Indian Journal of Science and Technology, Vol 9(37), DOI: 10.17485/ijst/2016/v9i37/98001, October 2016
ISSN (Print) : 0974-6846
ISSN (Online) : 0974-5645
Minimum Vertex Cover of Circulant Graph having
Hamiltonian Path and its Application
Atowar-ul Islam1*, Jayanta Kr Choudhury2 and Bichitra Kalita3
Department of Computer Science and IT, Cotton College, Guwahati - 781001, Assam, India;
[email protected]
2
Department of Mathematics, Swadeshi College of Commerce, Guwahati - 781007, Assam, India;
[email protected]
3
Department of Computer Application (MCA), Assam Engineering College, Guwahati - 781013, Assam, India;
[email protected]
1
Abstract
Objective: The Minimum Vertex Cover of a circulant graph for m≥2 obtained from the complete graph K2m+1 and K2m+2 have
been discussed. Methods: Minimum Vertex Cover is a NP Complete problem. Various properties to find out the Minimum
Vertex Cover of different types of circulant graphs of even and odd values of m≥2 have been studied. Findings: After
studied the Minimum vertex cover we find two theorems for the graph K2m+1 and K2m+2 and results are also observed. An
algorithm also developed to find the Minimum Vertex Cover. Application: An application of minimum vertex cover has
been cited to avoid the deadlock condition in process graph with an example.
Keywords: Circulant Graph, Complete Graph, Deadlock, Minimum Vertex Cover, Process, Resources
1. Introduction
Circulant graphs have many applications in different
areas like telecommunication network, VLSI design
and distributed computing. Circulant graph is a natural
extension of a ring with increased connectivity. In1
discussed the decomposition of complete graph K2m+1
for m≥2. They established two theorem for different m
values of m≥2 relating to 2m+1 is prime, 2m+1=3.3n for
n≥1 and 2m+1=3s for s≥5 is also prime. They have also
established an algorithm for travelling salesman problem
when the weight of edges is non-repeated of the complete
graph K2m+1 for m≥2 with the help of circulant graph.
In2 discussed the essential and sufficient conditions
under which a directed circulant graph G of order n and
with k jumps can be decomposed into k pair wise arc
disjoint anti-directed Hamiltonian cycles, each induced
by two jumps. In addition, the essential condition for
complete decomposition of G into arbitrary anti-directed
Hamiltonian cycles has been discussed. It has been found
that the cyclic decomposition of circulant graphs are into
almost bipartite graphs. In3 proved that any 4-regular
* Author for correspondence
connected Cayley graph on a finite abelian group can
be decomposed into two Hamiltonian cycles. In4 proved
that the complete graph K2n can be decomposed into n−2
n-suns, a Hamiltonian cycle and a perfect matching, when
n is even and for odd cases, the decomposition is n−1 n–
suns and a perfect matching. They also discussed that a
spanning tree decomposition of even order complete
graph using the labeling scheme of n-suns decomposition.
In5 investigated the decomposition of complete graph
Kncycles Ct’ s and stars Sk’ s and studied efficient and
sufficient conditions for existence of such type of
decomposition to exist. In6 showed that each bipartite
graph H which decomposes Kk and Kn also decomposes
Kkn. In7 considered the problem of decomposing a
complete graph into the Cartesian product of two complete
graphs Kr and Kc and they found a common method of
constructing such decomposition using various kinds of
combinatorial designs. The Traveling Salesman Problem
(TSP) in the circulant weighted undirected graph case
have been discussed by8. They discussed an upper bound
and a lower bound for the Hamiltonian graph and
analyzed the two stripe case. They studied short SCTSP
Minimum Vertex Cover of Circulant Graph having Hamiltonian Path and its Application
(Symmetric Circulant Traveling Salesman Problem)
and they presented an upper bound, a lower bound and
a polynomial time 2-approximation algorithm for the
general case of SCTSP (Symmetric Circulant Traveling
Salesman Problem). In9 discussed the minimum vertex
cover of Regular planar Sub-graph H(2m+2,3m+3) and
K(2m+2,4m+4) for m≥2 and J(2m+2,5m+5) for m≥5
obtained from the complete graph K2m+2. An algorithm
has been developed to find the minimum vertex cover
of these types of regular planar sub-graph. Finally the
application of minimum vertex cover has been discussed
to reduce the power consumption of sensor network.
In this paper, we present two theorems where the first
one explain for decomposition of complete graphs K2m+1
into circulant graph C 2m+1 (j) for m≥2 where j is the jump,
1 ≤ j ≤ m into m edge disjoint circulant graphs and the
second theorem explain the decomposition of complete
graphs K2m+2 , C2m+1(j) for m≥2 where j is the jump, 1 ≤ j
≤ m into m edge disjoint circulant graphs. An algorithm
also has been developed to determine the Minimum
vertex cover of circulant edge disjoint graph K2m+1 and
K2m+2 for m≥2 and finally this algorithm has been used
to avoid the dead lock in process synchronization. In10
discussed straight line drawing is a mapping of an edge
into a straight line segment but they mainly discussed
elaborately the slope number of complete graph. They
consider the edges are straight line segments of a complete
graph instead of to obtain the number of slopes. They
also interpret the characterization of slopes in complete
graph according to an odd and even number of edges and
investigated in detail. In application slope number is used
to find out different layout methods for the same graph.
In11 introduced the concept of K-Hamiltonian connected
graphs. They established every (k+1) Hamiltonian
connected graph is k-ordered for k≥3. Some necessary
and sufficient conditions for a graph to be k-ordered are
discussed. A lot of sufficient conditions have been given
for Hamiltonian graphs. They also discussed about two
well-known graphs which are Dirac and Ore. In12 analyzed
the three Hamiltonian structures associated with the
Hamiltonian description of the system of equations WBK
and probe Jacobi identity. They discuss the compatibility
of the three structures using the method of functional
multi-vectors. By considering multipliers of order three
they construct conservation laws of the system. From
the Hamiltonian operators and conserved densities they
deduce Hamiltonian symmetries of the WBK equations.
The paper is organized as follows: The Section 1
2
Vol 9 (37) | October 2016 | www.indjst.org
includes the introduction which contains the works of
other researcher. Section 2 includes the definition. Section
3 contains two theorems which are stated and proved.
Section 4 includes an algorithm. Section 5 includes the
experimental results. Section 6 includes the application of
the algorithm. Section 7 includes the conclusion.
2. Definition
Definition of circulant matrix: A circulant graph is a
graph which has a circulant adjacency matrix. Examples
of circulant graphs are the cycle Cn, the complete graph
Kn, and the complete bipartite graph Kn, n.
Circulant Matrix: Every n x n matrix C of the form
is called a circulant matrix. The matrix C is completely
determined by its first row because other rows are
rotations of the first row. C is symmetric if Cn-1 = Ci for
i = 1, 2, 3…….., n-1. Further, C is an non adjacency matrix
if C0 = 0 and Cn-1 = Ci∈{0,1}.
The graph of Figure 1 is a circulant graph having jump1 and Figure 2 is also a circulant graph having jump-2.
The circulant matrix of the circulant graph of Figure
1 and Figure 2 are shown in Figure 3 and Figure 4
respectively.
Figure 1. C5(1).
Figure 2. C5(2).
Indian Journal of Science and Technology
Atowar-ul Islam, Jayanta Kr Choudhury and Bichitra Kalita
Figure 3. Circulant matrix of C5(1).
Figure 4. Circulant matrix of C5(2).
Definition of Vertex Cover: A vertex cover of an
undirected graph is a subset of its vertices such that for
every edge (u, v) of the graph, either u or v is in vertex
cover and the set covers all the edges of the given graph.
The Minimum Vertex Cover is a vertex cover having
the smallest possible number of vertices of a given graph
and which covers all the edges. For example the graph
Figure 5 has Minimum Vertex cover is 4 having {a, f, d, e}
which covers all the edges.
Figure 5.
edges are also 5 and if the graph contains 6 vertices than
the graph contains also 6 edges. The circulant graph of
complete graph K2m+1 contains the odd number of vertices
and edges and the circulant graph of complete graph K2m+2
contains the even number of vertices and edges and the
graph also having Hamiltonian path. After studying the
Minimum vertex cover of these two types of graphs we
have find two theorems.
Theorems
Theorem 1: The minimum vertex cover of K2m+1 is m+1
when m≥2.
Proof: It has been proved that [1] the complete graph K2m+1
for m≥2 can be decomposed to m edge disjoint circulant
graph C2m+1(j) where j is the jump and 1≤ j≤m. The graph
are also bipartite graph and contains Hamiltonian path.
We now proceed to prove that the minimum vertex cover
of circulant graph having odd vertices and odd edges
is m+1 for m≥2. The result is true for m = 2 (Figure 6.)
having the circulant graph C5(1) C5(2) and minimum
vertex cover is 3 and for m = 3 (Figure 7.) having the
circulant graph C7(1) , C7(2) and C7(3) and minimum
vertex cover is 4 respectively for m≥2.
Figure 6 shows two Circulant graphs C5(1) , C5(2) and
minimum vertex cover for both the graph is 3. Figure 7
shows three Circulant graphs C7(1) , C7(2) and C7(3) and
minimum vertex cover for all the graph is 4.
Figure 6.
3. Present Work
3.1.1 Theoretical Investigation
In this paper Minimum Vertex Cover for decomposition
circulant graph for m≥2 obtained from the complete graph
K2m+1 and K2m+2 have been discussed. The total number of
vertices and edges are equal for both (K2m+1 and K2m+2)
of circulant graph. If a graph contains 5 vertices then
Vol 9 (37) | October 2016 | www.indjst.org
Figure 7.
Indian Journal of Science and Technology
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Minimum Vertex Cover of Circulant Graph having Hamiltonian Path and its Application
Let the circulant graph of complete graph ( K2k+1 ) has
minimum vertex cover k+1 when m = k. We now show
that the graph K2m+1 have minimum vertex cover k+1+1.
Now when we put m = k+1, then the form of the given
graph K2m+1 is K2(k+1)+1= K2k+3 . But our theorem states for
the values of m≥2. Hence m = k+1≥2 =>k≥1 which is true
for the graph K2k+3 for k≥1. Hence the theorem has been
found.
Theorem 2: The minimum vertex cover of K2m+2 is m+1
when m≥2.
Proof: It has been proved that [4] K2m+2 graph are the
circulant graph of complete graph for m≥2.The graph are
also bipartite graph and contains Hamiltonian path. We
now proceed to prove that the minimum vertex cover of
circulant graph having even number of vertices and even
number of edges is m+1 for m≥2. The result is true for
m = 2 (Figure 8.) having the circulant graph C6(1) C6(2)
and minimum vertex cover is 3 and for m = 3 (Figure 9.)
having the circulant graph C8(1), C8(2) and C8(3) and
minimum vertex cover is 4 respectively for m≥2.
Figure 8.
4. Algorithm
Step-1: Start
Step-2: While e ∈ E = NULL do, (E is the set of edges).
Step 3: Calculate the degree of all the vertices.
Step-4: Select a vertexfrom Max(d(Vm) where
m,n=1,2,3…………and V is the set of vertices.
Step-5: If degree(vm) = degree(vn) then select any one of
them.
Step-6: Select the incident edges(e ∈ E) of Max(d(Vm)
from which vertices are selected in the Step-3.
Step-7: Observe, whether the selected vertices of incident
edges of the graph covers all the edges or not. If not
covers all the edges then proceed to next step. Step-8:
Degree(vm,vn) = (degree(vm,vn) - no of connected adjacent
E(vm,vn)(which are already selected))
Step-9: If the vertex considered in step-4does not cover
all the edges, then goto Step-10 otherwise go to Step-13.
Step-10: Select another vertex which is containing the
maximum degree of the graph. The new selected vertex
should be the non adjacent of previous selected but it
should be continuously connected of the previous selected
vertex and observed that its covers all the edges or not.
Step-11: Continue Step-4 to Step-10 until and unless the
selection of vertices covers all the edges
Step-12: End while
Step-13: Stop.
5. Experimental Results
Some experimental result of Circulat Graph of Complete
graphs K2m+1 and K2m+2 is given below.
Figure 9.
Let the circulant graph of complete graph ( K2k+2 ) has
minimum vertex cover k+1 when m = k. We now show
that the graph K2m+2 have minimum vertex cover k+1+1.
Now when we put m = k+1, then the form of the given
graph K2m+1 is K2(k+1)+2= K2k+4. But our theorem states for
the values of m≥2.Hence m=k+1≥2 =>k≥1 which is true
for the graph K2k+4 for k≥1. Hence the theorem.
4
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Table 1. Shows minimum vertex cover of K2m+1 graph
for m≥2
Value(m)
m=2
m=3
m=4
m=5
m=6
m=7
m=8
m=9
Graph (K2m+1)
Minimum Vertex Cover
Vertices=5, Edges=5
3
Vertices=7, Edges=7
4
Vertices=9, Edges=9
5
Vertices=11, Edges=11
6
Vertices=13, Edges=13
7
Vertices=15, Edges=15
8
Vertices=17, Edges=17
9
Vertices=19, Edges=19
10
Indian Journal of Science and Technology
Atowar-ul Islam, Jayanta Kr Choudhury and Bichitra Kalita
Table 2. Shows minimum vertex cover of K2m+2 graph
for m≥2
Value(m)
m=2
m=3
m=4
m=5
m=6
m=7
m=8
m=9
Graph (K2m+2)
Minimum Vertex Cover
Vertices=6, Edges=6
3
Vertices=8, Edges=8
4
Vertices=10, Edges=10
5
Vertices=12, Edges=12
6
Vertices=14, Edges=14
7
Vertices=16, Edges=16
8
Vertices=18, Edges=18
9
Vertices=20, Edges=20
10
Note: In the complete graph K2m+1 and K2m+2 when we decompose the graph
into different circulant graph according to different jump we find some edge
disjoint graph for m≥2. In the edge disjoint graph the Minimum Vertex
Cover is exceptional then the other graph which is not consider in the
theorem. As an example when we decompose the K2m+2 graph for m = 2
(jump-2) and K2m+2 for m = 4 (jump-3) there is edge disjoint graph for each
whose Minimum Vertex Cover is 4 and 6 instead of 3 and 5 respectively.
6. Application
Let us consider our circulant graph is a process graph
where vertices are processes and edges are resources.
Before implementation of Minimum vertex cover
algorithm the graphs are circualant graph and which
are disjoint. The process of circulant graphs shares the
resources by other processes. Before implementation of
Minimum vertex cover algorithm all the Circulant graph
K2m+1 and K2m+2 m≥2 contains all the condition of dead
lock. A dead lock system must satisfied the following 4
conditions.
• Mutual exclusion.
• Hold and wait.
• No preemption.
• Circular wait.
For example we consider Figure 10(a) and 10(b) as
a process graph. The graph Figure 10(a) and10(b) have
a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r and s are the
vertices and e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12, e13, e14,
e15, e16, e17, e18 and e19 are the edges which are considered
as processes and resources in process graph respectively.
In our process graph one resources are shared by two
processes. In Figure 10(a) where the ‘a’ process is executed
its needs e1 and e19 resources which are shared by ‘b’ and ‘s’
processes also. In Figure 10(b) if process ‘a’ is executed its
needs also e1 and e19 resources which are shared by ‘c’ and
‘r’ processes. Similarly all the other process is also shared
the resources by other processes. So we can say that
Vol 9 (37) | October 2016 | www.indjst.org
resources are shareable mode and the graph Figure 10(a)
and 10(b) satisfy all the other condition of dead lock.
Figure 10(a) Minimum vertex cover is 10 for C19(1),
Figure 10(b) Minimum vertex cover is 10 for C19(2). In
the above graph red color vertices are the minimum
vertex cover.
But when we use our Minimum Vertex Cover
algorithm in our circulant or process graph then its avoid
the dead lock condition which clarify the process graph
of Figure 10(a) and 10(b). In process synchronization
when process are executed its first come to ready queue
and then executed the process according to the minimum
vertex cover algorithm. In process scheduling there are
some algorithm which are also avoid dead lock condition
as an example Banker’s algorithm, Resource allocation
algorithm etc. But our proposed algorithm is different
from the above mention algorithm. First we consider our
Figure 10(a) graph for executing the process. The ready
queue of the process graph is describing below along the
execution of process using our proposed algorithm.
Figure 10.
There is a problem which includes that there is no way
to know the length of the next CPU burst. We may not
know the length of the CPU burst, but we may be able to
predict its value. We expect that the next CPU burst will
be similar in length to the previous one. In ready queue
process execution time is predicted by the processor. The
next CPU burst is generally predicted as an exponential
average of the measure length of previous CPU burst. The
predicted Ready Queue is given below.
Ready Queue
In the ready queue the symbols a, b, c, d, e, f, g, h, i,
Indian Journal of Science and Technology
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Minimum Vertex Cover of Circulant Graph having Hamiltonian Path and its Application
Table 3. Ready queue for the graph 10(a)
0
2
4
6
8
a
b
c
d
e
10 12 14 16 18 20 22 24 26 28 30 32 34 36
f
g
h
i
j, k, l, m, n, o, p, q, r, s are process and 0,2,4…..38 are
the processing time which will be given by processor
according their process execution time. Here process
time is prediction time and these are not actual time. This
is only an imagination because ready queue always make
by processor which are not predictable by user.
In process execution if we use our minimum
vertex cover algorithm in Figure 10(a) the algorithm
will be work as follows. First the algorithm calculates
the degree of all the vertices (here degree define the
resources). Thereafter selectany one vertex which degree
is maximum. In the graph 10(a) first select the vertices
‘a’ or process and after selection the vertex ‘a’ select the
resources which are incident on it which are e1 and e19.
In our algorithm resources are used only one time. When
process ‘a’ is executed then e1 and e19 resources are used
then the resources of ‘b’ and ‘s’ is decreased by one then
the resources of ‘b’ is 1 and resources of ‘s’ is also 1. When
‘a’ process is executed no other process is executed in
our minimum vertex cover algorithm. After execution
of process ‘a’ in ready queue or process graph of Figure
10(a) the ‘b’ process is not executed but the ‘c’ process
is executed because of ‘b’ process resource is one and c
process resources is two. When ‘c’ process is executed
e2 and e3 resources are used and then the resources of
‘d’ will be decreased by one and similarly the ‘b’ process
resources also decreases by 1 and resources of ‘b’ process
is 0. It means that the ‘b’ process have no resources. In the
ready queue, after execution of ‘c’ process ‘d’ process will
come but ‘d’ process is not executed e process is executed
because ‘d’ process have resource 1 and ‘e’ process have
resources 2. So our Minimum vertex cover algorithm
executes ‘e’ process. When ‘e’ process is executed e4 and
e5 resources are used and resources of ‘d’ process and ‘f ’
process are decrease by one and resource of ‘d’ process
is 0 and resource of ‘f ’ process is 1. In this way from the
process graph of Figure 10(a) or ready queue the process
a, c, e, g, i, k, m, o, q, s process are executed.
Now applying the same algorithm for the graph of
Figure 10(b) the process a, e, i, m, q, b, f, j, n, r processes
are executed until and unless all the resources are used.
In this way for both the graph the process are executed
one by one until and unless all the resources are used
6
Vol 9 (37) | October 2016 | www.indjst.org
j
k
l
m
n
o
p
q
r
s
using our Minimum vertex cover algorithm. So we can
implement our algorithm in the circulant graph k2k+1,k2k+2
for m≥2 and executed the process until and unless all
the resources are used. In process synchronization in
critical section only one process is executed at a time and
other process will be in waiting state. After analyzing the
execution of process it is seen that no same resources
are shared at the time of process execution. So when
executing the process there is no mutual exclusion, no
hold and wait and no preemption. So our algorithm helps
to avoid deadlock in process synchronization.
But there is a limitation that in our algorithm that all
the processes are not executed. Some of the process are
always in ready queue but are not executed and used all
the resources by other process.
7. Conclusion
Here we have discussed Minimum Vertex Cover of
circulant graphs which are obtained from complete graphs
K2m+1and K2m+2 for m ≥2. An application is also developed
through Minimum vertex cover. However, one can study
to find out the Minimum vertex cover of circulant graph
to remove dead lock in process synchronization. It is
applicable for the entire circulant graph which we have
studied here.
8. References
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