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Fuzzy graph-based structures such as fuzzy automata, fuzzy labeled transitionsystems, fuzzy Kripke models, fuzzy social networks and fuzzy interpretationsin fuzzy description logics are useful in various applications. Given twostates, two actors or two individuals $x$ and $x'$ in such structures $G$ and$G'$, respectively, the similarity degree between them can be defined to be$Z(x,x')$, where $Z$ is the greatest fuzzy bisimulation between $G$ and $G'$w.r.t. some t-norm-based fuzzy logic. Such a similarity measure has theHennessy-Milner property of fuzzy bisimulations as a strong logical foundation.A fuzzy bisimulation between a fuzzy structure $G$ and itself is called a fuzzyauto-bisimulation of $G$. The greatest fuzzy auto-bisimulation of animage-finite fuzzy graph-based structure is a fuzzy equivalence relation. It isuseful for classification and clustering.  In this paper, we design an efficient algorithm with the complexity$O((m\log{l} + n)\log{n})$ for computing the fuzzy partition corresponding tothe greatest fuzzy auto-bisimulation of a finite fuzzy labeled graph $G$ underthe G\"odel semantics, where $n$, $m$ and $l$ are the number of vertices, thenumber of non-zero edges and the number of different fuzzy degrees of edges of$G$, respectively. Our notion of fuzzy partition is novel, defined only forfinite sets with respect to the G\"odel t-norm, with the aim to facilitate thecomputation of the greatest fuzzy auto-bisimulation. By using that algorithm,we also provide an algorithm with the complexity $O(m\cdot\log{l}\cdot\log{n} +n^2)$ for computing the greatest fuzzy bisimulation between two finite fuzzylabeled graphs under the G\"odel semantics. This latter algorithm is better(has a lower complexity order) than the previously known algorithms for theconsidered problem. Our algorithms can be restated for the other mentionedfuzzy graph-based structures.

Computing the Fuzzy Partition Corresponding to the Greatest Fuzzy Auto-Bisimulation of a Fuzzy Graph-Based Structure