1
Representation of K-Map in to Graph and its relationship by Boolean
Function
1
Hussein Abdel Wasi Hussein, 2 Dr. Mohit James
M.Sc. Dept. Mathematics & statistics, SHIATS, Allahabad, [email protected]
2
Assistant Professor. Dept. Mathematics & statistics, SHIATS, Allahabad, [email protected]
1

vertices associated with each edge, or its edges may be
Abstract-- In computer science, graphs are used to represent
directed from
networks of communication, data organization, computational
(mathematics) for more detailed definitions and for other
devices, the flow of computation, etc. For instance, the link
structure of a website can be represented by a graph, in which
one
vertex
to
another;
see graph
variations in the types of graph that are commonly
considered. Graphs are one of the prime objects of study
the vertices represent web pages and directed edges
represent links from one page to another. A similar approach
can be taken to problems in travel, biology, computer chip
design,
and
many
other
fields.
The
development
in discrete mathematics [6]. Refer to the glossary of graph
theory for basic definitions in graph theory [7].
In the most common sense of the term, a graph is an ordered
of algorithms to handle graphs is therefore of major interest in
pair G = (V, E)
computer science. The transformation of graphs is often
Comprising
formalized
systems.
set E of edges or lines, which are 2-element subsets of
Complementary to graph transformation systems focusing on
V (i.e., an edge is related with two vertices, and the relation
and
represented
by graph
rewrite
rule-based in-memory manipulation of graphs are graph
databases geared
towards transaction-safe, persistent storing
and querying of graph-structured data.
a set V of vertices or nodes together
with
a
is represented as an unordered pair of the vertices with
respect to the particular edge)[8].
The
vertices
belonging
to
an
edge
are
called
the ends, endpoints, or end vertices of the edge. A vertex
I. INTRODUCTION
In This manuscript we will transform K – map to graph and
may exist in a graph and not belong to an edge [9].
II. Motivation:
vice versa. We will take three cases of K – map case of two
This application can be used to read the graph in order to
variables and three and four variables [2]. The inverse case
find out k-map and throughout k-map we can find out
read this graph to a known the number of variables and
Boolean functions.
linkages between them and the degree of correlation (either
0 or 1) and transform in to K – map and then knowing only
the Boolean expiration [3]. The examples in following
III. Representation K-map as graph
explain the next action steps.
In this application will transform K – map to directed graph
In mathematics and computer science, graph theory is the
and vice versa. We will take three cases of K – map case of
study of graphs, which are mathematical structures used to
two variables and three and four variables [5]. The inverse
model pairwise relations between objects. A "graph" in this
case, read directed graph to a known the number of variables
context is made up of "vertices" or "nodes" and lines
and linkages between them and the degree of correlation
called edges that connect them. A graph may be undirected
[4] , meaning that there is no distinction between the two
(either 0 or 1) and transform into K – map and then knowing
only the electronic circuit [1]. The examples in following
explain the next action steps.
2
Case 1: two variables
Example1: Consider the Boolean functions
a- F= xy +xy'
b- F= xy + x'y + x'y'
Has the following k-map
Case 1: a
y
0
1
0
1
1
1
0
0
x
Case 1: b
y
0
1
x
Fig. 1
Case 2: Three Variables
0
1
C
1
BA
00
1
0
01
1
11
10
Example2:
Consider the Boolean function
F=C'B'A'+C'BA'+CB'A'+C'BA+CBA
0
1
0
1
1
1
1
0
1
1
This produces the truth map appearing below:
We can Representation this K- map as simple graph as
following:
We can Representation k-map as graph as following. , a &b
3
Example 4:
Fig .2
Case 3: Four Variables
Fig .4
Now to convert this graph into k – map, we not the links set
Example3: Simplify the Boolean function:
F(x‚y‚z‚w)=∑(0,1,2,4,5,6,8,9,12,13,14)
in graph, now:
yz
We can expression the above k-map as graph
y'z'
y'z
Yz
yz'
w'x'
1
1
0
1
w'x
1
1
0
1
Wx
1
1
0
1
wx'
1
1
0
0
wx
Fig .3
Result: number of vertices equal 2 multiply number of
variables i.e. Let number variables = x then number of
AB= 00
CD=01
AB= 01
CD=01
AB= 11
CD=01
AB= 10
CD=01
vertices = 2.x.
Inverse case:
In this case transform a directed graph to k- map by Read
the links vertices. For example let we take case of four
variables.
And,
AB= 00
CD=11
AB= 01
CD=11
4
AB= 11
CD=11
AB= 10
CD=11
logic hardware. The pendant edges 1 and 7 must be ncluded
in every covering of the .
The pendant edges 1 and 7 must be included in every
covering of the graph. Therefore, the terms
But these values in k-map we get
AB
00
01
11
10
00
0
0
0
0
01
1
1
1
1
11
1
1
1
1
10
0
0
0
0
CD
K-map for above graph
Fig.5
x y z
IV. Minimization of Boolean Functions:
And xyz are essential.
Two additional edges 3 and 6 (or 4 and 5 or 3 and 5) will
An important step in the logical design of a digital machine
cover the remainder. Thus a simplified version of F is
is to minimize Boolean functions before implementing them.
F  x y z  xyz  w y z  wyz
Suppose we are interested in building a logical circuit that
gives the following function F of four Boolean variables
w,x,y, and z.
F  w x y z  w x y z  w x y z  w x y z  w x y z  w x y z  w x y z,
This expression can again be represented by a graph of four
vertices, as shown in Fig.5. [above]
The essential term xyz and xyz cannot be covered by any
edge, and hence cannot be minimized further. One edge will
cover the remaining two vertices in Fig.5 [below]. Thus the
Where + denotes logical OR, y denotes x AND y, and x
minimized Boolean expression is:
denotes Not x. Let us represent each of the seven terms in F
F  x y z  xyz  w y.
by a vertex, ad join every pair of vertices that differ only in
one variable. Such a graph is shown in Fig.5. An edge
between two vertices represents a term with three variables.
A minimal cover of this graph will represent a simplified
form of F, performing the same function as F, but with less
V. Conclusion
Ongoing through the chapters of this thesis we conclude that
the graph theory works properly in real life problem and it is
5
easy to implement on real life problem. the result obtained
by graph theory is more clear so it precisely describe the
nature of solution rather than existing method. In this
continuation we are also describing directed graph and its
relationship with Krnough Map.
VI. REFERENCES
[1] Hamdey A. Taha. Operations research An introduction .
Low Price Edition, 2007
[2] Udit Agarwal, Dhanpat Rai&co(Pvt.) Ltd. Discrete
Mathematical
Structures,
Educational
&
Technical
Publishers.
[3] Richard Bellman, Kenneth L. Cooke and Jo Ann
Lockert, Algorithms Graphs and Computers, Academic
press New York and London, 1970
[4] Narsingh Deo. Graph Theory with applications to
engineering and computer science. Prentice-Hall, 1974
[5] Franklin Kenter. Discrepancy inequalities for directed
graphs
Original
Research
Article
Discrete
Applied
Mathematics. Volume 176, 30 October 2014.
[6] Min-Sheng Lin, Sheng-Huang Su, Counting maximal
independent sets in directed path graphs Information
Processing Letters. Volume 114, Issue 10, October 2014.
[7] Javier Marenco, Marcelo Mydlarz, Daniel Severín,
Topological additive numbering of directed acyclic graphs
Information
Processing
Letters, In
Press,
Corrected
Proof, Available online 21 September 2014.
[8] Kaveh Khoshkhah, Hossein Soltani, Manouchehr Zaker,
Dynamic monopolies in directed graphs: The spread of
unilateral influence in social networks Discrete Applied
Mathematics, Volume 171, 10 July 2014.
[9] Anna Khmelnitskaya, Dolf Talman, Tree, web and
average web values for cycle-free directed graph games
European Journal of Operational Research, Volume 235,
Issue 1, and 16 May 2014.