Retno Maharesi
Fakultas Teknologi Industri jurusan Teknik Informatika
UNIVERSITAS GUNADARMA Jakarta
Hamiltonian Cycle
Hamiltonian Cycle in a simple graph G(V,E)
is the longest circuit(s) passing through the vertex set V, hence all of its vertices
degree are equal to 2.
In this occasion we consider the problem of
Planar Graph
Planar graph is a graph whose visualization on
a plane does not crossing any arc e Є E.
There are some testing procedures to indicate
whether a certain graph is planar or not.
Examples: platonic graph, cubic graph, K4, etc Planar graph has an application in rendering
operation in the feld of Computer graphic, as an instance, (Epstein,2007) worked on TSP
problem applied on a cubic graph model to get a fast algorithm to be used as a rendering
Enumerating Hamiltonian Cycles in a
Simple Graph
A 2-connected Simple graph is a graph whose minimal vertex’s degree is 2, hence deleting
those edges causing the graph to have an isolated vertex.
The problem of enumerating Hamiltonian cycles in a 2-connected simple graph can be diferentiated
into 3 cases, those are:
1. Complete graph: Can be easily enumerated and obtaining the generating function for the sequence indicating the counted Hamiltonian cycles in a
complete graph for number of nodes n = 3, 4, 5 ….
Enumerating Hamiltonian Cycles in
a Simple Graph
2. Nearly complete graph: can be enumerated using an exact formulae obtained through ECO method. In this case a simple graph is treated as a complete graph whose some of few arcs be deleted. Hence the complexity in applying the formulae becomes greater as the more arcs be deleted.
3. Planar graph: We will use the graph induced by planar cycle bases in order to be able to enumerate the Hamiltonian cycles contained in a planar graph.
Vector Representation of a Graph
Graph G(V, E) can be represented as a
vector in a vector space of dimension
R
Ewith binary operations: simetrics
difference and dot product on a Field of
integer modulo 2.
Cycle space is a subspace of vector space
obtained formed by edge set E hence
there are m cycle bases which construct
any cycle in the cycle space.
Counting the Hamiltonian Cycles
Based on the m cycle bases contained in a2-connected graph, one can enumerate all cycles
containing in the graph as, showed in proposition 1, which can be proven using closeness property of
binary operation between any number of cycle bases.
Proposition 1: The number of cycles containing in a
2-connected graph G(V, E) can be expresses by the formulae:
Fundamental Cycle bases
Following (Leydold and Stadler, 1998)
fundamental cycle bases are obtained by
taking circuit part as a result of addition e E\ T, into T(G) or written as:
Example: below picture the planar
fundamental cycle bases is top right picture
Graph induced by Fundamental Bases
Definitioin: Graph induced by fundamental cycles Gf
(V’, E’) is a weighted graph whose vertex set V’ are the representations of fundamental cycle bases of G(V,E) and the edge set E’ are the arcs which connect pairs of fundamental cycles. The weights assigned to the arcs indicating the number of edges belonging to the pairs of connected vertices.
Length of combined fundamental bases based on the induced graph
Where i ≤ m ,with is the weight of graph
induced by the j-th and k-th fundamental
cycle bases, and are the set of
edges corespond to
The proof of this proposition is not yet
properly written, we only showed the
correctness of the above equation on
an instance of 2-connected graph,
Graph induced by Planar Cycle Bases
As showed by Proposition 2, it will be nice if we try to use the graph induced by planar cycle bases, due to its constant weight values, i.e = 1 for all edges.
Propositioin 3: Graph induced by fundamental planar cycle bases of G(V, E) i.e Gf is a planar
subgraph whose all its weight values are equal to 1.
Planar Cyle Bases
Algorithm to obtain the bases may follow the
technique presented (Deo,et al. 1982), in an attemp to get minimal fundamental planar
cycle bases. Here we proposed by adding one by one the vertex with great degrees.
input: incidence matrix of G(V, E)
Sort descending order sthe vertices degrees
of V
Find the spanning tree dan co tree of G(V, E)
based on step 2 by adding the vertex one by one until all vertices being visited.
From step 4, G\T is determined
Find all fundamental cycles: by adding
to T and get the circuit resulting.
Results obtained from Graph
induced by planar cycle bases
Proposition 4: If Gf is an induced graph of planar cycle bases then any combination of i out of m vertices in Gf represent operation simmetrics
diference applied to i planar cycle bases. As a result the length of i cycles is determined by the weight values of connecting arcs belong to the connected subgraph containing combination those vertices as given by the following formulae: =
Results obtained from Graph
induced by planar cycle bases
Proof: Every arc of planar graph G(V, E) is
belong to two diferent cycles. Hence
according to Proposition 4, every arc on Gf
has weight value equals to 1. As a result a certain arc cannot connect pair of other cycles. Length of i combination of cycle
bases then can be obtained using simmetric diference operation of the two sets
to get:
Results obtained from Graph
induced by planar cycle bases
Proposition 5: If Gf If Gf is an induced graph of planar cycle
bases then vertices combination on Gf which tree contains
all the Hamiltonian cycles belong to G.
Proof is shown on the base of 3 possibly types of
connection among the vertices in the induced graph, those are
0-connected type
cyclical connected type, and tree- connected type.
A Formulae obtained based on
Proposition 3-5
Based on the prevoiu three Propositions,
the following formulae
can be use to determine the number of combination of cycle bases in which all
Hamiltonian cycles are built by those i<m cycle bases after examination of the
I
mplementation of the formulae for enumerating Hamiltonian Cycles in some planar graphImplementation of the formulae for enumerating Hamiltonian Cycles in some planar graph
We already try to enumerate
Hamiltonian cycles contained in a
platonic graph using Matlab
program mainly through the use of
nchoosek function for creating all
of i combination of cycle bases . A
more efcient algorithm as a
substitution of nchoosek function is
due to (Vajnowski, 2008), as a
loopless algorithm can be designed
for generating the combination
How many are Hamiltonian cycles
in the graph and what are they ?
Number of vertex=20,
Number of edges =30
Number of cycles=11
Enumerating Results
Using a Matlab code we obtain contains 30
Hamiltonian cycles in the platonic graph
An Algorithm for enumerating HC
in A Platonic Graph
1. input 11 cycle bases in the form of vector of size
1x30, whose elements are 0 or 1 depending on the edge presence in the cycle.
2. obtaining the induced cycle bases graph through the
cycle adjacency matrix.
3. using equation : 6 (5)-2(5)=20, where length of the
cycle bases all are same, i.e, 5, then the combination consists of 6 cycle bases.
4. generate 11C6 combination objects
5. search among the combinations whose type of
connection is a tree graph, by applying 11C2
combination and fnd the objects which satisfy: for
HAMILTONIAN CYCLE OBTAINED BY COMBINATION
References
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