PRESENTATION NAME

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Partitioning the Labeled Spanning Trees of an
Arbitrary Graph into Isomorphism Classes
Austin Mohr
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Outline

• Problem Description
• Generating Spanning Trees
• Testing for Isomorphism
• Partitioning Spanning Trees
• Some Results
• Finding a Closed Formula for I(Ks,t)

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Problem Description
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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 edge
• uv?T1 if and only if the edge f(u)f(v)?T2

Problem Description
Reference pg. 3 - 4
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Spanning Trees of K2,3
Problem Description
Reference pg. 5
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Generating Spanning Trees
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Definitions
Let G be a graph on n vertices, H?G, e be an edge
of G, and T be a spanning tree of G.

• 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 Trees
Reference pg. 6
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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 Trees
Reference pg. 7
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Tree of trees for K2,3
Reference pg. 8
Generating Spanning Trees
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Definitions
Let G be a graph on n vertices, e be an edge of
G, and T be a spanning tree of G.

• 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 Trees
Reference pg. 8
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Finding the Children of a Tree
Reference pg. 11
Generating Spanning Trees
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Testing for Isomorphism
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Rooted Tree Isomorphism
We first consider the simpler problem of
determining when two rooted trees are isomorphic.
Testing for Isomorphism
Reference pg. 14
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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. 14
Testing for Isomorphism
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Sample Run of Algorithm for Rooted Trees
Reference pg. 17
Testing for Isomorphism
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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. 18
Testing for Isomorphism
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Definitions
Let u and v be vertices of a graph G.

• 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. 18
Testing for Isomorphism
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Finding the Center of a Tree

• Theorem (Jordan) The center of a tree is either
  a vertex or an edge.
• Jordans 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 - 19
Testing for Isomorphism
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Algorithm for General Tree Isomorphism
Reference pg. 21
Testing for Isomorphism
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Partitioning Spanning Trees
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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 partitioned

Reference pg. 22
Partitioning Spanning Trees
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Reference pg. 23
Partitioning Spanning Trees
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Some Results
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Finding a Closed Formula for I(Ks,t)
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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. 28
Finding a Closed Formula for I(Ks,t)
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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(nk-1, n).
• At least two buckets nonempty C(nk-1, n) - k

Reference pg. 28 - 29
Finding a Closed Formula for I(Ks,t)
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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. 29
Finding a Closed Formula for I(Ks,t)
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Configurations of K2,t
Center in 2-set No such tree
Reference pg. 32
Finding a Closed Formula for I(Ks,t)
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Configurations of K2,t
Center in t-set p2(t-1) 1 trees
Reference pg. 32 - 33
Finding a Closed Formula for I(Ks,t)
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Configurations of K2,t
Center is an edge Only one such tree
Reference pg. 33
Finding a Closed Formula for I(Ks,t)
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Summing Across the Sets

• Summing across the disjoint sets yields
• I(K2,t) 0 p2(t-1) 1 1 p2(t-1), t?2.
• Similarly, we can find
• I(K3,t) sumk2 to t-2(p2(k)) p3(t-1) 2,
  t?4.

Reference pg. 29
Finding a Closed Formula for I(Ks,t)
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Nicer Formulas

• Using the generating function for pk(n), we can
  simplify the formulas to
• I(K2,t) ?t/2?, t?2
• I(K3,t) 1/3(t2 t 1), t?4

Reference pg. 36 - 41
Finding a Closed Formula for I(Ks,t)
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Questions?