Let G = (V,E) be a simple, undirected graph. Given an integer r ≥ 1, we say that G is r-twin-free (or r-identifiable) if the balls B(v, r) for v ∈ V are all different, where B(v, r) denotes the set of all vertices which can be linked to v by a path with at most r edges. These graphs are precisely the ones which admit r-identifying codes. We show that if a graph G is r-twin-free, then it contain...

(a) (i) Since T is a connected graph with ≥ 2 vertices, there are edges in T . Hence, any maximal path in T has length ≥ 1, and thus two distinct end vertices. Let v0, vk be the end vertices of such a path P = (v0, v1, . . . , vk). As T is acyclic, the only neighbor of v0 on P is v1 and the only neighbor of vk on P is vk−1. Because P was chosen to be maximal, both v0 and vk have no neighbors ou...

We consider the individual points on a Martin-Löf random path of Brownian motion. We show (1) that Khintchine’s law of the iterated logarithm holds at almost all points; and (2) there exist points (besides the trivial example of the origin) having effective dimension < 1. The proof of (1) amounts essentially to showing for almost all times t, the path f is random relative to t and the dimension...

We extend the work of T. Lyons [Lyo98] and T. Lyons and Z. Qian [LQ02] to define integrals and solutions of differential equations along product of p and q rough paths, with 1/p+1/q > 1. We use this to write an Itô formula at the level of rough paths, and to see that any rough path can always be interpreted as a product of a p-geometric rough path and a p/2-geometric rough path.

Path-width of matroids naturally generalizes better known path-width of graphs, and is NP-hard by a reduction from the graph case. While the term matroid path-width was formally introduced by Geelen–Gerards–Whittle [JCTB 2006] in pure matroid theory, it was soon recognized by Kashyap [SIDMA 2008] that it is the same concept as long-studied so called trellis complexity in coding theory, later na...

An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with s, t ∈ V (G) and e ∈ E(G) to have (1) a Hamiltonian s-path, (2) a Hamiltonian st-path, (3) a Hamiltonian sand st-paths containing e when G has connectivity one, and (4) a Hamiltonian cycle containi...

Given a point s and a set of h pairwise disjoint polygonal obstacles of totally n vertices in the plane, we present a new algorithm for building an L1 shortest path map of size O(n) in O(T ) time and O(n) space such that for any query point t, the length of the L1 shortest obstacleavoiding path from s to t can be reported in O(log n) time and the actual shortest path can be found in additional ...

For two vertices s and t in a graph G = (V,E), the next-to-shortest path is an st-path which length is minimum amongst all st-paths strictly longer than the shortest path length. In this paper we show that, when the graph is undirected and all edge lengths are nonnegative, the problem can be solved in linear time if the distances from s and t to all other vertices are given.

Given an edge-weighted graph G and two distinct vertices s and t of G, the next-to-shortest path problem asks for a path from s to t of minimum length among all paths from s to t except the shortest ones. In this article, we consider the version where G is directed and all edge weights are positive. Some properties of the requested path are derived when G is an arbitrary digraph. In addition, i...

An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with s, t ∈ V (G) and e ∈ E(G) to have (1) Hamiltonian s-path, (2) a Hamiltonian st–path, (3) a Hamiltonian sand st-paths containing e when G has connectivity one, and (4) a Hamiltonian cycle containing...