We study the problem of space-efficient polynomial-time algorithms for directed stconnectivity (STCON). Given a directed graph G, and a pair of vertices s, t, the STCON problem is to decide if there exists a path from s to t in G. For general graphs, the best polynomial-time algorithm for STCON uses space that is only slightly sublinear. However, for special classes of directed graphs, polynomi...

The concept of a line digraph is generalized to that of a directed path graph. The directed path graph → Pk(D) of a digraph D is obtained by representing the directed paths on k vertices of D by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in D form a directed path on k + 1 vertices or form a directed cycle on k vertices in D. In this introductory paper ...

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 path. Thus |δ D(U)| ≥ 1 and |δ − D(U)| ≥ 1, implying |δG(U)| ≥ 2. ⇒: 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 may orie...

The stability of directed Min-Max optimal paths in cases of change in the random media is studied. Using analytical arguments it is shown that when small perturbations are applied to the weights of the bonds of the lattice, the probability that the new Min-Max optimal path is different from the original Min-Max optimal path is proportional to t‖ , where t is the size of the lattice, and ν‖ is t...

We study acceleration methods for point-to-point shortest path and constrained shortest path computations in directed graphs, in particular in road and railroad networks. Our acceleration methods are allowed to use a preprocessing of the network data to create auxiliary information which is then used to speed-up shortest path queries. We focus on two methods based on Dijkstra’s algorithm for sh...

Much of the literature on multi-agent path finding focuses on undirected graphs, where motion is permitted in both directions along a graph edge. Despite this, travelling on directed graphs is relevant in navigation domains, such as pathfinding in games, and asymmetric communication networks. We consider multi-agent path finding on strongly biconnected directed graphs. We show that all instance...

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satis5es the following two conditions: (i) For every pair of di7erent vertices u, v∈N , there is no monochromatic directed path between th...

Example 1 Undirected Hamiltonian Path (UHP) and Undirected Hamiltonian Circuit (UHC) are the same problems as the corresponding problems for directed graphs, but the new problems are for undirected graphs. Claim 2 Given a polynomial time algorithm for Undirected Hamiltonian Path (UHP) there is a polynomial time algorithm for Directed Hamiltonian Path. 1. A DHP builds the triple (H, p, q), where...

Proof. As the edges in GC represent causal relations, a path of length 0 (no edge) is not considered a causal relation, and existence of a directed path from a variable back to itself would contravene the causal DAG assumption, hence: irreflexive and acyclic (or, more accurate, asymmetric). Transitivity follows immediately, by concatenation, from the sequence 〈X, .., Y, .., Z〉, in which each no...

Directed edge path graphs are the intersection graphs of directed paths in a directed tree, viewed as sets of edges. They were studied by Monma and Wei (J. Comb. Theory B 41 (1986) 141–181) who also gave a polynomial time recognition algorithm. In this work, we show that the clique graphs of these graphs are exactly the two sections of the same kind of path families, and give a polynomial time ...