We address the problem of very efficient reasoning and update in ≤, <-chains, where a ≤, <-chain is a directed acyclic graph such that there is a directed path between every pair of vertices, and edges are consistently labelled by ≤ or <. It is easy to show that, subsequent to an O(n) labelling scheme, queries concerning implied ≤ and < relations can be answered in O(1) time. However this schem...

Max-Cost Paths in DAGS Let G = (V,E) be a weighted directed graph. Dijkstra’s algorithm constructs a shortest path tree rooted at (any vertex) s in O(|E| log |V |) time (or O(|E|+ |V | log |V |) with more sophisticated data structures) as long as the algorithm does not contain any negative cycles. Surprisingly, the corresponding max-cost problem, i.e., finding a maximum cost simple path tree ro...

We call the digraphD anm-coloured digraph if its arcs are coloured withm colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. Let D be an m-coloured digraph. A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u, v ∈ N there is no monochromati...

By a new type of finite size scaling analysis on the square lattice, and by renormalization group calculations on hierarchical lattices we investigate the effects of dilution on optimal undirected self-avoiding paths in a random environment. The behaviour of the optimal paths remains the same as for directed paths in undiluted medium, as long as forbidden bonds are not exceeding the percolation...

How difficult is it to find a path between two vertices in finite directed graphs whose independence number is bounded by some constant k? The independence number of a graph is the largest number of vertices that can be picked such that there is no edge between any two of them. The complexity of this problem depends on the exact question we ask: Do we only wish to tell whether a path exists? Do...

Robustness of ecological flow networks under random failure of arcs is considered with respect to two different functionalities: coherence and circulation. In our previous work, we showed that each functionality is associated with a natural path notion: lateral path for the former and directed path for the latter. Robustness of a network is measured in terms of the size of the giant laterally c...