For a tree T , let iT (t) be the number of independent sets of size t in T . It is an open question, raised by Alavi, Malde, Schwenk and Erdős, whether the sequence (iT (t))t≥0 is always unimodal. Here we answer the question in the affirmative for some recursively defined families of trees, specifically paths with auxiliary trees dropped from the vertices in a periodic manner. In particular, ex...

Let P be a path between two points s and t in a polygonal subdivision T with obstacles and weighted regions. Given a relative error tolerance ε ∈ (0, 1), we present the first algorithm to compute a path between s and t that can be deformed to P without passing over any obstacle and the path cost is within a factor 1+ ε of the optimum. The running time is O( 3 ε2 knpolylog(k, n, 1 ε )), where k ...

We consider two network interdiction problems: one where a network user tries to traverse a network from a starting vertex s to a target vertex t along the shortest path while an interdictor tries to eliminate all short s-t paths by destroying as few vertices (arcs) as possible, and one where the network user, as before, tries to traverse the network from s to t along the shortest path while th...

We consider the s-t-path TSP: given a finite metric space with two elements s and t, we look for a path from s to t that contains all the elements and has minimum total distance. We improve the approximation ratio for this problem from 1.599 to 1.566. Like previous algorithms, we solve the natural LP relaxation and represent an optimum solution x∗ as a convex combination of spanning trees. Gao ...

Let P be a path between two points s and t in a polygonal subdivision T with obstacles and weighted regions. Given a relative error tolerance ε ∈ (0, 1), we present the first algorithm to compute a path between s and t that can be deformed to P without passing over any obstacle and the path cost is within a factor 1 + ε of the optimum. The running time is O( 3 ε2 kn polylog(k, n, 1 ε )), where ...

A many-to-many k-disjoint path cover of a graph joining two disjoint vertex subsets S and T of equal size k is a set of k vertex-disjoint paths between S and T that altogether cover every vertex of the graph. It is classified as paired if each source in S is required to be paired with a specific sink in T , or unpaired otherwise. In this paper, we develop Ore-type sufficient conditions for the ...

A parity path in a vertex colouring of a graph is a path along which each colour is used an even number of times. Let χp(G) be the least number of colours in a proper vertex colouring of G having no parity path. It is proved that for any graph G we have the following tight bounds χ(G) ≤ χp(G) ≤ |V (G)| − α(G) + 1, where χ(G) and α(G) are the chromatic number and the independence number of G, re...

Given a point s and a set of h pairwise disjoint polygonal obstacles with a total of n vertices in the plane, after the free space is triangulated, we present an O(n + h log h) time and O(n) space algorithm for building a data structure (called shortest path map) of size O(n) such that for any query point t, the length of the L1 shortest obstacle-avoiding path from s to t can be reported in O(l...

Abs t rac t . Path keys are secrets established between communicating devices that do not share a pre-distributed key. They are required by most key pre-distribution schemes for sensor networks, because topology is unknown before deployment and storing complete pairwise-unique keys is infeasible for low-cost devices such as sensors. Unfortunately, path keys have often been neglected by existing...

In aqueous H2SO4 media, the chromic acid oxidation of D-galactose in the presence and absence of 2,2 ́ bipyridine (bpy) has been carried out under the conditions, [D-galactose]T >> [Cr(VI)]T at different temperatures. The monomeric species of Cr (VI) has been found to be kinetically active in the absence of bpy whereas in the bpy-catalysed path, the Cr(VI) -bpy complex has been suggested as the ...