We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph, which are straightforward generalizations of strongly connected components. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs only rather simple O(mn)-time algorithms were known. We use a hierarchical sparsifi...

Let G be a simple graph on n vertices having edge-connectivity /(.' (G) > a and minimum degree o(G) We say G is k-critical if /(.' (G) = k and /(.' (G e) < k for every edge e of G. In this paper we prove that a k-critical graph has 1<' (G) o(G). We descri be a number of classes of k-cri tical graphs and consider the problem of determining the edge-maximal ones.

We study the Edge Disjoint Paths (EDP) problem in undirected graphs: Given a graph G with n nodes and a set T of pairs of terminals, connect as many terminal pairs as possible using paths that are mutually edge disjoint. This leads to a variety of classic NP-complete problems, for which approximability is not well understood. We show a polylogarithmic approximation algorithm for the undirected ...

A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graph...

For an interger l > 1, the l-edge-connectivity λl(G) of G is defined to be the smallest number of edges whose removal leaves a graph with at least l components, if |V (G)| ≥ l; and λl(G) = |V (G)| if |V (G)| ≤ l. A graph G is (k, l)-edge-connected if the l-edge-connectivity of G is at least k. A sufficient and necessary condition for G to be minimally (k, k − 1)-edgeconnected is obtained in the...

A subset F of edges in a connected graph G is a h-extra edge-cut if G − F is disconnected and every component has more than h vertices. The h-extra edge-connectivity λ(G) of G is defined as the minimum cardinality over all h-extra edge-cuts of G. A graph G, if λ(G) exists, is super-λ if every minimum h-extra edge-cut of G isolates at least one connected subgraph of order h + 1. The persistence ...

Fix g > 1. Every graph of large enough tree-width contains a g × g grid as a minor; but here we prove that every four-edge-connected graph of large enough tree-width contains a g × g grid as an immersion (and hence contains any fixed graph with maximum degree at most four as an immersion). This result has a number of applications.

Let G be a k-connected graph of order n with |E(G)|>(n−k 2 )+ k2. Then for (k = 1, n 3), (k = 2, n 10), and (k = 3, n 16), G is hamiltonian. The bounds are tight and for k = 1, (k = 2, n 12), and (k = 3, n 18) the extremal graphs are unique. A general bound will also be given for the number of edges in a nonhamiltonian k-connected graph, but the bound is not tight. © 2006 Elsevier B.V. All righ...

In this paper, we study matching integral graphs of small order. A graph is called matching integral if the zeros of its matching polynomial are all integers. Matching integral graphs were first studied by Akbari, Khalashi, etc. They characterized all traceable graphs which are matching integral. They studied matching integral regular graphs. Furthermore, it has been shown that there is no matc...