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Keywords:
fuzzy graph; G-V fuzzy matroid; perfect $[0,1]$-matroid; connectedness; connected degree; fuzzifying matroid
fuzzy graph; G-V fuzzy matroid; perfect $[0,1]$-matroid; connectedness; connected degree; fuzzifying matroid
Summary:
This paper studies the connected degree of G-V fuzzy matroids. The concept is generalized from the concept of matroid connectedness through plausible reasoning and has good expansibility and practicability. It is proved that a G-V fuzzy matroid $M$ is disconnected if and only if its connected degree is the minimum (that is, $Con(M)=0$), that is to say, it is connected if and only if $Con(M)\neq 0$. It is also proved that the connected degree $Con(M(\widetilde{G}))=1$ of the fuzzy cycle matroid $M(\widetilde{G})$ of a fuzzy graph $\widetilde{G}$ if and only if the fuzzy graph $\widetilde{G}$ is 2-connected. Finally, combined with the existing connected degree theory of fuzzifying matroids, and relying on the transformation relationship between closed G-V fuzzy matroids and fuzzifying matroids, the consistency of the definitions of the connected degrees of the two types of matroids is demonstrated, revealing the inherent unity of different fuzzy matroid models in connectedness analysis.
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