enumerating unlabeled cactus graphs - princeton …mbahrani/index_files/talk-cacti.pdf ·...

Enumerating UnlabeledCactus Graphs

Dec 13, 2016

Princeton University Department of Computer Science

Maryam Bahrani

Under the Direction of Dr. Jérémie Lumbroso

Trees

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Trees

binary tree

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Trees

binary tree

left-leaning red-black tree** Images reprinted with permission from Robert Sedgewick and Kevin Wayne (Algorithms lecture sides) 1/13

Trees

binary treetrie*

left-leaning red-black tree** Images reprinted with permission from Robert Sedgewick and Kevin Wayne (Algorithms lecture sides) 1/13

Trees

binary treetrie*

left-leaning red-black tree*

binomial heap

* Images reprinted with permission from Robert Sedgewick and Kevin Wayne (Algorithms lecture sides)

4

814

18

12

2215

21

11

23 24

25

13

17

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Symbolic Method on TreesA binary tree is — either a leaf — or an internal node, and a left subtree, and a right subtree

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Symbolic Method on Trees

TTT T•T = [ ( ) symbolic specification

A binary tree is — either a leaf — or an internal node, and a left subtree, and a right subtree

•

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TTT T•T = [ ( ) symbolic specification

generating functionT (z) = 1 + T (z)⇥ z ⇥ T (z)


•

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TTT T•T = [ ( )

1, 1, 2, 5, 14, 42, 132, 429, 1430, . . .

symbolic specification

generating function

exact enumeration

T (z) = 1 + T (z)⇥ z ⇥ T (z)


•

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TTT T•T = [ ( )

1, 1, 2, 5, 14, 42, 132, 429, 1430, . . .


generating function

exact enumeration

T (z) = 1 + T (z)⇥ z ⇥ T (z)


Number of binary trees with 5 internal nodes

•

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TTT T•T = [ ( )

1, 1, 2, 5, 14, 42, 132, 429, 1430, . . .


generating function

exact enumeration

T (z) = 1 + T (z)⇥ z ⇥ T (z)

Recursivity


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Decomposing General Graphs

?

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Generalizing Trees

Directed Acyclic Graphs(DAGs)

block graphs(clique trees)

cactus graphs(cacti)

k-trees

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Generalizing Trees

Directed Acyclic Graphs(DAGs)

block graphs(clique trees)

cactus graphs(cacti)

k-trees

Focus of this talk

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Cactus GraphsA graph is a cactus iff every edge is part of at most one cycle.

cactus

not cactus

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Cactus Graphs

cactus

not cactus

A graph is a cactus iff every edge is part of at most one cycle.

pure3-cactus

mixed

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Cactus Graphs

cactus

not cactus labeledcactus

unlabeledcactus

A graph is a cactus iff every edge is part of at most one cycle.1

23

4

5

6 7

8

9

pure3-cactus

mixed

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Cactus Graphs

cactus

not cactus

unlabeledcactus

planecactus

labeledcactus

A graph is a cactus iff every edge is part of at most one cycle.1

23

4

5

6 7

8

9

from Enumeration of m-ary Cacti (Bóna et al.)

pure3-cactus

mixed

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Prior Work

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Prior Work

— derived functional equations for non-plane, mixed, unlabeled cacti.

— promised to provide “a more systematic treatment of the general case of pure k-cacti” in a subsequent paper (it appears they never published such a paper)

On the Number of Husimi Trees Harary and Uhlenbeck (1952):

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Prior Work

non-generalizablemethods

hard toextract




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Prior Work

— enumerated pure, plane, unlabeled cacti.


hard toextract




Enumeration of m-ary cacti Miklós Bóna et al. (1999):

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Prior Work

— enumerated pure, plane, unlabeled cacti.


hard toextract

only plane cacticomplicated methods




Enumeration of m-ary cacti Miklós Bóna et al. (1999):

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New ResultExact enumeration of unlabeled, non-plane, pure n-cacti.

n = 3

n = 4

n = 5

n = 6

0, 0, 1, 0, 1, 0, 2, 0, 4, 0, 8, 0, 19, 0, 48, 0, 126, 0, 355, 0, 1037, . . .

0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 7, 0, 0, 25, 0, 0, 88, 0, 0, 366, 0, . . .

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 8, 0, 0, 0, 31, 0, 0, 0, 132, . . .

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 13, 0, 0, 0, 0, 67, . . .

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n = 3

n = 4

n = 5

n = 6

0, 0, 1, 0, 1, 0, 2, 0, 4, 0, 8, 0, 19, 0, 48, 0, 126, 0, 355, 0, 1037, . . .

0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 7, 0, 0, 25, 0, 0, 88, 0, 0, 366, 0, . . .

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 8, 0, 0, 0, 31, 0, 0, 0, 132, . . .

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 13, 0, 0, 0, 0, 67, . . .

Number of pure 3-cacti with 9 vertices

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n = 3

n = 4

n = 5

n = 6

0, 0, 1, 0, 1, 0, 2, 0, 4, 0, 8, 0, 19, 0, 48, 0, 126, 0, 355, 0, 1037, . . .

0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 7, 0, 0, 25, 0, 0, 88, 0, 0, 366, 0, . . .

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 8, 0, 0, 0, 31, 0, 0, 0, 132, . . .

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 13, 0, 0, 0, 0, 67, . . .

The first non-zero term is always 1 (corresponding to polygon)

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New ResultExact enumeration of unlabeled, non-plane, pure n-cacti.n = 3

n = 4

n = 5

n = 6

0, 0, 1, 0, 1, 0, 2, 0, 4, 0, 8, 0, 19, 0, 48, 0, 126, 0, 355, 0, 1037, . . .

0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 7, 0, 0, 25, 0, 0, 88, 0, 0, 366, 0, . . .

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 8, 0, 0, 0, 31, 0, 0, 0, 132, . . .

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 13, 0, 0, 0, 0, 67, . . .

Our approach is simpler and more general than Bóna et al.:

— easily extendable to mixed cacti

e.g. plane 5-cacti: 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 17, 0, 0, 0, 102, . . .

— can easily be extended to derive their result (plane cacti)

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Tree Decomposition of Graphs

G = Z ⇥ (P + SC)

P = Seq=4 (Z + SX)

SX = Z ⇥ Seq>1 (P)SC = Cyc>2 (P)split

decompositionsymbolic

specification

computer algebrasystem (CAS)

0, 0, 1, 0, 1, 0, 2, 0,

4, 0, 8, 0, 19, 0, 48,

0, 126, 0, 355, 0,

1037, . . .

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Tree Decomposition of Graphs

G = Z ⇥ (P + SC)

P = Seq=4 (Z + SX)

SX = Z ⇥ Seq>1 (P)SC = Cyc>2 (P)

symbolicspecification

computer algebrasystem (CAS)

0, 0, 1, 0, 1, 0, 2, 0,

4, 0, 8, 0, 19, 0, 48,

0, 126, 0, 355, 0,

1037, . . .

splitdecomposition

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The Split DecompositionGives a graph-labeled tree representation of a graphvia a series of split operations— Can read adjacencies from alternated paths.

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The Split DecompositionGives a graph-labeled tree representation of a graphvia a series of split operations— Can read adjacencies from alternated paths.

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Decomposition base cases:

prime nodes:

degenerate nodes:

cliqueK

starS

cycleP

e.g.

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SubtletiesWhere do we start decomposing from?— unlabeled structures have symmetries— different set of symmetries for different starting points (“roots”)

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Subtleties

Dissymmetry theorem (from species theory):— allows us to correct for symmetries of trees

Cycle-pointing (based on Pólya theory):— allows us to correct for symmetries of graphs

Where do we start decomposing from?— unlabeled structures have symmetries— different set of symmetries for different starting points (“roots”)

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VerificationVerifying the enumeration:

— proof of the characterization

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— manual generation of small instances

— proof of the characterization

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— proof of the characterization — manual generation of small instances

— brute force generation of small instances

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ConclusionSummary

Next Steps

— Derived an exact enumeration for cactus graphs (previously unknown)

— Parameter analysis

— Consider other kinds of prime nodes (e.g. bipartite nodes are prime nodes for parity graphs and were studied asymptotically by Jessica Shi ’18)

— Random sampling

— Further extended the split decomposition introduced by Chauve et al. (2014), and extended by Bahrani and Lumbroso (2016)

— For the first time studied a graph class with a split decomposition tree that contains prime nodes

Note— This work is related to independent work project by Sam Pritt ’17, who developed software for extracting enumerations from symbolic equations

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Thank you!

enumerating unlabeled cactus graphs - princeton …mbahrani/index_files/talk-cacti.pdf ·...

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