Fuzzy Threshold Graphs Sovan Samanta and Madhumangal Pal

Abstract- In this paper, fuzzy threshold graphs, fuzzy alternating

-cycles, threshold dimension of fuzzy graphs and fuzzy Ferrers digraphs are defined. We show that fuzzy threshold graphs are fuzzy split graphs. Forbidden configurations of fuzzy Ferrers digraph are described. Also some basic theorems related to the stated graphs have been presented.

Keywords- Fuzzy Ferrers digraph, fuzzy graph, fuzzy threshold dimension, fuzzy threshold graph.

1. INTRODUCTION

Threshold graphs play an important role in graph theory as well as in several applied areas such as psychology, computer science, scheduling theory, etc. These graphs can be used to control the flow of information between processors, much like the traffic lights used in controlling the flow of the traffic.

Chvatal and Hammer [3] coined the name “threshold graphs" and studied the graphs for their application inset packing problems. Ordman [10] found the use of graphs in resource allocation problems. Chvatal and Hammer defined threshold graph as follows. A graph is a threshold graph when there exists non-negative reals

and such that if and only if

is stable set where . So is a threshold graph whenever one can assign vertex weights such that a set of vertices is stable if and only if its total weight does not exceed a certain threshold. The threshold dimension, of a

graph is the minimum number of threshold subgraphs

of that cover the edge set of . Threshold

partition number, denoted by , is the minimum number

of edge disjoint threshold subgraphs needed to cover .

Formally, an edge cover of a graph is a set of edges such that each vertex is incident with at least one edge in . The set is said to cover the vertices of .

Ferrers digraph [11] is a digraph related to threshold graph. A digraph is said to be a Ferrers digraph

if it does not contain vertices , not necessarily

distinct, satisfying and

. For a digraph , the

underlying loop less graph , where

.

A split graph is a graph in which the vertices can be partitioned into a clique and an independent set.

Alternating -cycle of a graph is a

configuration consisting of distinct vertices such

that and . By

considering the presence or absence of edges ,

we see that the vertices of alternating -cycle induce a path

, a square , or a matching .

For the graph with vertices and

distinct positive vertex degrees ,

(even no vertex of degree exists) and ,

, degree partition is the sequence

for .

1.1 Fuzzy graphs

The fuzzy systems have been used with success in last few 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. A fuzzy set on a set is characterized by a mapping

, called the membership function. We shall

denote a fuzzy set as . Fuzzy graph theory was

introduced by Azriel Rosenfeld [12] in . Though it is very young, it has been growing fast and has numerous applications in various fields. A fuzzy graph is

a non empty crisp set together with a pair of functions and such that for all ;

. So is a fuzzy relation on . A

fuzzy digraph is similarly defined except

need not be symmetric. We call the pairs

fuzzy arcs to emphasize that the

symmetry is not required. The fuzzy graph is

called a partial fuzzy subgraph of if and .

Similarly, the fuzzy graph is called fuzzy

subgraph of induced by if , for all

and for all . The degree

of a vertex is . The minimum degree of is

. The maximum degree of is

. The strength of connectedness

between two vertices and is

where

. An edge is a fuzzy bridge of if deletion of reduces the strength of connectedness between the pair of vertices. A vertex is a fuzzy cut vertex of if deletion of reduces the strength of connectedness between other pair of vertices. The order of fuzzy graph is . We use the

notation of order of as . The size of fuzzy graph is

. In a fuzzy graph an arc is said to be

strong arc [5] or strong edge, if and

the node is said to be strong neighbour of . If is

not strong arc then is called isolated node or isolated vertex. is said to fuzzy dominating set of if for every , there exists in such that

. The minimum scalar cardinality of

is called fuzzy domination number and it is denoted by

. Scalar cardinality of a fuzzy set is . Two

nodes in a fuzzy graph are said to be fuzzy independent if there is no strong arc between them. Let be a node in fuzzy

graph then N is a strong arc} is called open

neighborhood of and N [u] = N(u) {u} is called closed neighborhood of .

A path in fuzzy graph is a sequence of distinct

nodes such that .

Here is called the length of the path. The strength of

is defined as . In other words, strength of a

path is the weight of the weakest arc of the path. If the path

has length , then its strength is . We call a cycle if

and . We recall that a graph without cycle is called acyclic or forest, a connected forest is tree. We call a fuzzy graph a forest if the graph consisting of its nonzero arcs is a forest. We call fuzzy graph a fuzzy forest if it has a fuzzy spanning subgraph which is a forest, where for all

arcs not in the subgraph , we have

. Thus if but ,

there is a path in between and whose strength is

greater than . is a cycle if and only if

is a cycle. is a fuzzy cycle if

and only if is a cycle and there does

not exist unique , such that

. is a

clique if is a clique. is fuzzy clique if it is a clique and every cycle in it is a fuzzy cycle.

1.2 Review of previous works

Andelic and Simic [1] presented some results on the threshold graphs. Bhutani, Moderson and Rosenfeld [2] discussed on degrees of end nodes and cut nodes in fuzzy graphs. Mathew and Sunitha [4] defined types of arcs in a fuzzy graph. Mordeson and Nair [5] have given the details of fuzzy graphs and hypergraphs. Nagoorgani and Malarvizhi [6] decribed the isomorphism properties of strong fuzzy graphs. Nagoorgani and Radha [7] proved some results on regular fuzzy graphs. Nair and Cheng [8] discussed cliques and fuzzy cliques in fuzzy graphs. Natarajan and Ayyasawamy [9] described on strong (weak) domination in fuzzy graphs.

1.3 Our works

In this paper, we define fuzzy threshold graph and investigated some of its important properties. Here we use the notation Cn,

Pn, Kn , K for cycle, path and complete graph of length in fuzzy graph and fuzzy clique respectively.

2. FUZZY THRESHOLD GRAPHS

Definition 1 A fuzzy graph is called a fuzzy

threshold graph if there exists non negative real number

such that if and only if is stable set in .

Example 1 Let be a fuzzy graph with vertex

set such that

. Also let and all cycles of the graph are fuzzy cycles. This graph is an example of fuzzy threshold graph (see Fig. 1).

Fig. 1: Example of fuzzy threshold graph.

Now, we define fuzzy alternating cycle.

Definition 2 Let be a fuzzy graph and

. Also let ,

be positive and . This configuration of four vertices is called fuzzy alternating

-cycle. This fuzzy alternating -cycle induces a path P

(when one of is zero and other is non-

zero), a square C (when both are non-

zero) or a matching 2K2 (when both are zero).

Fuzzy alternating -cycle may also induce a fuzzy C if the arcs membership values don't have exactly one

minimum value. The fuzzy alternating -cycle and its induced fuzzy graphs are shown in Fig. 2.

Definition 3 A strong alternating -cycle is an alternating -

cycle if fuzzy C4 be induced from it.

P4

C4 2K2

Fig. 2: Induced subgraphs of fuzzy alternating -cycle.

Degree partition is very important term in graph theory. Here we define degree partition in a fuzzy graph.

Definition 4 Let be a fuzzy graph whose

distinct positive vertex degrees are , and let

(even if no isolated vertex exists), .

Let D } for non negative integer . The sequence

D0 , D1,…, Dp is called degree partition of the fuzzy graph .

Theorem 1 A fuzzy threshold graph does not have a strong fuzzy alternating -cycle.

Proof. Let be a fuzzy threshold graph. Let, if

possible, has a strong fuzzy alternating -cycle. So there

exists vertices with , and

. As the graph is fuzzy threshold graph

with threshold , then

, [as the

construct strong fuzzy alternating -cycle]

, [as

].

These inequalities are inconsistent. Hence does not construct a strong fuzzy alternating -cycle. So a fuzzy threshold graph does not have a strong fuzzy alternating

-cycle. □

Theorem 2 A fuzzy threshold graph is a fuzzy split graph. Proof. Let be a fuzzy threshold graph and K =(K, , ) , K V, (x)= (x) for all x K, (x,y)= (x,y)

for all x,y , be the largest clique in . If be a

strong arc in , then by maximality of , there exists

distinct vertices , in such that .

These vertices create a strong fuzzy alternating -cycle. So contradiction arises. To avoid strong fuzzy alternating -cycle, must be stable set. Hence is a fuzzy split graph. □

Theorem 3 If is a fuzzy threshold graph, then can be constructed from the one vertex graph by repeatedly adding a fuzzy isolated vertex or a fuzzy dominating vertex.

Proof. In Theorem 2, we proved that a fuzzy threshold graph is a fuzzy split graph. It is enough to show that has a fuzzy isolated vertex or a fuzzy dominating vertex, if for such a vertex is removed, then the graph will be still fuzzy split graph. Let S be a stable set of the fuzzy threshold graph =

( K , S). Let S be nonempty. If S contains fuzzy isolated

vertices only, then the result holds. If S has no fuzzy isolated

vertices, then vertex u S with smallest neighborhood has

some neighbor v K. As K is a fuzzy clique, so the vertex is dominating vertex of . □

Comparability in fuzzy graph is important topic. We now define strong comparability of vertices in fuzzy graph.

Definition 5 Let be a fuzzy graph. Two

vertices and are said to be strong comparable if there exists a path from to or to whose every arc is strong arc. We now give the definition of threshold dimension in fuzzy graph.

Definition 6 The threshold dimension of a fuzzy

graph is the minimum number of fuzzy

threshold subgraphs T , T , , T of that cover the edge

set of that is if T1 T 2, … Tk then

.

Fig. 3: A fuzzy graph with and .

Since every fuzzy edge along with fuzzy isolated

vertices is a fuzzy threshold subgraph, the threshold dimension of fuzzy graph is well defined and is bounded by the number of edges of the graph. We denote as stability number

of a fuzzy graph , i.e., the order of the largest stable set of .

Theorem 4 For every fuzzy graph on

vertices we have . Furthermore, if is

triangle-free, then S where S is the stable set with largest number of vertices.

Proof. Let S = (S, ) be a stable set with largest number of vertices of the fuzzy graph with vertices. For each vertex

, we consider the star centered at . Each such star is a fuzzy threshold graph. If we add one or more weak fuzzy arc of stable set to the stars then they satisfy the condition of fuzzy threshold graph. So all such stars together with weak arcs of stable sets covers the edge set of . Thus

. Again we know that and

, , being the crisp sets. Thus

.

We know |S|=|supp(S)|. So S. If in addition is fuzzy triangle free, then every fuzzy

threshold graph is a star or star together with weak edges. So S . Hence S

. □

Fig. 4: Forbidden configurations in fuzzy Ferrers digraph.

Fuzzy threshold partition is as important as threshold dimension. Definition 7 The fuzzy threshold partition number

of the fuzzy graph is the minimum number of fuzzy threshold subgraphs, not containing common strong arcs, cover edge set of .

Example 2 Here we give an example of fuzzy graph whose fuzzy threshold dimension number is and fuzzy partition number is . The graph has vertices with membership values

d(0.2), d1(0.2) and strong arcs shown in Fig. 3.

Theorem 5 If is a fuzzy triangle free graph, then

.

Proof. Let be a fuzzy graph. We know that the

edge set can be covered by number of stars. If a strong arc belongs to more than one stars then we delete it from all but remain one of the stars. This gives a fuzzy threshold partition of size . □

Fuzzy Ferrers digraph is an important graph related to fuzzy threshold graph. Definition 8 A fuzzy digraph is said to be a

fuzzy Ferrers digraph if it does not contain vertices , not necessarily distinct, satisfying ,

are non zero and , are zero. Forbidden configurations are shown in Fig. 4. Here solid arrow represents that edge membership values are non zero and dotted arrow represents that edge membership values are zero.

Definition 9 Let be a fuzzy digraph. The

underlying fuzzy graph of is denoted by U and is

defined as U where min

for all .

Theorem 6 If is a symmetric fuzzy Ferrers digraph, then

its underlying fuzzy undirected loop less fuzzy graph U may or may not be fuzzy threshold graph. Proof. If is a symmetric fuzzy Ferrers digraph, then its

underlying fuzzy undirected loop less fuzzy graph U has

no fuzzy alternating cycle. So it may contain fuzzy cycle.

Hence U may be fuzzy threshold graph. If U does

not have strong fuzzy alternating cycle then U must

be fuzzy threshold graph using the result of theorem . □

3. CONCLUSION In this paper we have presented the relations between fuzzy threshold graphs and fuzzy split graphs, fuzzy threshold dimension and fuzzy partition number. Also we discussed some properties of fuzzy Ferrers digraphs. These graphs will help to solve resourse allocation problems in fuzzy approach. Also these graph are helpful to control the flow of information using fuzzy properties. We hope our study will enable us to extend the fuzzy graph classes like fuzzy difference graphs, fuzzy matroidal graphs, fuzzy matrogenic graphs.

REFERENCES

[1] M. Andelic and S.K. Simic, Some notes on the threshold graphs, Discrete Mathematics, ,

, . [2] K. R. Bhutani, J. Moderson and A. Rosenfeld, On

degrees of end nodes and cut nodes in fuzzy graphs, Iranian Journal of Fuzzy Systems, , ,

. [3] V. Chvatal and P. L. Hammer, Set-packing problems and

threshold graphs, CORR , University of Waterloo, Canada, .

[4] S. Mathew and M. S. Sunitha, Types of arcs in a fuzzy graph, Information Sciences, , , .

[5] J. N. Mordeson and P. S. Nair, Fuzzy graphs and hypergraphs, Physica Verlag, .

[6] A. Nagoorgani and J. Malarvizhi, Isomorphism properties of strong fuzzy graphs, International Journal of Algorithms, Computing and Mathematics, ,

, . [7] A. Nagoorgani and K. Radha, On regular fuzzy graphs,

Journal of Physical Sciences, , , . [8] P. S. Nair and S. C. Cheng, Cliques and fuzzy cliques in

fuzzy graphs, IFSA World Congress and 20th NAFIPS International Conference, , , .

[9] C. Natarajan and S. K. Ayyasawamy, On strong (weak) domination in fuzzy graphs, World Academy of Science, Engineering and Technology, , , .

[10] E. T. Ordman, Threshold coverings and resource

allocation, In th Southeastern Conference on Combinatorics, Graph Theory and Computing, ,

[11] U. N. Peled and N. V. Mahadev, Threshold graphs and

related topics, North Holland, . [12] A. Rosenfeld, Fuzzy graphs, in: L.A. Zadeh, K.S. Fu, M.

Shimura (Eds.), Fuzzy Sets and Their Applications, Academic Press, New York, , .

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 .