ∈ V ; z ∈ Zd }.
E ∗ = { (u, z)(v, z + s(uv)) | uv ∈ E; z ∈ Zd }.
Periodische Graphen und ein Resultat von Tutte – p.2/22
Periodic graphs
G = (V, E, s) a finite, directed multigraph.
s: E → Zd a “shift”-function.
Define the d-periodic graph G∗ = (V ∗ , E ∗ ) by:
V ∗ = { (v, z) | v ∈ V ; z ∈ Zd }.
E ∗ = { (u, z)(v, z + s(uv)) | uv ∈ E; z ∈ Zd }.
Convention: Assume periodic graphs connected.
Periodische Graphen und ein Resultat von Tutte – p.2/22
Periodic graphs
G = (V, E, s) a finite, directed multigraph.
s: E → Zd a “shift”-function.
Define the d-periodic graph G∗ = (V ∗ , E ∗ ) by:
V ∗ = { (v, z) | v ∈ V ; z ∈ Zd }.
E ∗ = { (u, z)(v, z + s(uv)) | uv ∈ E; z ∈ Zd }.
Convention: Assume periodic graphs connected.
(Test in O(d2 · |E|) [C OHEN , M EGIDDO 1990])
Periodische Graphen und ein Resultat von Tutte – p.2/22
Example 1:
(0,1)
(0,1)
1
(1,−1)
2
1
(−3,1)
3
4
(−1,0)
3
2
4
(−2,1)
G (the orbit graph)
G∗ (the p-graph)
Periodische Graphen und ein Resultat von Tutte – p.3/22
Example 2:
(0,1)
(0,0)
(1,0)
1
2
(0,0)
3
4
1
(0,0)
2 4
3
(0,0)
The orbit graph.
The p-graph.
Periodische Graphen und ein Resultat von Tutte – p.4/22
Example 3: diamond
(0,0,0)
(1,0,0)
1
2
(0,1,0)
(0,0,1)
The orbit graph.
The p-graph.
Periodische Graphen und ein Resultat von Tutte – p.5/22
Isomorphisms
Given G = (V, E, s) and G0 = (V 0 , E 0 , s0 ).
Periodische Graphen und ein Resultat von Tutte – p.6/22
Isomorphisms
Given G = (V, E, s) and G0 = (V 0 , E 0 , s0 ).
f ∗ : V ∗ → V 0∗ a graph isomorphism.
Periodische Graphen und ein Resultat von Tutte – p.6/22
Isomorphisms
Given G = (V, E, s) and G0 = (V 0 , E 0 , s0 ).
f ∗ : V ∗ → V 0∗ a graph isomorphism.
f : Zd → Zd a group isomorphism.
Periodische Graphen und ein Resultat von Tutte – p.6/22
Isomorphisms
Given G = (V, E, s) and G0 = (V 0 , E 0 , s0 ).
f ∗ : V ∗ → V 0∗ a graph isomorphism.
f : Zd → Zd a group isomorphism.
f ∗ (v, z) = f ∗ (v, 0) + f (z) for all v ∈ V ; z ∈ Z.
Periodische Graphen und ein Resultat von Tutte – p.6/22
Isomorphisms
Given G = (V, E, s) and G0 = (V 0 , E 0 , s0 ).
f ∗ : V ∗ → V 0∗ a graph isomorphism.
f : Zd → Zd a group isomorphism.
f ∗ (v, z) = f ∗ (v, 0) + f (z) for all v ∈ V ; z ∈ Z.
Then f ∗ is called a periodic isomorphism.
Periodische Graphen und ein Resultat von Tutte – p.6/22
Isomorphisms
Given G = (V, E, s) and G0 = (V 0 , E 0 , s0 ).
f ∗ : V ∗ → V 0∗ a graph isomorphism.
f : Zd → Zd a group isomorphism.
f ∗ (v, z) = f ∗ (v, 0) + f (z) for all v ∈ V ; z ∈ Z.
Then f ∗ is called a periodic isomorphism.
Knowing f and f ∗ -images for V × {0} suffices.
Periodische Graphen und ein Resultat von Tutte – p.6/22
Isomorphisms
Given G = (V, E, s) and G0 = (V 0 , E 0 , s0 ).
f ∗ : V ∗ → V 0∗ a graph isomorphism.
f : Zd → Zd a group isomorphism.
f ∗ (v, z) = f ∗ (v, 0) + f (z) for all v ∈ V ; z ∈ Z.
Then f ∗ is called a periodic isomorphism.
Knowing f and f ∗ -images for V × {0} suffices.
Periodic drawings defined similarly.
Periodische Graphen und ein Resultat von Tutte – p.6/22
Barycentric drawings
Place each vertex in
the center of gravity
of its neighbors:
Periodische Graphen und ein Resultat von Tutte – p.7/22
Barycentric drawings
Place each vertex in
the center of gravity
of its neighbors:
1 X
p(w)
p(v) =
d(v)
∗
vw∈E
where
p = placement,
d = degree.
Periodische Graphen und ein Resultat von Tutte – p.7/22
Barycentric drawings
[T UTTE 1960/63]:
Pick and realize a
convex outer face.
Place rest
barycentrically.
Periodische Graphen und ein Resultat von Tutte – p.8/22
Barycentric drawings
[T UTTE 1960/63]:
Pick and realize a
convex outer face.
Place rest
barycentrically.
G planar, 3-connected
⇒ convex
planar drawing.
Periodische Graphen und ein Resultat von Tutte – p.8/22
Periodic version
Place one vertex, choose
linear map Zd → Rd .
Periodische Graphen und ein Resultat von Tutte – p.9/22
Periodic version
Place one vertex, choose
linear map Zd → Rd .
Theorem:
This defines a unique
barycentric placement.
Periodische Graphen und ein Resultat von Tutte – p.9/22
Periodic version
Place one vertex, choose
linear map Zd → Rd .
Theorem:
This defines a unique
barycentric placement.
Corollary:
All barycentric
placements of a p-graph
are affinely equivalent.
Periodische Graphen und ein Resultat von Tutte – p.9/22
Sketch of proof
(c.f. [R ICHTER -G EBERT 1996])
Barycentric placements are critical points of
X
W (p) :=
||p(w, s(vw)) − p(v, 0)||2 .
vw∈E
Periodische Graphen und ein Resultat von Tutte – p.10/22
Sketch of proof
(c.f. [R ICHTER -G EBERT 1996])
Barycentric placements are critical points of
X
W (p) :=
||p(w, s(vw)) − p(v, 0)||2 .
vw∈E
Because G∗ is connected:
||p|| large ⇒ ∃ long edge ⇒ W (p) large.
Periodische Graphen und ein Resultat von Tutte – p.10/22
Sketch of proof
(c.f. [R ICHTER -G EBERT 1996])
Barycentric placements are critical points of
X
W (p) :=
||p(w, s(vw)) − p(v, 0)||2 .
vw∈E
Because G∗ is connected:
||p|| large ⇒ ∃ long edge ⇒ W (p) large.
W quadratic ⇒ exactly one critical point.
Periodische Graphen und ein Resultat von Tutte – p.10/22
Sketch of proof
(c.f. [R ICHTER -G EBERT 1996])
Barycentric placements are critical points of
X
W (p) :=
||p(w, s(vw)) − p(v, 0)||2 .
vw∈E
Because G∗ is connected:
||p|| large ⇒ ∃ long edge ⇒ W (p) large.
W quadratic ⇒ exactly one critical point.
Remark: Minimizes square-sum of edge lengths.
Periodische Graphen und ein Resultat von Tutte – p.10/22
Caveat
The barycentric placement need not be injective:
Periodische Graphen und ein Resultat von Tutte – p.11/22
Caveat
The barycentric placement need not be injective:
If it is, the p-graph is called stable.
Periodische Graphen und ein Resultat von Tutte – p.11/22
Symmetries
Uniqueness ⇒
each periodic automorphism induces an
affine symmetry of the barycentric placement.
Periodische Graphen und ein Resultat von Tutte – p.12/22
Symmetries
Uniqueness ⇒
each periodic automorphism induces an
affine symmetry of the barycentric placement.
∃ a metric turning all these into isometries.
Periodische Graphen und ein Resultat von Tutte – p.12/22
Symmetries
Uniqueness ⇒
each periodic automorphism induces an
affine symmetry of the barycentric placement.
∃ a metric turning all these into isometries.
⇒ The automorphism group maps
onto a crystallographic space group.
Periodische Graphen und ein Resultat von Tutte – p.12/22
Symmetries
Uniqueness ⇒
each periodic automorphism induces an
affine symmetry of the barycentric placement.
∃ a metric turning all these into isometries.
⇒ The automorphism group maps
onto a crystallographic space group.
For stable graphs the map is an isomorphism.
Periodische Graphen und ein Resultat von Tutte – p.12/22
Symmetries
Uniqueness ⇒
each periodic automorphism induces an
affine symmetry of the barycentric placement.
∃ a metric turning all these into isometries.
⇒ The automorphism group maps
onto a crystallographic space group.
For stable graphs the map is an isomorphism.
⇒ Stable p-graphs can be drawn fully symmetric.
Periodische Graphen und ein Resultat von Tutte – p.12/22
Convex drawings
[T HOMASSEN 1980]: Any “sufficiently nice”
infinite, planar, 3-connected graph has a
topologically unique planar embedding, which
can be made convex. (c.f. [T UTTE 1960/63]).
Periodische Graphen und ein Resultat von Tutte – p.13/22
Convex drawings
[T HOMASSEN 1980]: Any “sufficiently nice”
infinite, planar, 3-connected graph has a
topologically unique planar embedding, which
can be made convex. (c.f. [T UTTE 1960/63]).
[D. 2001]: The barycentric placement of any
planar, 3-connected, 2-periodic graph induces
a convex planar embedding.
Periodische Graphen und ein Resultat von Tutte – p.13/22
Convex drawings
[T HOMASSEN 1980]: Any “sufficiently nice”
infinite, planar, 3-connected graph has a
topologically unique planar embedding, which
can be made convex. (c.f. [T UTTE 1960/63]).
[D. 2001]: The barycentric placement of any
planar, 3-connected, 2-periodic graph induces
a convex planar embedding.
Corollary: These graphs are necessarily
stable, so they have a convex planar
embedding with full symmetry.
Periodische Graphen und ein Resultat von Tutte – p.13/22
Outline of proof
(following [R ICHTER -G EBERT 1996])
(0) Consider placements with each vertex in
relative interior of neighbors (call these good).
Periodische Graphen und ein Resultat von Tutte – p.14/22
Outline of proof
(following [R ICHTER -G EBERT 1996])
(0) Consider placements with each vertex in
relative interior of neighbors (call these good).
(1) Suffices to prove for triangulations.
Periodische Graphen und ein Resultat von Tutte – p.14/22
Outline of proof
(following [R ICHTER -G EBERT 1996])
(0) Consider placements with each vertex in
relative interior of neighbors (call these good).
(1) Suffices to prove for triangulations.
(2) No vertex has all neighbors on a line.
Periodische Graphen und ein Resultat von Tutte – p.14/22
Outline of proof
(following [R ICHTER -G EBERT 1996])
(0) Consider placements with each vertex in
relative interior of neighbors (call these good).
(1) Suffices to prove for triangulations.
(2) No vertex has all neighbors on a line.
(3) Small perturbation is still good. (by (2))
Periodische Graphen und ein Resultat von Tutte – p.14/22
Outline of proof
(following [R ICHTER -G EBERT 1996])
(0) Consider placements with each vertex in
relative interior of neighbors (call these good).
(1) Suffices to prove for triangulations.
(2) No vertex has all neighbors on a line.
(3) Small perturbation is still good. (by (2))
(4) No degenerate triangle ⇒ embedding.
Periodische Graphen und ein Resultat von Tutte – p.14/22
Outline of proof
(following [R ICHTER -G EBERT 1996])
(0) Consider placements with each vertex in
relative interior of neighbors (call these good).
(1) Suffices to prove for triangulations.
(2) No vertex has all neighbors on a line.
(3) Small perturbation is still good. (by (2))
(4) No degenerate triangle ⇒ embedding.
(5) No degenerate triangle. (by (3),(4))
Periodische Graphen und ein Resultat von Tutte – p.14/22
Outline of proof
(following [R ICHTER -G EBERT 1996])
(0) Consider placements with each vertex in
relative interior of neighbors (call these good).
(1) Suffices to prove for triangulations.
(2) No vertex has all neighbors on a line.
(3) Small perturbation is still good. (by (2))
(4) No degenerate triangle ⇒ embedding.
(5) No degenerate triangle. (by (3),(4))
(6) Placement induces embedding. (by (4),(5))
Periodische Graphen und ein Resultat von Tutte – p.14/22
Proof of (2)
(non-degenerate neighborhoods)
Claim: If all neighbors of some
vertex v are on a straight line L,
then the graph contains a K3,3 .
a
v
b
c L
Periodische Graphen und ein Resultat von Tutte – p.15/22
Proof of (2)
(non-degenerate neighborhoods)
Claim: If all neighbors of some
vertex v are on a straight line L,
then the graph contains a K3,3 .
a
v
b
c L
(a) ∃ 3 disjoint paths from v to vertex not on L.
Periodische Graphen und ein Resultat von Tutte – p.15/22
Proof of (2)
(non-degenerate neighborhoods)
Claim: If all neighbors of some
vertex v are on a straight line L,
then the graph contains a K3,3 .
a
v
b
c L
(a) ∃ 3 disjoint paths from v to vertex not on L.
(b) Consider first vertices with neighbors off L.
Periodische Graphen und ein Resultat von Tutte – p.15/22
Proof of (2)
(non-degenerate neighborhoods)
Claim: If all neighbors of some
vertex v are on a straight line L,
then the graph contains a K3,3 .
a
v
b
c L
(a) ∃ 3 disjoint paths from v to vertex not on L.
(b) Consider first vertices with neighbors off L.
(c) Each has neighbor above L.
Periodische Graphen und ein Resultat von Tutte – p.15/22
Proof of (2)
(non-degenerate neighborhoods)
Claim: If all neighbors of some
vertex v are on a straight line L,
then the graph contains a K3,3 .
a
v
b
c L
(a) ∃ 3 disjoint paths from v to vertex not on L.
(b) Consider first vertices with neighbors off L.
(c) Each has neighbor above L.
(d) These can all be connected above L.
Periodische Graphen und ein Resultat von Tutte – p.15/22
Proof of (2)
(non-degenerate neighborhoods)
Claim: If all neighbors of some
vertex v are on a straight line L,
then the graph contains a K3,3 .
a
v
b
c L
(a) ∃ 3 disjoint paths from v to vertex not on L.
(b) Consider first vertices with neighbors off L.
(c) Each has neighbor above L.
(d) These can all be connected above L.
(e) Repeat construction below L.
Periodische Graphen und ein Resultat von Tutte – p.15/22
Proof of (2)(d)
(connection above L)
Finite version:
Ascending path must reach outer polygon.
Periodische Graphen und ein Resultat von Tutte – p.16/22
Proof of (2)(d)
(connection above L)
Finite version:
Ascending path must reach outer polygon.
Periodic version:
Ascending path must contain translates.
Periodische Graphen und ein Resultat von Tutte – p.16/22
Proof of (2)(d)
(connection above L)
Finite version:
Ascending path must reach outer polygon.
Periodic version:
Ascending path must contain translates.
∃ path above L from start vertex to translate.
Periodische Graphen und ein Resultat von Tutte – p.16/22
Proof of (2)(d)
(connection above L)
Finite version:
Ascending path must reach outer polygon.
Periodic version:
Ascending path must contain translates.
∃ path above L from start vertex to translate.
Connect a, b, c above L by combining
translates of such paths and of paths on L.
Periodische Graphen und ein Resultat von Tutte – p.16/22
Proof of (2)(d)
(connection above L)
Finite version:
Ascending path must reach outer polygon.
Periodic version:
Ascending path must contain translates.
∃ path above L from start vertex to translate.
Connect a, b, c above L by combining
translates of such paths and of paths on L.
Alternative: construct an artificial "outer polygon".
Periodische Graphen und ein Resultat von Tutte – p.16/22
Proof of (4)
(no degenerate triangle ⇒ embedding)
Map each triangle of embedding affinely to
corresponding one in barycentric placement.
Periodische Graphen und ein Resultat von Tutte – p.17/22
Proof of (4)
(no degenerate triangle ⇒ embedding)
Map each triangle of embedding affinely to
corresponding one in barycentric placement.
Factor out translations → map ϕ: T 2 → T 2 .
Periodische Graphen und ein Resultat von Tutte – p.17/22
Proof of (4)
(no degenerate triangle ⇒ embedding)
Map each triangle of embedding affinely to
corresponding one in barycentric placement.
Factor out translations → map ϕ: T 2 → T 2 .
Unsigned angles sum up to #triangles · π.
Periodische Graphen und ein Resultat von Tutte – p.17/22
Proof of (4)
(no degenerate triangle ⇒ embedding)
Map each triangle of embedding affinely to
corresponding one in barycentric placement.
Factor out translations → map ϕ: T 2 → T 2 .
Unsigned angles sum up to #triangles · π.
Around a vertex v, sum is α(v) ≥ 2π.
Periodische Graphen und ein Resultat von Tutte – p.17/22
Proof of (4)
(no degenerate triangle ⇒ embedding)
Map each triangle of embedding affinely to
corresponding one in barycentric placement.
Factor out translations → map ϕ: T 2 → T 2 .
Unsigned angles sum up to #triangles · π.
Around a vertex v, sum is α(v) ≥ 2π.
But #triangles = 2#vertices, so α(v) = 2π.
Periodische Graphen und ein Resultat von Tutte – p.17/22
Proof of (4)
(no degenerate triangle ⇒ embedding)
Map each triangle of embedding affinely to
corresponding one in barycentric placement.
Factor out translations → map ϕ: T 2 → T 2 .
Unsigned angles sum up to #triangles · π.
Around a vertex v, sum is α(v) ≥ 2π.
But #triangles = 2#vertices, so α(v) = 2π.
All angles have same sign (no flips).
Periodische Graphen und ein Resultat von Tutte – p.17/22
Proof of (4)
(no degenerate triangle ⇒ embedding)
Map each triangle of embedding affinely to
corresponding one in barycentric placement.
Factor out translations → map ϕ: T 2 → T 2 .
Unsigned angles sum up to #triangles · π.
Around a vertex v, sum is α(v) ≥ 2π.
But #triangles = 2#vertices, so α(v) = 2π.
All angles have same sign (no flips).
ϕ is covering ⇒ lift to R2 is homeomorphism.
Periodische Graphen und ein Resultat von Tutte – p.17/22
Isomorphism testing
Goal: find a unique representation for each
periodic graph.
Periodische Graphen und ein Resultat von Tutte – p.18/22
Isomorphism testing
Goal: find a unique representation for each
periodic graph.
⇒ isomorphism testing in O(|E| · d).
Periodische Graphen und ein Resultat von Tutte – p.18/22
Isomorphism testing
Goal: find a unique representation for each
periodic graph.
⇒ isomorphism testing in O(|E| · d).
Idea: generate a small characteristic
collection of representations and pick the
lexicographically smallest.
Periodische Graphen und ein Resultat von Tutte – p.18/22
Ordered traversals
For G∗ locally stable:
Periodische Graphen und ein Resultat von Tutte – p.19/22
Ordered traversals
For G∗ locally stable:
Place start vertex,
choose map Zd → Rd .
1
(0,0)
Periodische Graphen und ein Resultat von Tutte – p.19/22
Ordered traversals
For G∗ locally stable:
Place start vertex,
choose map Zd → Rd .
Do a BFS.
1
(0,0)
Periodische Graphen und ein Resultat von Tutte – p.19/22
Ordered traversals
For G∗ locally stable:
Place start vertex,
choose map Zd → Rd .
Do a BFS.
(−1,1)
(1,1)
2
4
1
(0,−2)
(0,0)
3
Sort neighbors by
position.
Periodische Graphen und ein Resultat von Tutte – p.19/22
Ordered traversals
For G∗ locally stable:
Place start vertex,
choose map Zd → Rd .
6
5
2
9
1
Do a BFS.
Sort neighbors by
position.
4
3
7
8
⇒ unique vertex numbering for each
set of start conditions.
Periodische Graphen und ein Resultat von Tutte – p.19/22
The traversal
construction
Mimik the traversal on the unlabelled orbit graph,
use first vertex of each Zd -orbit as representative:
1
1
Periodische Graphen und ein Resultat von Tutte – p.20/22
The traversal
construction
Mimik the traversal on the unlabelled orbit graph,
use first vertex of each Zd -orbit as representative:
(0,0)
1
2
2
1
Periodische Graphen und ein Resultat von Tutte – p.20/22
The traversal
construction
Mimik the traversal on the unlabelled orbit graph,
use first vertex of each Zd -orbit as representative:
(0,0)
(0,0)
3
1
2
2
1
3
Periodische Graphen und ein Resultat von Tutte – p.20/22
The traversal
construction
Mimik the traversal on the unlabelled orbit graph,
use first vertex of each Zd -orbit as representative:
(0,0)
(0,0)
3
4
(0,0)
1
2
2
4
1
3
Periodische Graphen und ein Resultat von Tutte – p.20/22
The traversal
construction
Mimik the traversal on the unlabelled orbit graph,
use first vertex of each Zd -orbit as representative:
(0,0)
(0,0)
3
4
(0,0)
1
(−1,0)
2
2
4
1
3
Periodische Graphen und ein Resultat von Tutte – p.20/22
The traversal
construction
Mimik the traversal on the unlabelled orbit graph,
use first vertex of each Zd -orbit as representative:
(0,0)
(0,0)
3
4
(0,0)
1
(−1,0)
(0,1)
2
2
4
1
3
Periodische Graphen und ein Resultat von Tutte – p.20/22
The traversal
construction
Mimik the traversal on the unlabelled orbit graph,
use first vertex of each Zd -orbit as representative:
(0,0)
(0,0)
3
4
(0,0)
1
(−1,0)
(0,1)
2
2
4
1
3
Periodische Graphen und ein Resultat von Tutte – p.20/22
The traversal
construction
Mimik the traversal on the unlabelled orbit graph,
use first vertex of each Zd -orbit as representative:
(0,0)
(0,0)
3
4
(0,0)
1
(0,−1)
(−1,0)
(0,1)
2
2
4
1
3
Periodische Graphen und ein Resultat von Tutte – p.20/22
Characteristic
traversals
Wanted: Small list of start conditions.
Periodische Graphen und ein Resultat von Tutte – p.21/22
Characteristic
traversals
Wanted: Small list of start conditions.
Idea: Map edge directions to unit vectors,
use source of first as start vertex.
Periodische Graphen und ein Resultat von Tutte – p.21/22
Characteristic
traversals
Wanted: Small list of start conditions.
Idea: Map edge directions to unit vectors,
use source of first as start vertex.
⇒ At most (2 · |E|)d characteristic traversals.
Periodische Graphen und ein Resultat von Tutte – p.21/22
Characteristic
traversals
Wanted: Small list of start conditions.
Idea: Map edge directions to unit vectors,
use source of first as start vertex.
⇒ At most (2 · |E|)d characteristic traversals.
Traversal costs O(d · |E| · log |E|) (sorting).
Periodische Graphen und ein Resultat von Tutte – p.21/22
Characteristic
traversals
Wanted: Small list of start conditions.
Idea: Map edge directions to unit vectors,
use source of first as start vertex.
⇒ At most (2 · |E|)d characteristic traversals.
Traversal costs O(d · |E| · log |E|) (sorting).
Plus: precise arithmetic, basis conversion.
Periodische Graphen und ein Resultat von Tutte – p.21/22
Characteristic
traversals
Wanted: Small list of start conditions.
Idea: Map edge directions to unit vectors,
use source of first as start vertex.
⇒ At most (2 · |E|)d characteristic traversals.
Traversal costs O(d · |E| · log |E|) (sorting).
Plus: precise arithmetic, basis conversion.
Final bound still polynomial.
Periodische Graphen und ein Resultat von Tutte – p.21/22
Characteristic
traversals
Wanted: Small list of start conditions.
Idea: Map edge directions to unit vectors,
use source of first as start vertex.
⇒ At most (2 · |E|)d characteristic traversals.
Traversal costs O(d · |E| · log |E|) (sorting).
Plus: precise arithmetic, basis conversion.
Final bound still polynomial.
⇒ the isomorphism problem
for locally stable p-graphs is in P.
Periodische Graphen und ein Resultat von Tutte – p.21/22
Thanks for your attention!
Periodische Graphen und ein Resultat von Tutte – p.22/22