Diestel and Müller showed that the connected tree-width of a graph G, i. e., the minimum width of any tree-decomposition with connected parts, can be bounded in terms of the tree-width of G and the largest length of a geodesic cycle in G. We improve their bound to one that is of correct order of magnitude. Finally, we construct a graph whose connected tree-width exceeds the connected order of a...

In [1] Johnson, Robertson, Seymour and Thomas define the notion of directed tree-width dtw(D) of a directed graph D. They ask whether dtw(D) ≥ k− 1 implies that D has a haven of order k. A negative answer is given. Furthermore they define a generalisation of the robber and cops game of [3] to digraphs. They ask whether it is true that if k cops can catch the robber on a digraph, then they can d...

In this paper, we consider the width invariant trapezoidal and triangularapproximations of fuzzy numbers. The presented methods avoid the effortful computation of Karush-Kuhn-Tucker Theorem. Some properties of the new approximation methods are presented and the applicability of the methods is illustrated by examples. In addition, we show that the proposed approximations of fuzzy numbers preserv...