Arc disjoint fuzzy graphs

Abstract

Examines a special kind of fuzzy graph called an arc-disjoint graph. Let S be a finite set, /spl sigma/ a function from S into the closed interval [0,1], and /spl mu/ a function of S/spl times/S into [0,1] such that /spl mu/(x,y)=/spl mu/(y,x) for all x,y in S. Then the pair G=(/spl sigma/,/spl mu/) is called a fuzzy graph if /spl mu/(x,y)/spl les/min{/spl sigma/(x),/spl sigma/(y)}. Let G=(/spl sigma/,/spl mu/) be a fuzzy graph. Define the function /spl mu//sup /spl infin// of S/spl times/S into [0,1] by /spl forall/x,y/spl isin/S, /spl mu//sup /spl infin//(x,y)=sup{/spl mu//sup i/(x,y) | i=1, 2, ...}, where /spl mu//sup i/ is the max-min composition of /spl mu/ with itself i times (i=1, 2, ...). G is called arc-disjoint if no two cycles share a common arc. We show that if G is arc-disjoint, then G is a fuzzy forest iff in any cycle C of G, there is an arc (x,y) such that /spl mu/(x,y)</spl mu/(u,v) where (u,v) is an arc of C other than (x,y). We also show that if G is arc-disjoint, then G is a fuzzy forest iff there is at most one arc-disjoint strongest path between any two nodes of G. We then turn our attention to the cycle rank of a graph. We show that if G is arc-disjoint and G=C/sub 1//spl cup/.../spl cup/C/sub n/, where the C/sub i/ are cycles, then m(C/sub 1//spl cup/.../spl cup/C/sub n/)=m(C/sub 1/)...m(C/sub n/)=n, where m(/spl middot/) denotes the cycle rank of a graph. We also show that if G is a cycle vector, then G is arc-disjoint iff m(G)=the number of cycles of G.