Proof: ⇐: Fix a strongly-connected orientation D. For any non-empty U ⊂ V , we may choose u ∈ U and v ∈ V \U . Since D is strongly connected, there is a directed u-v path and a directed v-u δ in path. Thus |δ out (U)| ≥ 1 and | D (U)| ≥ 1, implying |δG(U)| ≥ 2. D ⇒: Since G is 2-edge-connected, it has an ear decomposition. We proceed by induction on the number of ears. If G is a cycle then we m...

All graphs considered here are directed and have no loops. For a directed graph D, let V(D) denote the set of vertices of D, 2E(D) its set of arcs, and z(D) the chromatic number of D. D is symmetric iff (x, y)~E(D)~(y, x)EE(D). A directed walk of length k in D is a sequence o f k arcs (not necessarily distinct), e~, e2, ..., ek such that the initial vertex of ei+l is the terminal vertex of e, f...

An asteroidal triple is a set of three independent vertices in a graph such that any two vertices in the set are connected by a path which avoids the neighbourhood of the third. A classical result by Lekkerkerker and Boland [10] showed that interval graphs are precisely the chordal graphs that do not have asteroidal triples. Interval graphs are chordal, as are the directed path graphs and the p...

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If x is a vertex of a digraph D, then we denote by d(x) and d−(x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is de6ned by ig(D) = max{d+(x); d−(x)} − min{d+(y); d−(y)} over all vertices x and y o...

We study distance properties of a general class of random directed acyclic graphs (DAGs). In a DAG, many natural notions of distance are possible, for there exists multiple paths between pairs of nodes. The distance of interest for circuits is the maximum length of a path between two nodes. We give laws of large numbers for the typical depth (distance to the root) and the minimum depth in a ran...

The basic idea of re-flowable document understanding and automatic typesetting is to generate logical documents by judging the hierarchical relationship of physical units and logical tags based on the identification of logical paragraph tags in re-flowable document. In order to overcome the shortages of conventional logical structure reconstruction methods, a novel logical structure reconstruct...

We investigate the hardness of approximating the longest path and the longest cycle in directed graphs on n vertices. We show that neither of these two problems can be polynomial time approximated within n1-ε for any ε > 0 unless P = NP. In particular, the result holds for digraphs of constant bounded outdegree that contain a Hamiltonian cycle. Assuming the stronger complexity conjecture that S...

Weconsider a directed graph on the integerswith a directed edge fromvertex i to j present with probability n−1, whenever i < j , independently of all other edges. Moreover, to each edge (i, j) we assign weight n−1(j − i). We show that the closure of vertex 0 in such a weighted random graph converges in distribution to the Poisson-weighted infinite tree as n → ∞. In addition, we derive limit the...

Sequencing activities over time is a fundamental optimization problem. The problem can be modeled using a directed network in which activities are represented by nodes and pairs of activities that can be performed consecutively are represented by arcs. A sequence of activities then corresponds to a path in the directed network, and an optimal sequence of activities, assuming appropriately chose...

Let G be a directed graph and H a subgraph of G. A D(G,H, λ) design is a multiset D of subgraphs of G each isomorphic to H so that every directed 2-path in G lies in exactly λ subgraphs in D. In this paper, we show that there exists a D(K∗ n,n, −→ C 2n, 1) design for every n ≥ 2, where K∗ n,n is the complete bipartite directed graph and −→ C 2n is a directed Hamilton cycle in K∗ n,n.