We propose a conjecture regarding the lower bound for the number of edges in locally k-connected graphs and we prove it for k = 2. In particular, we show that every connected locally 2-connected graph is M3-rigid. For the special case of surface triangulations, this fact was known before using topological methods. We generalize this result to all locally 2-connected graphs and give a purely com...

A locally connected quadratic Siegel Julia set has a simple explicit topological model. Such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution. We constructively produce parameter values for Siegel quadratics for which the Julia sets are non-computable, yet locally connected.

Given a compact and connected metric space (continuum) X, we study topological and dynamical properties of the hyperspace of normal fuzzy sets F1(X) equipped with the Hausdorff, endograph or sendograph metric. Among the many results we show that it is contractible, path connected, locally contractible, locally path connected, locally simply connected and locally connected. For the endograph met...

This work was supported by TUBITAK 1001 program (no:118E722), Isik University BAP (No: 16A202), and NVIDIA hardware donation of a Tesla K40 GPU unit.