Enumerating Hamiltonian cycles in a planar graph using Fundamental cycle bases
Enumerating Hamiltonian cycles in a planar
graph using Fundamental cycle bases
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
enumerating Hamiltonian cycles in a simple
2- connected graph G(V,E), which is also a
planar graph using planar cycle bases
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
algorithm
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.
Application: A Hamiltonian cycle correspond to a
Gray codes of a class of combinatoric objects.
Vector Representation of a Graph
Graph G(V, E) can be represented as a
vector in a vector space of dimension
R E with 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.
The dimension of the cycle space follows
a simple formulae: m=|E |–|V| +1, which
is also known as cyclomatic number.
Counting the Hamiltonian Cycles
Based on the m cycle bases contained in a 2-
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 2connected graph G(V, E) can be expresses by the
formulae:
where m = the number of fundamental cycle bases
and 1 ≤ i ≤m.
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.
Proposition 2: If Gf is a graph induced by fundamental
cycles of G(V,E) then any combination of i vertices of
Gf represent the combination of i cycle bases out of m
cycle fundamental cycle bases with length of cycle is
determined by the weight values belong to the arcs
of connected subgraph containing the combination of
those vertices and satisfying formulae:
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,
i.e.using binary operation:
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 G f is a planar
subgraph whose all its weight values are
equal to 1.
The proof is obtained by implementing Euler
theorem that every 2 faces in G(V,E) pass
through one arc.
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
Results obtained from Graph
induced by planar cycle bases
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:
Proposition 4: If
=
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 G f
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.
By using the fact that simmetrics diference operation on
the cycle space to be closed then above statement was
proven. Only does not produce an isolated vertices the
tree type of connection hence other types of connection
cannot produce Hamiltonian cycles.
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
graph using Fundamental cycle bases
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
enumerating Hamiltonian cycles in a simple
2- connected graph G(V,E), which is also a
planar graph using planar cycle bases
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
algorithm
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.
Application: A Hamiltonian cycle correspond to a
Gray codes of a class of combinatoric objects.
Vector Representation of a Graph
Graph G(V, E) can be represented as a
vector in a vector space of dimension
R E with 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.
The dimension of the cycle space follows
a simple formulae: m=|E |–|V| +1, which
is also known as cyclomatic number.
Counting the Hamiltonian Cycles
Based on the m cycle bases contained in a 2-
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 2connected graph G(V, E) can be expresses by the
formulae:
where m = the number of fundamental cycle bases
and 1 ≤ i ≤m.
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.
Proposition 2: If Gf is a graph induced by fundamental
cycles of G(V,E) then any combination of i vertices of
Gf represent the combination of i cycle bases out of m
cycle fundamental cycle bases with length of cycle is
determined by the weight values belong to the arcs
of connected subgraph containing the combination of
those vertices and satisfying formulae:
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,
i.e.using binary operation:
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 G f is a planar
subgraph whose all its weight values are
equal to 1.
The proof is obtained by implementing Euler
theorem that every 2 faces in G(V,E) pass
through one arc.
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
Results obtained from Graph
induced by planar cycle bases
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:
Proposition 4: If
=
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 G f
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.
By using the fact that simmetrics diference operation on
the cycle space to be closed then above statement was
proven. Only does not produce an isolated vertices the
tree type of connection hence other types of connection
cannot produce Hamiltonian cycles.
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