Let G = (V, A) be a digraph with diameter D 6= 1. For a given integer 2 ≤ t ≤ D, the t-distance connectivity κ(t) of G is the minimum cardinality of an x → y separating set over all the pairs of vertices x, y which are at distance d(x, y) ≥ t. The t-distance edge connectivity λ(t) of G is defined similarly. The t-degree of G, δ(t), is the minimum among the out-degrees and in-degrees of all vert...

This paper studies the NP-hard problem of /nding a minimum size 2-edge connected spanning subgraph (2-ECSS). An algorithm is given that on an r-edge connected input graph G=(V; E) /nds a 2-ECSS of size at most |V |+(|E|−|V |)=(r−1). For r-regular, r-edge connected input graphs for r = 3, 4, 5 and 6, this gives approximation guarantees of 4 ; 4 3 ; 11 8 and 7 5 , respectively. c © 2003 Elsevier ...

We study the following problem: Given a set V of n vertices and a set E of m edge pairs, we define a graph family G(V, E) as the set of graphs that have vertex set V and contain exactly one edge from every pair in E . We want to find a graph in G(V, E) that has the minimal number of connected components. We show that, if the edge pairs in E are non-disjoint, the problem is NP-hard. This is true...

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...

We consider the classical minimum Travelling Salesman Problem on the class of 3-edge-connected cubic graphs. More specifically we consider their (shortest path) metric completions. The well-known conjecture states that the subtour elimination LP relaxation on the min TSP yields a 4/3 approximation factor, yet the best known approximation factor is 3/2. The 3-edge-connected cubic graphs are inte...

We provide a complete structural characterization of K2,4-minor-free graphs. The 3-connected K2,4minor-free graphs consist of nine small graphs on at most eight vertices, together with a family of planar graphs that contains K4 and, for each n ≥ 5, 2n − 8 nonisomorphic graphs of order n. To describe the 2-connected K2,4-minor-free graphs we use xy-outerplanar graphs, graphs embeddable in the pl...

Canonical orderings and their relatives such as st-numberings have been used as a key tool in algorithmic graph theory for the last decades. Recently, a unifying link behind all these orders has been shown that links them to well-known graph decompositions into parts that have a prescribed vertex-connectivity. Despite extensive interest in canonical orderings, no analogue of this unifying conce...

In this paper, we obtain some upper and lower bounds for the general extended energy of a graph. As an application, we obtain few bounds for the (edge) Zagreb energy of a graph. Also, we deduce a relation between Zagreb energy and edge-Zagreb energy of a graph $G$ with minimum degree $delta ge2$. A lower and upper bound for the spectral radius of the edge-Zagreb matrix is obtained. Finally, we ...

Tutte observed that every nowhere-zero k-flow on a plane graph gives rise to a kvertex-coloring of its dual, and vice versa. Thus nowhere-zero integer flow and graph coloring can be viewed as dual concepts. Jaeger further shows that if a graph G has a face-k-colorable 2-cell embedding in some orientable surface, then it has a nowhere-zero k-flow. However, if the surface is nonorientable, then a...