Enumerating Unlabeled Cactus Graphs Dec 13, 2016 Princeton University Department of Computer Science Maryam Bahrani Under the Direction of Dr. Jérémie Lumbroso
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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Symbolic Method on Trees
TTT T•T = [ ( ) symbolic specification
generating functionT (z) = 1 + T (z)⇥ z ⇥ T (z)
A 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 = [ ( )
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . .
symbolic specification
generating function
exact enumeration
T (z) = 1 + T (z)⇥ z ⇥ T (z)
A binary tree is — either a leaf — or an internal node, and a left subtree, and a right subtree
•
2/13
Symbolic Method on Trees
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)
A binary tree is — either a leaf — or an internal node, and a left subtree, and a right subtree
Number of binary trees with 5 internal nodes
•
2/13
Symbolic Method on Trees
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)
Recursivity
A binary tree is — either a leaf — or an internal node, and a left subtree, and a right subtree
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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
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5
6 7
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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
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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
— 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):
7/13
Prior Work
— enumerated pure, plane, unlabeled cacti.
non-generalizablemethods
hard toextract
— 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):
Enumeration of m-ary cacti Miklós Bóna et al. (1999):
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Prior Work
— enumerated pure, plane, unlabeled cacti.
non-generalizablemethods
hard toextract
only plane cacticomplicated methods
— 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):
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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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, . . .
Number of pure 3-cacti with 9 vertices
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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, . . .
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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VerificationVerifying the enumeration:
— manual generation of small instances
— proof of the characterization
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VerificationVerifying the enumeration:
— 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!