Fuzzy Graph Theory by Sunil Mathew John N. Mordeson & Davender S. Malik
Author:Sunil Mathew, John N. Mordeson & Davender S. Malik
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham
Suppose that (iv) is true and (v) is not true.
Fig. 4.23Theorem 4.3.1 ()
In this case, there exists three distinct vertices , and x such that x lies on every strongest strong path. Let P be a strongest strong path and u and y be the two neighbors of x in P. Let C be the cycle mentioned in (iv), containing ux and xy and Q be the path in it, which is different from . Let a be the last vertex at which Q intersects subpath of P and b, the first vertex at which Q intersects subpath of P moving from y to u, along Q. Consider the path S given by the union of subpath of P, subpath of Q and subpath of P. The strengths of subpath and subpath of P are clearly greater than or equal to because P is a strongest strong path. The strengths of the two strongest strong paths constituting C is . Because the membership values of both the edges are greater than or equal to , and so is the strength of the subpath of Q. Hence, . So, S is a strongest strong path, a contradiction as it does not contain x. It follows that our assumption is wrong. Therefore, for any three distinct vertices , and x there is a strongest strong path which does not contain x (Fig. 4.23).
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