Since decades transitive graphs are a topic of great interest. The study of s-edge transitive (undirected) graphs goes back to Tutte [13], who showed that finite cubic graphs cannot be s-edge transitive for s> 5. Weiss [14] proved several years later that the only finite connected s-edge transitive graphs with s = 8 are the cycles. Considering directed graphs the situation is much more involved...

The main subject of the thesis is connectivity augmentation: we would like to make a given graph k-connected by adding a minimum number of new edges. There are four basic problems in this field, since one might consider both edgeand node-connectivity augmentation in both graphs and digraphs. The thesis wishes to contribute to three out of these four problems: directedand undirected node-connect...

A properly connected coloring of a given graph G is one that ensures that every two vertices are the ends of a properly colored path. The proper connection number of G is the minimum number of colors in such a coloring. We study the proper connection number for edge and vertex colorings, in undirected and directed graphs, respectively. More precisely, we initiate the study of the complexity of ...

We classify the finite connected-homogeneous digraphs, as well as the infinite locally finite such digraphs with precisely one end. This completes the classification of all the locally finite connected-homogeneous digraphs.

In this paper we initiate a systematic study of the Reduced Degree Spanning Tree problem, where given a digraph D and a nonnegative integer k, the goal is to construct a spanning out-tree with at most k vertices of reduced out-degree. This problem is a directed analog of the wellstudied Minimum-Vertex Feedback Edge Set problem. We show that this problem is fixed-parameter tractable and admits a...

We show that planar embeddable 3-connected Laman graphs are generically non-soluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity. Formally, a Laman graph is a maximally independent graph, that is, one that satisfies the vertex-edge count 2v − 3 = e together with a corresponding inequality for ...

We show that planar embeddable 3-connected CAD graphs are generically non-soluble. A CAD graph represents a configuration of points on the Euclidean plane with just enough distance dimensions between them to ensure rigidity. Formally, a CAD graph is a maximally independent graph, that is, one that satisfies the vertex-edge count 2v − 3 = e together with a corresponding inequality for each subgr...

Day and Tripathi [K. Day, A. Tripathi, Unidirectional star graphs, Inform. Process. Lett. 45 (1993) 123–129] proposed an assignment of directions on the star graphs and derived attractive properties for the resulting directed graphs: an important one is that they are strongly connected. In [E. Cheng, M.J. Lipman, On the Day–Tripathi orientation of the star graphs: Connectivity, Inform. Process....