2. Incidence
We introduce the notion of the degree of incidence of a vertex and an edge in a fuzzy graph in fuzzy graph theory. We concentrate on incidence, where the edge is adjacent to the vertex. We determine results concerning bridges, cutvertices, cutpairs, fuzzy incidence paths, and a fuzzy incidence tree for fuzzy incidence graphs. In [
5,
6], Dinesh introduced the notion of the degree of incidence of a vertex and an edge in fuzzy graph theory. Basic results concerning fuzzy graphs can be found in [
2,
11].
Let V be a finite set and let E denote a subset of the power set of such that every set in E contains exactly two elements. For in we write Then The elements of V are called vertices, and the elements of E are edges. The pair is called a graph.
Definition 1. Let be a graph. Then is called an incidence graph, where .
We note that, if and then is an incidence graph even though
Definition 2. Let be an incidence graph. If then is called an incidence pair or simply a pair. If then and are called adjacent edges.
Definition 3. An incidence subgraph of an incidence graph is an incidence graph having its vertices, edges, and pairs in
It is important to note that, if is an incidence graph, then is a bipartite graph since , where we interpret as an undirected edge between u and and so between and u also. This leads us to the following definition.
Definition 4. is a fuzzy incidence subgraph of if is a fuzzy subgraph of where for all for all and for all
Note that, in the previous definition, we have not required to be a fuzzy subgraph of If we do require it, then we may, at times, want to consider near fuzzy incidence subgraphs of where and and is allowed. See the following definition.
Definition 5. Let be a fuzzy incidence graph. Then is a near fuzzy incidence subgraph of G if and for all
Note that, in the previous definition, for is not required.
Definition 6. Let be an incidence graph. The sequencesare called incidence walks. Then and are called closed if and respectively. If the vertices and edges are distinct, then they are called incidence paths. If and are closed incidence paths, then they are called incidence cycles. The shortest incidence cycles have three vertices, three edges, and six pairs. By the definition of a cycle, all pairs of vertices and edges are distinct. Thus, from the definition of an incidence path, if is on the path, so are but not an incidence pair of the form with
Let be a graph and an incidence graph. Then We will assume in the following that Let (Note that, since Incidence pairs of the form , where , are not allowed here.
Let be a graph and a fuzzy subgraph of Let be an incidence graph. Let be a fuzzy incidence graph. Define as follows: if and if Since we can consider as a fuzzy subgraph of That is, the elements of are the vertices and the elements of are the edges. This interpretation will aid in the understanding of the proofs to follow.
Definition 7. An incidence graph in which all pairs of vertices and all pairs of edges are joined by an incidence path is said to be connected.
Definition 8. An incidence graph having no cycles is called an incidence forest. If it is connected, then it is called an incidence tree.
Since a tree is connected, all pairs of vertices are connected by an incidence path. By the definition of an incidence path, if is on the path so are but no incidence pair of the form with is on the path.
A component in an incidence graph is a maximally connected incidence subgraph. Recall that the definition of connectedness uses a path which, for incidence graphs, involves and for every in the path. Thus the removal of a pair can increase the number of components in an incidence graph. For example, consider the incidence graph is connected, but is not. H has two components, namely and
Definition 9. If the removal of an edge in an incidence graph increases the number of connected components, then the edge is called an incidence bridge.
Definition 10. If the removal of a vertex in an incidence graph increases the number of connected components, then the vertex is called an incidence cutvertex.
Definition 11. If the removal of an incidence pair in an incidence graph increases the number of connected components, then the incidence pair is called an incidence cutpair.
Consider the incidence graph is not a cutpair since remains connected since there is a path from u to v going through
3. Fuzzy Incidence
Definition 12. Let ⊗ be a function of the closed interval into itself. If ⊗ satisfies the following properties, it is called a t-norm. For all,
- (1)
(boundary condition);
- (2)
implies (monotonicity);
- (3)
(commutativity);
- (4)
(associativity).
If ⊗ is a t-norm, we write for If for all then ⊗ is called subidempotent.
In the following ⊗ denotes a t-norm.
Definition 13. Let be a graph and σ be a fuzzy subset of V and μ a fuzzy subset of Let Ψ be a fuzzy subset of If for all and then Ψ called a fuzzy incidence of with respect to
Definition 14. Let be a graph and be a fuzzy subgraph of with respect to If Ψ is a fuzzy incidence of then is called a fuzzy incidence subgraph graph of G with respect to
Definition 15. Two vertices and joined by an incidence path in a fuzzy incidence graph are said to be connected.
Definition 16. The incidence strength of a fuzzy incidence graph is defined to be Supp
Example 1. Let be an incidence graph and be a fuzzy incidence graph associated with where and Thenis an incidence walk, but not an incidence path since is repeated. The sequenceis an incidence path. The vertices and are connected. The incidence strength of this sequence is when the t-norm minimum is used. Definition 17. The fuzzy incidence graph is an incidence cycle if (Supp Supp Supp is a cycle.
Definition 18. The fuzzy incidence graph is a fuzzy incidence cycle if it is an incidence cycle and no unique Supp exists such that Supp
Example 2. Let be the fuzzy incidence cycle with and defined as follows:Consider the incidence walk Then it is a fuzzy incidence cycle since no unique exists such that the incidence strength of Note also that there is no unique such that There is no unique v such that instead of
Definition 19. The fuzzy incidence graph is an incidence tree if (Supp Supp Supp is an incidence tree and is an incidence forest if (Supp Supp Supp is an incidence forest.
Definition 20. Let be a fuzzy incidence graph. Then is a fuzzy incidence subgraph of G if and A fuzzy incidence subgraph H of G is a fuzzy incidence spanning subgraph of G if and
Definition 21. Let be a fuzzy incidence graph. Define to be the incidence strength of the incidence path from u to of greatest incidence strength.
Let
and
Then a path from
to
would be
Since the strength of the path is The strength of the strongest such path is what is meant in the previous definition.
Let and By we mean the strength of the strongest path from u to not including Let with and let By we mean the strength of the strongest path from u to (Note that cannot be included by the definition of incidence path, where and w are distinct.)
Definition 22. Let be a fuzzy incidence graph with respect to Then where
is an incidence cutpair;
is not a weakest pair of any incidence cycle.
Proof. Suppose is not a cutpair. Then
Suppose Then there is an incidence path from x to not involving that has incidence strength This incidence path together forms an incidence cycle of which is the weakest pair.
Suppose that is a cutpair with respect to Then there is such that If Q is the strongest incidence path in G from x to then Q must contain Hence Let P be the strongest incidence path from x to Then , and the incidence strength of is thus ≥ the incidence strength of Hence, the incidence path must contain and thus P must contain Hence, Thus, is the strongest incidence path from x to Hence, where strict inequality holds since is the strength of a strongest incidence path in ☐
Example 3. Let Let for all and Let and Then where and, elsewhere, . Thus, holds, but doesn’t.
Proposition 1. Let be an incidence graph and let ⊗ and be t-norms. Suppose that, for all If is a fuzzy incidence with respect to then is a fuzzy incidence graph with respect to
Proposition 2. Let be a fuzzy incidence graph with respect to t-norms ⊗ and , where . Let If is an incidence cutpair with respect to , then is an incidence cutpair with respect to
Proposition 3. Let be a fuzzy incidence graph with respect to Then a pair is said to be effective if
Proposition 4. Let be a fuzzy incidence graph with respect to If the pair is effective, then
Proposition 5. Let be a fuzzy incidence graph with respect to Then is called incidence complete with respect to ⊗ if for all
A fuzzy incidence graph is called a partial incidence fuzzy subgraph of if and The fuzzy incidence subgraph is called a fuzzy incidence subgraph of if for all for all and for all
We say that the partial fuzzy incidence subgraph spans if and In this case, we call H a spanning incidence fuzzy subgraph of For any fuzzy subset τ of V and ν of E such that and the partial fuzzy incidence subgraph induced by τ and ν is the maximal partial fuzzy incidence subgraph that has vertex and edge set This is the partial fuzzy incidence graph where for all
A (crisp) incidence graph that has no incidence cycles is called acyclic or an incidence forest . A connected forest is called an incidence tree . A fuzzy incidence graph is called an incidence forest if the graph consisting of its nonzero pairs is a forest and an incidence tree if this graph is connected. We call the fuzzy incidence graph a fuzzy incidence forest if it has a partial fuzzy spanning incidence subgraph that is an incidence forest, where for all pairs not in we have In other words, if is in but not in there is an incidence path in F between x and whose incidence strength is greater than It is clear that an incidence forest is a fuzzy incidence forest.
Definition 23. Let ⊗ be a t-norm. A fuzzy incidence graph is a fuzzy incidence tree with respect to ⊗ if has a partial fuzzy incidence subgraph that is an incidence tree and not in
Example 4. Consider the fuzzy incidence graph in Example 3. We note that is an incidence cycle, but not a fuzzy incidence cycle since there is a unique such that It is also the case that is not a fuzzy incidence tree since if it had a fuzzy incidence spanning tree then but this is not the case since
Theorem 1. A fuzzy incidence graph with respect to ⊗ is a fuzzy incidence tree if and only if it has a unique maximum fuzzy incidence spanning tree.
Proof. Suppose is a fuzzy incidence fuzzy tree with respect to Then, G is a fuzzy incidence tree with respect to Hence, G has a unique maximum fuzzy spanning subgraph with respect to Let be a fuzzy incidence spanning subgraph of G with respect to Then and and not in That is, F is a partial fuzzy incidence spanning subgraph of G with respect We see that if F is a maximum with respect to i.e., then F is a maximum with respect to ∧ and so is unique for ⊗ since it is unique for ☐
The following known theorem motivates our next definition.
Theorem 2. [
2]
Let be a cycle. is a fuzzy cycle with respect to ∧ if and only if is not a fuzzy tree with respect to Definition 24. Let ⊗ be a t-norm. A fuzzy incidence is a fuzzy incidence cycle with respect to ⊗ if is an incidence cycle and there is no unique such that and is not a fuzzy incidence tree with respect to
We have that of Example 3 is not a fuzzy incidence cycle with respect to
We see that a fuzzy incidence cycle with respect to ⊗ is a fuzzy incidence cycle with respect to
Example 5. Let for all and Let is a fuzzy incidence cycle with respect to ⊗ since is not a fuzzy incidence tree with respect to
Example 6. Let for all and Let Then is a fuzzy incidence cycle with respect to ∧ and with respect to multiplication, but is not a fuzzy incidence tree with respect to ⊗ since
Theorem 3. Suppose is a fuzzy incidence graph with respect to Suppose that ⊗ is subidempotent. Suppose is a fuzzy incidence cycle with respect to If is not a fuzzy incidence cycle with respect to then there are two pairs such that
Proof. Suppose there are three or more pairs such that say. Then Thus, is not a fuzzy incidence tree with respect to ⊗ and hence is a fuzzy incidence cycle with respect to a contradiction. Since is a fuzzy incidence cycle with respect to there is no unique pair such that ☐
Theorem 4. Suppose that is a fuzzy incidence cycle with respect to Then is a fuzzy incidence cycle with respect to ⊗ if and only if is not a fuzzy incidence tree with respect to
Proof. By Definition 24, is a fuzzy incidence cycle with respect to ⊗ if and only if there is no unique such that and is not a unique fuzzy incidence tree with respect to Since is a fuzzy incidence cycle with respect to there is no unique such that Hence the desired result holds. ☐
Let be a fuzzy incidence graph and let w be a vertex. Otherwise, let be the fuzzy incidence subgraph defined by . If not, then all and . If not this, then for all and .
Theorem 5. Suppose that is a fuzzy incidence graph with respect to Suppose that is an incidence cycle. If a vertex is an incidence cutvertex, then it is a common vertex to incidence cutpairs.
Proof. Let w be an incidence cutvertex of such that such that w is on every strongest u–v incidence path since Since G is an incidence cycle, there exists only one strongest incidence path u–v incidence path containing w and all its pairs are incidence cutpairs. Thus, w is a common vertex of two incidence cutpairs. ☐
Example 7. The converse of the previous result is not true. Let G be as defined in Example 3. Then z is a common vertex of the incidence bridges and but z is not an incidence cutvertex since its deletion does not reduce the strength of connectedness between any pair of vertices and edges.
Theorem 6. If is an incidence cutpair, then
Proof. Suppose that is an incidence cutpair and that . Then there is a strongest incidence path P with incidence strength greater than and all pairs of this strongest incidence path have incidence strength greater than such that since is an incidence cutpair. Let Q be a strongest incidence path from x to including Then together with P is an incidence path from x to that is stronger than a contradiction. ☐
Definition 25. Suppose that is a fuzzy incidence graph with respect to G is called incidence complete if in
Proposition 6. Suppose that is a complete fuzzy incidence graph with respect to Then
- (1)
for all in
- (2)
G has no incidence cutvertices and no incidence bridges.
Proof. Let Since G is incidence complete, If P: is an incidence path from u to then the incidence strength of P is of the form, where
Suppose w is an incidence cutvertex (with respect to Then for some such that where for all z and otherwise. However, by , which is impossible.
Suppose is an incidence bridge. Then for some such that and elsewhere. However, by , which is impossible. ☐
Proposition 7. Let be a fuzzy incidence tree with respect to Then G is not incidence complete.
Proof. Suppose G is complete. Since G is a fuzzy incidence tree with respect to has a partial fuzzy incidence spanning subgraph ,, which is an incidence tree and not in but and so a contradiction of of the previous proposition. ☐
Example 8. Consider the fuzzy incidence graph of Example 3. Then all pairs are incidence cutpairs, and z is a common vertex of two incidence cutpairs but is not an incidence cutvertex since, if z is deleted, both and are deleted and so there are no two elements from whose incidence connectivity has been reduced by the deletion of
Theorem 7. Let be a fuzzy incidence tree with respect Then the internal vertices of the partial fuzzy incidence spanning subgraph are the cut vertices of
Proof. Let w be a vertex in G, which is not an end vertex of Then there are vertices such that and are in Since F is an incidence tree, there is a unique incidence path in F from to This incidence path must be If there is no incidence path from to then the removal of w disconnects G, and w is thus an incidence cutvertex in Suppose there is an incidence path P in connecting and , since is the only incidence in F connecting , and must contain a pair not in Thus, Hence, w is an incidence cutvertex. ☐
Theorem 8. Let be a fuzzy incidence graph with respect to Then G is a fuzzy incidence forest with respect to ⊗ if and only if in any incidence cycle of there is a pair such that where is the partial fuzzy incidence subgraph obtained by deletion of the pair from
Proof. Suppose is a pair in an incidence cycle C that satisfies the condition of the theorem and which is such that is the smallest of all such pairs. If no such incidence cycle exists, then G is an incidence forest and so is a fuzzy incidence forest. The partial fuzzy incidence subgraph satisfies the condition of the theorem. If there are no incidence cycles in we repeat this process until we obtain a partial fuzzy incidence subgraph F of G without incidence cycles. Clearly, F is an incidence forest. Let be an incidence cutpair of G not in Then is a pair that was previously deleted and there is an incidence path from x to that is stronger than and does not involve or any pairs deleted before the deletion of If this incidence path involves pairs that were deleted after it can be diverted around them using a stronger incidence path and hence stronger pairs. If this incidence path uses a pair that was deleted after the weakest such pair can be diverted around using a stronger incidence path and hence one with stronger pairs. This process ends in a finite number of steps resulting in Since G is a fuzzy incidence forest with respect to ☐
Conversely, suppose G is a fuzzy incidence forest. Let C be any incidence cycle in Then there is a pair of C not in where F is a partial fuzzy incidence spanning subgraph of Thus, where and ; otherwise, . Thus, G is a fuzzy incidence forest.
Example 9. Proposition 2.6 in [11] does not hold. Let be a fuzzy incidence forest with respect to ⊗
of Example 3. Then there is at most one strongest incidence between any two vertices, namely the pair itself. However, G is not a fuzzy incidence forest since where and Theorem 9. Let be a fuzzy incidence forest with respect to Then the pairs of are just the incidence cutpairs of
Proof. A pair not in F cannot be an incidence cutpair in G since Suppose that a pair is a pair in Suppose is not an incidence cutpair of Then there is an incidence path P in G from x to not involving and of incidence strength greater than or equal to . Since is an incidence cycle, P must involve pairs not in F, since F is an incidence forest and has no incidence cycles. Let be a pair in P not in Now can be replaced by an incidence path in F from u to of incidence strength greater than Now cannot contain since all its pairs are strictly greater than Thus, by replacing each such pair by a we obtain an incidence cycle in contradicting the fact that F is an incidence forest. ☐
Theorem 10. Let be a fuzzy incidence graph. Then G is a fuzzy incidence forest if and only if in any fuzzy incidence cycle of G, there is such that where is the fuzzy incidence subgraph obtained by deletion of from G and restricted to (or
Proof. If there are no incidence cycles, the result is trivially true. Suppose is an incidence pair in G that belongs to a fuzzy incidence cycle such that Let it be the pair with the least value among all such pairs Supp Delete If there are other incidence cycles, remove incidence pairs in a similar way. At each step, the incidence pair deleted will not have lesser incidence strength than those deleted earlier. After deletion, the remaining fuzzy incidence subgraph is a fuzzy incidence forest Therefore, there is an incidence path P from x to with more incidence strength than and not containing If incidence pairs deleted earlier are in then we can bypass them using an incidence path with more incidence strength. ☐
Conversely, if G is a fuzzy incidence forest and C any incidence cycle, then by definition, there is an of C not in F such that where F is as in the definition of a fuzzy incidence forest.
Example 10. Example 3 shows that following statement is not true. If there is at most one path with the most incidence strength between any vertex and edge of the fuzzy incidence graph then G is a fuzzy incidence forest.
Theorem 11. Let be an incidence cycle. Then G is a fuzzy incidence cycle if and only if G is not a fuzzy incidence tree.
Proof. Suppose is a fuzzy incidence cycle. Then there is at least two incidence pairs with Supp Supp Let be a spanning fuzzy incidence tree in Then there is a such that SuppSupp Hence, there is no incidence path in between u and of greater incidence strength than Thus, G is not a fuzzy incidence tree. ☐
Conversely, suppose that G is not a fuzzy incidence tree. Since G is an incidence cycle, it follows that in Supp there is a fuzzy incidence spanning subgraph , which is an incidence tree, and Supp Therefore, Ψ does not uniquely attain Supp. Hence, G is a fuzzy incidence cycle.