Post on 29-Jun-2015
description
Counting Bi-Point-Determining GraphsA Bijection
Ji Li
Combinatorics Seminar
Department of MathematicsBrandeis University
April 20th, 2007
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
Outline
1 Point-Determining Graphs
2 Bi-Point-Determining Graphs
3 A Bijection for Bi-Point-Determining Graphs
4 Generating Functions
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 2 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
Neighborhood of a Vertex
Definition
In a graph G , the neighborhood of a vertex v is the set of vertices adjacent to v ,the augmented neighborhood of a vertex is the union of the vertex itself and itsneighborhood.
Example
v
w1 w2 w3 w4
In the above figure, the neighborhood of vertex v is the set {w1, w2, w3, w4}, whilethe augmented neighborhood of v is the set {v , w1, w2, w3, w4}.
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 3 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
Point-Determining Graphs and
Co-Point-Determining Graphs
Definition
• A graph is called point-determining if no two vertices of this graph have thesame neighborhoods.
• A graph is called co-point-determining if its complement is point-determining.
• Equivalently, a graph is co-point-determining if no two vertices of this graphhave the same augmented neighborhoods.
Example
The graph on the left is co-point-determining, and the graph on the right ispoint-determining. These two graphs are complements of each other.
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 4 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
A Transformation
A transformation from a graph G to a point-determining graph P.
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The above figure illustrates the transformation from a graph G with vertex set [11]to a point-determining graph P with vertex set {{1, 9, 3}, {8}, {4, 7}, {6}, {2, 5}}.
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 5 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
Another Transformation
A transformation from a graph G to a co-point-determining graph Q.
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Here is another similar transformation from a graph G with vertex set [11] to aco-point-determining graph Q with vertex set {{1, 9, 3}, {8}, {4, 7}, {6}, {2, 5}}.
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 6 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
Outline
1 Point-Determining Graphs
2 Bi-Point-Determining Graphs
3 A Bijection for Bi-Point-Determining Graphs
4 Generating Functions
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 7 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
Bi-Point-Determining Graphs
Definition
A bi-point-determining graph is a graph that is both point-determining andco-point-determining.
Example
Listed in above are all unlabeled bi-point-determining graphs with no more than 5vertices.
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 8 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
Transform a Graph into a Bi-Point-Determining Graph?
Yes.
In this way, we end up with a graph in which
or the same augmented neighborhood, i.e.,
But, we would like to keep track of the procedure
no two vertices have the same neighborhood
On each step, we group vertices with the same
of vertices of the previous graph.
a bi−point−determining graph.
and get a new graph whose vertices are sets
neighborhoods or the same augmented neighborhoods,
so that the original graph can be reconstructed....
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J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 9 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
Alternating Phylogenetic Trees
Definition
• A phylogenetic tree is a rooted tree with labeled leaves and unlabeled internalvertices in which no vertex has exactly one child.
• An alternating phylogenetic tree is either a single vertex, or a phylogenetictree with more than one labeled vertex whose internal vertices are coloredblack or white, where no two adjacent vertices are colored the same way.
Example
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An alternatingphylogenetic tree on 9vertices, where the rootis colored black.
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 10 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
Outline
1 Point-Determining Graphs
2 Bi-Point-Determining Graphs
3 A Bijection for Bi-Point-Determining Graphs
4 Generating Functions
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 11 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
An Informal Description
The claim is —
The structure of alternating phylogenetic trees can be used to keep track of thetransformation of an arbitrary graph into a bi-point-determining graph.
We let
• G denote arbitrary graphs
• R denote arbitrarybi-point-determininggraphs
• S denote alternatingphylogenetic trees.
An illustration
G
R
S
S
S
The above figure means..
the structure of graphs is the structure of bi-point-determining graphssuperimposed with the structures of alternating phylogenetic trees.J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 12 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
An Illustration of the Bijection
Example
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V1 V2
V4V3
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V1
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 13 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
The Formal Description of the Bijection
For any finite set U, we construct a bijection between
the set of
graphs withvertex set U
and
the set of
triples of the form (π, R , γ) such that
• π is a partition of the set U, i.e.,π = {V1, V2, . . . , Vk}
• R is a bi-point-determining graph with vertex set π
• γ is a set of alternating phylogenetic trees{S1, S2, . . . , Sk}, where each Si is an alternatingphylogenetic tree with vertex set Vi .
Next, we will see..
• how to get a triple from an arbitrary graph
• how to construct a graph from a given triple
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 14 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
From a Graph to a Triple
Example
Whenever vertices with the same neighborhods are grouped,we connect the corresponding vertices/alternating phylogenetic
trees with a black node.
Whenever vertices with the same augmented neighborhoods
or vertices with the same augmented neighborhods.
are grouped, we connected the corresponding vertices/
alternating phylogenetic trees with a white node.
Vertices left untouched are not colored.
On each step, we group vertices with the same neighborhoods
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J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 15 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
From a Triple to a Graph
Example
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V1 Given a triple (π, R , γ), where
• π = {V1, V2, . . . } is apartition of U
• R is a bi-point-determininggraph on the blocks of π
• γ is a set {S1, S2, . . . } inwhich each Si is analternating phylogenetictree labeled on the set Vi .
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 16 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
From a Triple to a Graph: Continue
Example
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Then there is a unique graph G
with vertex set U such that...
Vertices v1 and v2 of G areadjacent if and only if exactlyone of the following twoconditions is satisfied:
a) v1 and v2 are labels ofvertices of Si for some i ,and the common ancestorof v1 and v2 in Si is coloredwhite.
b) v1 ∈ Vi , v2 ∈ Vj , and Vi
and Vj are adjacent verticesin the bi-point-determininggraph R .
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 17 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
The Common Ancestor
Definition
The common ancestor of two vertices a and b in a phylogenetic tree is defined tobe such that if we take the unique shortest path from a to b, say, w0w1 · · ·wl ,with w0 = a and wl = b, then the common ancestor of a and b is the unique wi
for which both wi−1 and wi+1 are children of wi .
Example
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The common ancestor of vertices 5 and 4 is colored black, while the commonancestor of vertices 5 and 3 is colored white.J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 18 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
Outline
1 Point-Determining Graphs
2 Bi-Point-Determining Graphs
3 A Bijection for Bi-Point-Determining Graphs
4 Generating Functions
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 19 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
This Bijection Gives Rise to Functional Equations
As illustrated..
G
R
S
S
S
We get
G = R ◦ S .
Which reads..
The structure of graphs is the structure of bi-point-determining graphs composedwith the structure of alternating phylogenetic trees.
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 20 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
The Exponential Generating Function
for Labeled Bi-Point-Determining Graphs
We write
• R(x) to be the exponential generating function of labeledbi-point-determining graphs;
• G(x) to be the exponential generating function of labeled graphs, which isgiven by
G(x) =∑
n≥0
2
(n2
)xn
n!.
Then we get
R(x) = G(2 log(1 + x) − x).
R(x) =x
1!+ 12
x4
4!+ 312
x5
5!+ 13824
x6
6!+ 1147488
x7
7!+ 178672128
x8
8!+ · · · .
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 21 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
The Ordinary Generating Function
for Unlabeled Bi-Point-Determining Graphs
We write
R̃(x) to be the (ordinary) generating function of unlabeled bi-point-determininggraphs. Then
R̃(x) = ZG (x − 2x2, x2 − 2x4
, . . . ).
Here ZG is the so-called cycle index of the structures of graphs G , and theformula for ZG is known.
We write down the beginningterms of R̃(x)
R̃(x) = x + x4 + 6x5 + 36x6
+ 324x7 + 5280x8
+ · · · .
Compare with..
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 22 / 23
Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating Functions
The End
Thank you for your patience!
J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 23 / 23