A vertex dominating path in a graph is a path P such that every vertex outside P has a neighbor on P . In 1988 H. Broersma stated a result implying that every n-vertex kconnected graph G such that σ(k+2)(G) ≥ n− 2k − 1 contains a dominating path. We show that every n-vertex k-connected graph with σ2(G) ≥ 2n k+2 + f(k) contains a dominating path of length at most O(|T |), where T is a minimum do...

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.

A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. No efficient algorithm is known to find a shortest descending path from s to t in a polyhedral terrain. We give some properties of such paths. In the case where the face sequence is specified, we show that the shortest descending path is unique, and giv...

A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. No efficient algorithm is known to find a shortest descending path (SDP) from s to t in a polyhedral terrain. We give two approximation algorithms (more precisely, FPTASs) that solve the SDP problem on general terrains. Both algorithms are simple, robus...

We present an approximation algorithm that, given a simple, possibly nonconvex polyhedron P with n vertices in R 3 , and two points s and t on its surface @P , constructs a path on @P between s and t whose length is at most 7(1 + "), where is the length of the shortest path between s and t on @P , and " > 0 is an arbitararily small positive constant. The algorithm runs in O(n 5=3 log 5=3 n) tim...

(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...

For a connected graph G not isomorphic to a path, a cycle or a K1,3, let pc(G) denote the smallest integer n such that the nth iterated line graph Ln(G) is panconnected. A path P is a divalent path of G if the internal vertices of P are of degree 2 in G. If every edge of P is a cut edge of G, then P is a bridge divalent path of G; if the two ends of P are of degree s and t , respectively, then ...

We show that the number of t-ary trees with path length equal to p is exp ( h(t−1)t log t p log p (1 + o(1)) ) , where h(x)=−x log x−(1−x) log(1−x) is the binary entropy function. Besides its intrinsic combinatorial interest, the question recently arose in the context of information theory, where the number of t-ary trees with path length p estimates the number of universal types, or, equivalen...