57 Int. J Latest Trend Math
Vol-1 No. 2 December 2011
Fuzzy Tolerance Graphs Sovan Samanta and Madhumangal Pal Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore - 721 102, India. email:
[email protected] Abstract -In this paper, we define fuzzy tolerance, fuzzy tolerance graph, fuzzy bounded tolerance graph, fuzzy interval containment graph and regular representation of fuzzy tolerance graph, fuzzy unit tolerance graph and proper tolerance graph. Also some basic theorems related to the stated graphs have been presented. Keywords: Fuzzy graphs, intersection graphs, fuzzy tolerance graphs.
1. Introduction In mathematical area of graph theory, an intersection graph is a graph that represents the pattern of intersection of family of sets. An interval graph is the intersection of multiset of intervals on real line. Interval graphs are useful in resource allocation problem in operations research. Besides, interval graphs are used extensively in mathematical modeling, archaeology, developmental psychology, ecological modeling, mathematical sociology and organization theory. Tolerance graphs [5] are generalization of interval graphs in which each vertex can be represented by an interval and a tolerance such that an edge occurs if and only if the overlap of corresponding intervals is at least as large as the tolerance associated with one of the vertices. Hence a graph G = (V , E ) is a tolerance graph [5] if there is a set I= {Iv:v V} of closed real intervals and a set T={Tv:v V} of positive real numbers such that ( x, y) E if |Ix Iy| min{Tx,Ty} . The collection I,T of intervals and tolerances is called tolerance representation of the graph G . Tolerance graphs were introduced in order to generalize some well known applications of interval graphs. The main motivation was to model resource allocation and certain scheduling problems, in which resources, such as rooms and vehicles, can tolerate sharing among users. Tolerance graphs find in a natural way for applications in biology and bioinformatics. The tolerance graphs find numerous other applications in constrained-based temporal reasoning, data transmission through networks to efficiently scheduling aircraft and crews, as well as contributing to genetic analysis and studies of the brain. 1.1 Fuzzy sets The fuzzy systems have been used with success in last years, in problems that involve the approximate reasoning. The objective of the work is to specify the fuzzy systems with the addition of the interval theory in its components. Fuzzy graph theory was introduced by Azriel Rosenfeld [16] in 1975 . Though it is very young, it has been growing fast and has numerous applications in various fields. is characterized by a mapping m : X [0,1] , called the membership function. We shall denote a fuzzy set as A = ( X , m) . The support of A is supp A = {x X | m( x) 0} . The A fuzzy set
core of
A
A
on a set
X
is the crisp set of all members whose membership values are 1 .
A
is non trivial if supp
A
is nonempty.
The height of A is h( A) = max{m( x) | x X } . A is normal if h(A) = 1 . The membership function of the intersection of two fuzzy sets A and B with membership functions m A and mB respectively is defined as the minimum of the two individual membership functions. mAB = min(mA , mB ) . Let A = ( X , m) and
B = ( X , m' ) be two fuzzy sets on X then A B (fuzzy subset) if m(x) m' (x) for all x X . The family of all fuzzy subsets is denoted by (x) . A fuzzy set A = ( X , mA ) is said to beconvex if its membership value satisfies the following condition mA[x1 (1 ) x2 ] min[mA ( x1 ), mA ( x2 )] for any x1 , x2 X _______________________________________ International Journal of Latest Trends in Mathematics IJLTM, E-ISSN: 2049-2561 Copyright © ExcelingTech, Pub, UK (http://excelingtech.co.uk/)
and
58 Int. J Latest Trend Math
Vol-1 No. 2 December 2011
[0,1] . The families of sets most often considered in connection with intersection graphs are families of intervals of a linearly ordered set. Let X be a linearly ordered set. A fuzzy interval on X is normal, convex fuzzy subset of X . A fuzzy number is a real fuzzy interval. 1.2 Fuzzy graphs
= (V , , ) is a non empty set V together with a pair of functions : V [0,1] and : V V [0,1] such that for all x, y, ( x, y) ( x) ( y) . So is a fuzzy relation on . A fuzzy diagraph = ( , ) is similarly defined except : V V [0,1] need not be symmetric. We call the pairs (( x, y), ( x, y)) fuzzy arcs to emphasize that the symmetry is not required. McAllister uses a measure of A fuzzy graph [16]
fuzziness of the fuzzy intersection of two fuzzy sets to define two structures which together are called a fuzzy
(u, v) =
u and v is
intersection graph. The strength of connectedness between two vertices
sup{ k (u, v) | k = 1,2,...} where k (u, v) =
sup{ (u, u1 ) (u1 , u2 ) (uk 1 , v) | u1 , u2 ,, uk 1 V }
. An edge (u, v) is a fuzzy bridge of
deletion of (u, v) reduces the strength of connectedness between the pair of vertices. A vertex
if
u is a fuzzy cut
if deletion of u reduces the strength of connectedness between other pair of vertices. The order of S ( ) = (u, v) O( ) = (u ) ( u ,v ) > 0 uV fuzzy graph is . The size of fuzzy graph is . In a fuzzy graph an arc vertex of
(u, v) is said to be strong arc or strong edge, if (u, v) (u, v) and the node v is said to be strong neighbour of u . If (u, v) is not strong arc then v is called isolated node or isolated vertex. Given t [0,1] and a fuzzy set
A,
we define the t -cut level set of
{x supp A | m( x) t} . The t -cut level graph of is the crisp graph fuzzy subsets, the t -cut level family of is
A to be t = ( t , t ) .
At = For a family of the crisp set
t t = { A | A } . The fuzzy intersection (min operator) is use to define the fuzzy intersection graph of
a family of fuzzy sets. As in common to fuzzy set theory a crisp set is identified with a characteristic function. When context is clear two concepts are used interchangeably. Crisp definitions are extended using the concept of cut level sets. Significant results are discussed between fuzzy structure and the corresponding cut level sets.
= {A1 = ( X , m1 ), A2 = ( X , m2 ), An = ( X , mn )} be a V = {v1 , v2 , vn } finite family of fuzzy sets defined on a set X and consider as crisp vertex set . The fuzzy Int ( ) = ( V , , ) [0,1] : V intersection graph of is the fuzzy graph where is defined by Fuzzy intersection graph [7] is defined as:
(vi ) = h( Ai )
and
: V V [0,1] is defined by h( Ai Aj ), 0,
(vi , v j ) = Here
if i j if i = j
= {(vi , v j ) | (vi , v j ) > 0} is called the edge set of the fuzzy graph. An edge (vi , v j )
mi m j
has a zero
strength iff is zero function (empty intersection). A fuzzy interval graph [4] is the fuzzy intersection graph of a finite family of fuzzy intervals. 1.3 Review of Previous Works Nagoorgani and Radha [13] introduced regular fuzzy graphs. Nagoorgani and Vijayalakshmi [10] explained about insentive arc in domination of fuzzy graph . Nair and Chang [14] defined the concept of fuzzy clique. Natarajan and Ayyasawamy [15] described the concept on strong (weak) domination in fuzzy graphs. Nagoorgani
59 Int. J Latest Trend Math
Vol-1 No. 2 December 2011
and Vadivel [11] introduced relations between the parameters of independent domination and irredundance in fuzzy graph. Nagoorgani and Malarvizhi [12] proved the isomorphism properties of strong fuzzy graphs. Mathew and Sunitha [6] defined different types of arcs in a fuzzy graph. Bhutani et al. [2] discussed about the degrees of end nodes and cut nodes in fuzzy graphs. 1.4 Our works In this paper we define fuzzy tolerances, then fuzzy tolerance graphs and prove some basic results.
2
Fuzzy Tolerance Graphs
2.1 Fuzzy Tolerances Definition 1 Fuzzy tolerance of a fuzzy interval is denoted by and defined by an arbitrary fuzzy interval whose core length is a positive real number. If the real number is taken L and | ik ik 1 |= L where ik 1 , ik R then fuzzy tolerance is a fuzzy set of the interval
[ik 1 , ik ] .
The core and support of a fuzzy tolerance are defined as tolerance core and tolerance support respectively. The tolerance core and tolerance support of fuzzy tolerance are denoted by c( ) and s( ) respectively. Fuzzy tolerance may be a fuzzy number. The fuzzy tolerance is shown in Figure 1.
Figure 1: Fuzzy tolerance 2.2 Fuzzy tolerance graphs Fuzzy tolerance graph is the generalization of fuzzy interval graph which is defined below.
Definition 2 Let ={ 1,2,. . . ,n} be a finite family of fuzzy intervals on the real line and={1,
2,…,n}be the corresponding fuzzy tolerances. We consider as a crisp vertex set V = {v1 , v2 ,, vn } . The fuzzy tolerance graph is the fuzzy graph (V , , ) where : V [0,1] is defined by (vi)=h(i)=1 for all vi V and : V V [0,1] is defined by 1, s(I I ) min{s (T ), s (T )} i j i j (v i , v j ) = h(I i I j ), s(I i I j ) 0,
if c(I i I j ) min{c(T i ), c(T j )} otherwise if s(I i I j ) min{s (T i ), s (T j )} otherwise.
where c(i j) is the core of the intersection of the intervals I and j . min{c(i ),c(j )} is the minimum of the cores of the corresponding tolerances i and j. Also s(i j), min{s(i ),s(j ) are the support of the intersection of intervals i ,j and minimum support the of tolerances i, j respectively. If the fuzzy tolerances are not incorporated with the fuzzy intervals, the corresponding graph becomes
60 Int. J Latest Trend Math
Vol-1 No. 2 December 2011
fuzzy interval graph. The fuzzy tolerance graph have been shown in Figures 2, 3, 4 in three different cases. We consider two fuzzy intervals i ,j together with two fuzzy tolerances i , j . In the first case minimum of two tolerance cores min{c(i ),c(j )} is less than the intersection core c(i j) of the intervals I and j. In this case,edge membership value is 1 . If min{c(i ),c(j )} is greater than c(i j) then second case arises where membership value of the edge is less than 1 . Here minimum of two tolerance supports min{s(i ),s(j )} is lessthan the intersection support s(i j) of the intervals i, j. Otherwise, the third case arises where edge membership value is
0. If is a tolerance representation of a crisp graph G and v is a point instead of an interval then the corresponding vertex is isolated vertex of G . But it may not be true in the fuzzy sense.If < , > is a fuzzy tolerance representation of and f is a fuzzy number instead of fuzzy interval, then the corresponding vertex may not be isolated. Intersection of supports of a fuzzy interval and a fuzzy number may be greater than minimum support of their corresponding tolerances. So there is an edge of some positive strength. If it is not strong arc then the fuzzy number is isolated otherwise it is not isolated.
Figure 2: Fuzzy tolerance graph with edge membership value 1
Figure 3: Fuzzy tolerance graph with edge membership value less than 1
61 Int. J Latest Trend Math
Vol-1 No. 2 December 2011
Figure 4: Fuzzy tolerance graph with edge membership value 0
Example 1 Let us consider an example of three fuzzy intervals 1, 2, 3 with corresponding fuzzy tolerances 1, 2, 3. Let the core and support of 1 be [2,7] and [1,8] ; that of 2 are [4,8]and [2.5,9.5]; that of 3 are [9,13] and [6.5,15.5] . The core and support lengths of1, 2, 3 be respectively {1,1.2} , {0.8,1} , {0.5,2} . So h(1 2)=1, h( )=0.75, and h(1 )= 0.43. Clearly, the edge membership values of the edges (v1 , v2 ) , (v2 , v3 ) and (v1 , v3 ) are obtained as 1 , 0.5 and 0.08 respectively. The corresponding diagram is shown in Figure 5.
Figure 5: Example of a fuzzy tolerance graph 2.3 Fuzzy bounded tolerance graphs If a crisp graph G has a tolerance representation
I,T
, such that Tj is a fuzzy bounded tolerance representation then no fuzzy interval v , v
V is fuzzy number. Theorem 1 If is a fuzzy interval graph then is a fuzzy tolerance graph with constant core and constant support of tolerances. Proof. Let be a fuzzy interval graph with a representation v is assigned to the vertex v . Let the cores of fuzzy intervals x and y be denoted by c(x) and c(y) and that of supports be s(x) and s(y). Also c(x y)=c(x) c(y ) and s(x y)=s(x) s(y ). Let k1 and k 2 be positive real numbers such that k1 |c(x y)| and k2 |s(x y)| for all x, y V with k1 k2 . Thus the intervals {v |v V} together with tolerances with core k1 and support k 2 give a fuzzy tolerance representation.
Theorem 2 If is a fuzzy tolerance graph with constant core and constant support of tolerances then is fuzzy bounded tolerance graph. Proof. Let={ 1,2,. . . ,n} be a finite family of fuzzy intervals on real line and={1, 2,…,n}be the corresponding fuzzy tolerances. Let be a fuzzy tolerance graph with and be thecorresponding fuzzy interval and fuzzy tolerance representation. Let k1 and k 2 be two positive real numbers. Let c(i )=k1and s(i )=k2for i = 1,2,, n . If c(i)≥k1 and s(i)≥k2forall i = 1,2,..., n then is fuzzy bounded tolerance graph. If c(i) of a fuzzy unit or proper tolerance graph. We may assume all endpoints of core and support in this representation are distinct. We replace c( x) by |c( x)| for
x V when c(x) ≥ |c(x)| and s(x) replace by |s(x)| for each x V when s(x) ≥ |s(x)|, Since there are no containments of core and support of fuzzy intervals, this will not change any adjacencies. each
References [1] J. F. Allen, Maintaining knowledge about temporal intervals, Communications of the ACM, 26(11) ,
832 843,1983 . [2] K. R. Bhutani, J. Moderson and A. Rosenfeld, On degrees of end nodes and cut nodes in fuzzy graphs, 1(1) ,
57 64 , 2004 . [3] K. P. Bogart, P. C. Fishburn, G. Isaak and L. Langley, Proper and unit tolerance graphs, Discrete Applied Mathematics, 60,99 117,1995 . [4] W. L. Craine, Charcterizations of fuzzy interval graphs, Fuzzy Sets and Systems, 68,181 193 , 1994 . [5] M. C. Gulumbic and A. Trenk, Tolerance Graphs, Cambridge University Press, (2004) . [6] S. Mathew and M. S. Sunitha, Types of arcs in a fuzzy graph, Information Sciences, 179 , 1760 1768 , 2009 . [7] M. L. N. McAllister, Fuzzy intersection graphs, Comput. Math. Applic., 15 (10) , 871 886 , 1988 . [8] G. B. Mertzios and I. S. S. Zaks, The recognition of tolerance and bounded tolerance graphs, Symposiam on
67 Int. J Latest Trend Math
Vol-1 No. 2 December 2011
Theoritial Aspects of Computer Science, 585 596 , 2000 . [9] J. N. Mordeson and P. S. Nair, Fuzzy Graphs and Hypergraphs, Physica Verlag, 2000 .
[10] A. Nagoorgani and P. Vijayalaakshmi, Insentive arc in domination of fuzzy graph, Int. J. Contemp. Math. Sciences, 6(26) , 1303 1309,2011. [11] A. Nagoorgani and P. Vadivel, Relations between the parameters of independent domination and irredundance in fuzzy graph, International Journal of Algorithms, Computing and Mathematics, 2(1) , 15 18 , 2009 . [12] A. Nagoorgani and J. Malarvizhi, Isomorphism properties of strong fuzzy graphs, International Journal of Algorithms, Computing and Mathematics, 2(1) , 39 47 , 2009 . [13] A. Nagoorgani and K. Radha, On regular fuzzy graphs, Journal of Physical Sciences, 12 , 33 40 , 2008 . [14] P. S. Nair and S. C. Cheng, Cliques and fuzzy cliques in fuzzy graphs, IFSA World Congress and 20th NAFIPS International Conference, 4 , 2277 2280 , 2001. [15] C. Natarajan and S. K. Ayyasawamy, On strong (weak) domination in fuzzy graphs, World Academy of Science, Engineering and Technology, 67,526 528 , 2010 . [16] A. Rosenfeld, Fuzzy graphs, in; L.A. Zadeh, K.S. Fu, M.Shimura (Eds.), Fuzzy Sets and their Applications to Cognitive and Decision Processes, Academic Press, New York, 77 95 , 1975 . Madhumangal Pal is a Professor of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, India. He received University Silver Medal for rank second in B.Sc. (Honours) in the year 1988 from Vidyasagar University, India. He received University Gold Medal for rank first in M.Sc. in the year 1990 from the same university. He received Computer Division medal from Institution of Engineers (India) in the year 1996 for the best research work published in the Institution journal jointly with Prof. G.P.Bhattacharjee . He is also Editor-in-Chief of Journal of Physical Sciences. He is member of Editorial Board of International Journal of Computer Sciences, Systems Engineering and Information Technology, International Journal of Fuzzy Systems & Rough Systems, Advanced Modeling and Optimization, Romania International Journal of Logic and Computation, Malaysia. He is a reviewer of several international journals. He has written several books on Mathematics and Computer Science. His research interest includes computational graph theory, fuzzy matrices, game theory and regression analysis, parallel and genetic algorithms, etc.
Sovan Samanta received his B.Sc. degree in 2007 and M.Sc. degree in 2009 in Applied Mathematics from Vidyasagar University, India. He is now currently a research scholar in the Department of Applied Mathematics, Vidyasagar University since 2010 . His research interest includes fuzzy graph theory .
