Genetic Theory of Cubic Graphs

Genetic Theory of Cubic Graphs

Pouya Baniasadi

Hamiltonian Cycle, Traveling Salesman and RelatedOptimization Problems Workshop, 2012

Pouya Baniasadi Genetic Theory of Cubic Graphs

Joint work with:

Vladimir Ejov, Jerzy Filar, Michael Haythorpe

Earlier works containing ideas that can be seen as precursors.

1 D. Blanus̆a. “Problem cetiriju boja.” Glasnik Mat. Fiz. Astr.Ser. II, 1:31–42, 1946.

2 N. Wormald. “Enumeration of labelled graphs II: Cubicgraphs with a given connectivity.” Journal of the LondonMathematical Society, s2-20(1):1-7, 1979.

3 B. D. McKay and G. F. Royle. “Constructing the cubic graphson up to 20 vertices.” Ars Combinatoria, 12A:129–140, 1986.

4 G. Brinkmann. “Fast Generation of Cubic Graphs.” Journal ofGraph Theory, 23:139–149, 1996.

5 G. Nguyen. “Hamiltonian Cycle Problem, Markov DecisionProcesses and Graph Spectra”. PhD Thesis, UniSA, 2009.


The Hamiltonian Cycle Problem: An Introduction

The Hamiltonian Cycle Problem (HCP) is defined by thefollowing - deceptively simple - statement:

Given a graph, find a simple cycle that contains all vertices of thegraph (Hamiltonian cycle (HC)) or prove that one does not exist.

The HCP is NP-Complete and has become a challenge both in itsown right and because of its close relationship to the famousTravelling Salesman Problem (TSP).

A graph is cubic, if every vertex has exactly three edges emanatingfrom it.

The HCP is already NP-Complete for cubic graphs.























Studying the “Population” of Cubic Graphs

The philosophy underlying this presentation is that much canbe learned about the HCP, by studying the whole populationof cubic graphs on n vertices.

For small n there is reliable, public domain, software enablingus to explicitly enumerate all distinct, connected, cubic graphson n vertices.

Indeed, for n = 8, 10, 12, 14, 16, 18, there are5, 19, 85, 509, 4060, 41301 such graphs.

Unsurprisingly, perhaps, the structure of these classes ofgraphs is very rich.




















“Population” of 10-vertex Cubic Graphs

First a sample of four (out of 19) Hamiltonian graphs


“Population” of 10-vertex Cubic Graphs

Next, the only two (out of 19) non-Hamiltonian graphs

The first is a “bridge graph”, the second is the famousPetersen graph.


Rhetorical questions:

Can this richness of structure be, somehow, explained?

In particular, is it “inherited” from simpler structures?

Are there some cubic graphs that could be called “genes”?

If so, could all “non-genes” be “descendants” of genes?

If so, can this be of help in determining Hamiltonicity of agraph?






























Non-Hamiltonian cubic 12-vertex graphs


Tietze’s graph


Tietze’s graph

+


Tietze’s graph

+


Tietze’s graph

+


Tietze’s graph

+ =

Hence, we say that Tietze’s graph is a “descendant” of thePetersen graph, and the 4-vertex graph.


Tietze’s graph

+ =

Hence, we say that Tietze’s graph is a “descendant” of thePetersen graph, and the 4-vertex graph.


Crackers and Genes

An n-cracker is a set of n non-adjacent edges whoseremoval disconnects the graph, such that removal of anyproper subset of the n-cracker does not disconnect the graph.

Example of a 2-cracker.

Example of a cutset that is not a cracker.


Crackers and Genes





Crackers and Genes





Crackers and Genes

An “n-cracker” for n ∈ {1, 2, 3} will be called a “cubiccracker”.

A cubic graph without a cubic cracker will be called a“gene”!.

It will be seen that cubic cracker are created by simplebreeding operations.

Graphs obtained by these breeding operations are calleddescendants.


Crackers and Genes






Crackers and Genes






Crackers and Genes






Type 1 breeding

+


Type 1 breeding

+


Type 1 breeding

+ =

Creates a bridge graph, that is, a 1-cracker.

Descendant more complex than either parent.


Type 1 breeding

+ =




Type 1 breeding

+ =




Type 2 breeding

+


Type 2 breeding

+


Type 2 breeding

+ =

Descendant still more complex than either parent even thoughtotal number of vertices is the same.

Creates a 2-cracker


Type 2 breeding

+ =

Descendant still more complex than either parent even thoughtotal number of vertices is the same.

Creates a 2-cracker


Type 3 breeding

+


Type 3 breeding

+


Type 3 breeding

+ =

Descendant still more complex than either parent even thoughtotal number of vertices is reduced by 2.

Creates a 3-cracker


Type 3 breeding

+ =

Descendant still more complex than either parent even thoughtotal number of vertices is reduced by 2.

Creates a 3-cracker


Parthenogenesis

“Breeding” involving a single graph.

From biology: Parthenogenesis is a form of asexualreproduction that does not require the involvement of apartner.

Idea: Exploit the structure of an existing 1-cracker or2-cracker to create a more complex descendant.


Parthenogenesis





Parthenogenesis





Type 1 Parthenogenic operation

&



& =

Inserts a parthenogenic diamond,

Increases total size by 4 vertices.



&



& =

Inserts a parthenogenic bridge,




&



& =

Inserts a parthenogenic triangle,



Descendants

Formally, a descendant is any cubic graph obtained byapplying a breeding/parthenogenic operation to one or twocubic graphs.

Equivalently, a descendant is any graph containing one ormore cubic crackers.

Results: Any cubic graph is either a gene G or a descendant graphGD . Suppose it is a descendant graph, then,

(1) GD can be obtained from a set of genes by a finite sequenceof the six breeding operations described earlier.

(2) The set of genes which through such a sequence of the sixbreeding operations results in the descendant graph GD willbe called the ancestor genes of ΓD


Descendants







Descendants







Descendants







Ancestor Genes

There could be factorially many ways to decompose a graphinto a set of genes.

For example for an 80 node graph there could be up to 10billion decompositions.

Do we always obtain the same family of genes no matter howwe decompose the graph?

Theorem?

Every descendant possesses a unique set of ancestor genes.

If correct, this allows us to consider graph theoretic properties ofthe ancestor genes that are inherited by the descendant.


Ancestor Genes




Theorem?




Ancestor Genes




Theorem?




Ancestor Genes




Theorem?




Ancestor Genes




Theorem?




Uniqueness of ancestor genes


















Genetic Theory and Non-Hamiltonicity

It can be shown that breeding a non-Hamiltonian genecreates a non-Hamiltonian descendant! But...

When are descendants of Hamiltonian genesnon-Hamiltonian?

Type 1 breeding - non-Hamiltonian.

Type 2 & 3 breeding - usually (but not always) Hamiltonian.




















Mutants

A mutant is any gene that is non-Hamiltonian.

Extremely rare.

Nodes NHNB graphs/NH Mutants/NH Mutants/NHNB

10 50.00% 50.00% 100%

12 20.00% 0% 0%

14 17.14% 0% 0%

16 15.07% 0% 0%

18 13.86% 0.12% 0.86%

20 12.60% 0.05% 0.38%

22 11.41% 0.02% 0.21%


Mutants


Extremely rare.


10 50.00% 50.00% 100%

12 20.00% 0% 0%

14 17.14% 0% 0%

16 15.07% 0% 0%

18 13.86% 0.12% 0.86%

20 12.60% 0.05% 0.38%

22 11.41% 0.02% 0.21%


Mutants


Extremely rare.


10 50.00% 50.00% 100%

12 20.00% 0% 0%

14 17.14% 0% 0%

16 15.07% 0% 0%

18 13.86% 0.12% 0.86%

20 12.60% 0.05% 0.38%

22 11.41% 0.02% 0.21%


Mutants


Extremely rare.

Similar to, but more general than, snarks.

BH-Mutant

Zircon-Mutant


Mutants


Extremely rare.


BH-Mutant

Zircon-Mutant


Mutants


Extremely rare.


BH-Mutant

Zircon-Mutant


Inheritence of graph theoretic properties

The following properties can be inherited from the ancestor genes:

Non-Hamiltonicity (if any ancestor genes are mutants)

However, some non-bridge, non-Hamiltonian descendants existwith all Hamiltonian ancestor genes.

Bipartiteness (if and only if all genes are bipartite)

Planarity (if and only if all genes are planar)

There are undoubtedly other such graph theoretic properties thatcan be inherited.

Motivates a decomposition algorithm for graph theory problems -first identify ancestor genes, then solve the finite set of smallerproblems.
























































Identification of ancestors

Under mild technical conditions all six breeding operations areinvertible (ie., identify a unique parent or pair of parents).

Result: Identification all ancestor genes can always be achieved, inpolynomial time, by searching for crackers.


Identification of ancestors

Under mild technical conditions all six breeding operations areinvertible (ie., identify a unique parent or pair of parents).

Result: Identification all ancestor genes can always be achieved, inpolynomial time, by searching for crackers.



Any Questions?


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Genetic Theory of Cubic Graphs