A digraph is said to be super-connected if every minimum vertex cut is the out-neighbor set or in-neighbor set of a vertex. A digraph is said to be reducible, if there are two vertices with the same out-neighbor set or the same in-neighbor set. In this paper, we prove that a strongly connected arc-transitive oriented graph is either reducible or super-connected. Furthermore, if this digraph is ...

Entanglement is a complexity measure of digraphs that origins in fixedpoint logics. Its combinatorial purpose is to measure the nested depth of cycles in digraphs. We address the problem of characterizing the structure of graphs of entanglement at most k. Only partial results are known so far: digraphs for k = 1, and undirected graphs for k = 2. In this paper we investigate the structure of und...

The Road Coloring Conjecture is an old and classical conjecture posed in Adler and Weiss (1970); Adler et al. (1977). Let G be a strongly connected digraph with uniform out-degree 2. The Road Coloring Conjecture states that, under a natural (necessary) condition that G is “aperiodic”, the edges of G can be colored red and blue such that “universal driving directions” can be given for each verte...

For a given undirected graph G = (V;E; c G ) with edges weighted by nonnegative reals c G : E ! R + , let G (k) stand for the minimum amount of weights which needs to be added to make G k-edge-connected, and G (k) be the resulting graph obtained from G. This paper rst shows that function G over the entire range k 2 [0;+1] can be computed in O(nm+ n 2 logn) time, and then shows that all G (k) in...

One of Frank Boesch’s best known papers is ‘The strongest monotone degree conditions for n-connectedness of a graph’ [1]. In this paper, we give a simple sufficient degree condition for a graph to be k-edge-connected, and also give the strongest monotone condition for a graph to be 2-edge-connected.