partitioning the labeled spanning trees of an arbitrary graph into isomorphism classes austin mohr
TRANSCRIPT
![Page 1: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/1.jpg)
Partitioning the Labeled Spanning TreesPartitioning the Labeled Spanning Treesof an Arbitrary Graph into of an Arbitrary Graph into Isomorphism ClassesIsomorphism ClassesAustin Mohr
![Page 2: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/2.jpg)
Outline Problem Description Generating Spanning Trees Testing for Isomorphism Partitioning Spanning Trees Some Results Finding a Closed Formula for I(Ks,t)
![Page 3: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/3.jpg)
Problem DescriptionProblem Description
![Page 4: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/4.jpg)
Definitions Spanning tree T of graph G
› T is a tree with E(T)⊆E(G) and V(T)=V(G)
Isomorphic trees T1 and T2
› There exists a mapping f where the edgeuv∈T1 if and only if the edge f(u)f(v)∈T2
Problem DescriptionProblem DescriptionReference: pg. 3 - 4Reference: pg. 3 - 4
![Page 5: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/5.jpg)
Spanning Trees of K2,3
Problem DescriptionProblem DescriptionReference: pg. 5Reference: pg. 5
![Page 6: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/6.jpg)
Generating Spanning TreesGenerating Spanning Trees
![Page 7: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/7.jpg)
Definitions
Index of an edge› “Arbitrary” labeling of the edges of G
T*› Tree induced by the edge-subset {1,2,…,n-1}
top(H)/btm(H)› Edge of H with smallest/largest index
Cut(H,e)› Edges of G connecting the components of H\e
(T)› (T\f)∪g, f = btm(T), g = top(Cut(T,f))
Generating Spanning TreesGenerating Spanning TreesReference: pg. 6Reference: pg. 6
Let G be a graph on n vertices, H⊆G, e be an edge of G, and T be a spanning tree of G.
![Page 8: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/8.jpg)
Regarding (T) Let T be a spanning tree of G.
Then, (T) is a spanning tree of G.
Let T ≠ T* be a spanning tree of G with (T) = (T\f)∪g.Then, g∈T*∌f.› Means iteration of yields T*
Generating Spanning TreesGenerating Spanning TreesReference: pg. 7Reference: pg. 7
![Page 9: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/9.jpg)
“Tree of trees” for K2,3
Reference: pg. 8Reference: pg. 8 Generating Spanning TreesGenerating Spanning Trees
![Page 10: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/10.jpg)
Definitions
Pivot edge f of T› An edge such that T`\T = f for some child tree
T`
Cycle(T,e)› The set of edges of the unique cycle in T∪e
Generating Spanning TreesGenerating Spanning TreesReference: pg. 8Reference: pg. 8
Let G be a graph on n vertices, e be an edge of G, and T be a spanning tree of G.
![Page 11: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/11.jpg)
Finding the Children of a Tree
Reference: pg. 11Reference: pg. 11 Generating Spanning TreesGenerating Spanning Trees
![Page 12: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/12.jpg)
Testing for IsomorphismTesting for Isomorphism
![Page 13: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/13.jpg)
Rooted Tree Isomorphism
Testing for IsomorphismTesting for IsomorphismReference: pg. 14Reference: pg. 14
We first consider the simpler problem of determining when two rooted trees are isomorphic.
![Page 14: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/14.jpg)
Rooted Tree Isomorphism Given two rooted trees T1 and T2 on n
vertices, a mapping f: V(T1) → V(T2) is an
isomorphism if and only if for every vertex v∈V(T1), the subtree of T1 rooted at v is isomorphic to the subtree of T2 rooted at f(v).› Means we can start at the bottom of the tree
and work recursively toward the root
Reference: pg. 14Reference: pg. 14 Testing for IsomorphismTesting for Isomorphism
![Page 15: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/15.jpg)
Sample Run of Algorithm for Rooted Trees
Reference: pg. 17Reference: pg. 17 Testing for IsomorphismTesting for Isomorphism
![Page 16: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/16.jpg)
General Tree Isomorphism To generalize the algorithm, we need
a vertex u∈V(T1) and v∈V(T2) such that
f(u) = v for every isomorphism f.› If found, we root T1 at u, root T2 at v, and
use the previous algorithm› The center of each tree is suitable choice
Reference: pg. 18Reference: pg. 18 Testing for IsomorphismTesting for Isomorphism
![Page 17: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/17.jpg)
Definitions
d(u,v) (distance)› The number of edges in the shortest uv-path
eccentricity› Let v be a vertex of maximum distance from
u. Then, the eccentricity of u is d(u,v). center
› The subgraph of G induced by the vertices of minimum eccentricity
Reference: pg. 18Reference: pg. 18
Let u and v be vertices of a graph G.
Testing for IsomorphismTesting for Isomorphism
![Page 18: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/18.jpg)
Finding the Center of a Tree
Theorem (Jordan): The center of a tree is either a vertex or an edge.› Jordan’s proof also shows that we can find
the center by successively removing all the leaves from the tree until only a vertex or an edge remains.
Reference: pg. 18 - 19Reference: pg. 18 - 19 Testing for IsomorphismTesting for Isomorphism
![Page 19: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/19.jpg)
Algorithm for General Tree Isomorphism
Reference: pg. 21Reference: pg. 21 Testing for IsomorphismTesting for Isomorphism
![Page 20: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/20.jpg)
Partitioning Spanning TreesPartitioning Spanning Trees
![Page 21: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/21.jpg)
Partitioning Spanning Trees
Place T* in a subset S1
For each child T of T*› For each subset Si
If T is isomorphic to a tree in Si, place T in Si
Otherwise, create a new subset for T Find the children of the children of T*
and repeat Continue until all trees have been
partitionedReference: pg. 22Reference: pg. 22 Partitioning Spanning TreesPartitioning Spanning Trees
![Page 22: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/22.jpg)
Reference: pg. 23Reference: pg. 23 Partitioning Spanning TreesPartitioning Spanning Trees
![Page 23: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/23.jpg)
Some ResultsSome Results
![Page 24: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/24.jpg)
Finding a Closed Formula for Finding a Closed Formula for I(KI(Ks,ts,t))
![Page 25: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/25.jpg)
Definitions I(G)
› The number of isomorphism classes of the spanning trees of G
pk(n)› The number of partitions of the integer n
into at most k parts
Reference: pg. 28Reference: pg. 28 Finding a Closed Formula for Finding a Closed Formula for I(KI(Ks,ts,t))
![Page 26: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/26.jpg)
Useful Counting Tools The number of ways to arrange n
unlabeled balls into k unlabeled buckets is given by pk(n).
› At least two buckets nonempty: pk(n) - 1
The number of ways to arrange n unlabeled balls into k labeled buckets is given by C(n+k-1, n).› At least two buckets nonempty: C(n+k-1, n) -
kReference: pg. 28 - 29Reference: pg. 28 - 29 Finding a Closed Formula for Finding a Closed Formula for I(KI(Ks,ts,t))
![Page 27: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/27.jpg)
Configurations of Ks,t
A spanning tree of Ks,t belongs to one of three disjoint sets› The center is a vertex in the s-set› The center is a vertex in the t-set› The center is an edge between the two sets
We determine the number of nonisomorphic trees in each set and then sum to find I(Ks,t)
Reference: pg. 29Reference: pg. 29 Finding a Closed Formula for Finding a Closed Formula for I(KI(Ks,ts,t))
![Page 28: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/28.jpg)
Configurations of K2,t
Reference: pg. 32Reference: pg. 32 Finding a Closed Formula for Finding a Closed Formula for I(KI(Ks,ts,t))
Center in 2-setCenter in 2-setNo such treeNo such tree
![Page 29: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/29.jpg)
Configurations of K2,t
Reference: pg. 32 - 33Reference: pg. 32 - 33 Finding a Closed Formula for Finding a Closed Formula for I(KI(Ks,ts,t))
Center in Center in tt-set-setpp22(t(t-1-1)) – 1 trees – 1 trees
![Page 30: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/30.jpg)
Configurations of K2,t
Reference: pg. 33Reference: pg. 33 Finding a Closed Formula for Finding a Closed Formula for I(KI(Ks,ts,t))
Center is an edgeCenter is an edgeOnly one such treeOnly one such tree
![Page 31: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/31.jpg)
Summing Across the Sets Summing across the disjoint sets yields
I(KK2,t2,t) = 0 + pp22(t(t-1-1)) – 1 + 1 = – 1 + 1 = pp22(t(t-1-1), ), t2.. Similarly, we can find
I(KK3,t3,t) = sum{k=2 to t-2}(p2(k)) + p3(t-1) +2, t4.
Reference: pg. 29Reference: pg. 29 Finding a Closed Formula for Finding a Closed Formula for I(KI(Ks,ts,t))
![Page 32: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/32.jpg)
Nicer Formulas Using the generating function for pk(n),
we can simplify the formulas to:› I(K2,t) = ⌈t/2⌉, t2› I(K3,t) = [1/3(t2 + t + 1)], t4
Reference: pg. 36 - 41Reference: pg. 36 - 41 Finding a Closed Formula for Finding a Closed Formula for I(KI(Ks,ts,t))
![Page 33: Partitioning the Labeled Spanning Trees of an Arbitrary Graph into Isomorphism Classes Austin Mohr](https://reader037.vdocuments.us/reader037/viewer/2022103122/56649f4f5503460f94c70f50/html5/thumbnails/33.jpg)
Questions?Questions?