Let G be a graph. We denote p(G) and c(G) the order of a longest path and the order of a longest cycle of G, respectively. Let κ(G) be the connectivity of G, and let σ3(G) be the minimum degree sum of an independent set of three vertices in G. In this paper, we prove that if G is a 2-connected graph with p(G) − c(G) ≥ 2, then either (i) c(G) ≥ σ3(G) − 3 or (ii) κ(G) = 2 and p(G) ≥ σ3(G)− 1. Thi...

Two sharp lower bounds for the length of a longest cycle C of a graph G are presented in terms of the lengths of a longest path and a longest cycle of G − C, denoted by p and c, respectively, combined with minimum degree δ: (1) |C| ≥ (p + 2)(δ − p) and (2) |C| ≥ (c + 1)(δ − c + 1).

We consider the length L(n) of the longest path in a randomly generated Apollonian Network (ApN) An. We show that w.h.p. L(n) ≤ ne− log c n for any constant c < 2/3.

We propose a resynthesis method that modifies a given circuit to reduce the number of paths in the circuit and thus improve its path delay fault testability. The resynthesis procedure is based on replacing subcircuits of the given circuit by structures called comparison units. A subcircuit can be replaced by a comparison unit if it implements a function belonging to the class of comparison func...

Let D be a digraph. A path partition of D is called k-optimal if the sum of the k-norms of its paths isminimal. The k-norm of a path P ismin(|V (P)|, k). Berge’s path partition conjecture claims that for every k-optimal path partition P there are k disjoint stable sets orthogonal to P . For general digraphs the conjecture has been proven for k = 1, 2, λ − 1, λ, where λ is the length of a longes...

Given k disjoint regions in the plane or space, the bridge problem asks for k − 1 line segments (optimal bridges) which connect these regions so as to minimize the length of the longest path between two points in the final connected region. We present O(n) and O(n2) time algorithms for finding an optimal bridge between two convex regions in the plane and space, respectively. An important proper...

We consider the distributions of the lengths of the longest monotone and alternating subsequences in classes of permutations of size n that avoid a specific pattern or set of patterns, with respect to the uniform distribution on each such class. We obtain exact results for any class that avoids two patterns of length 3, as well as results for some classes that avoid one pattern of length 4 or m...

We give algorithms to find the optimal disjoint paths around a rectangle. The set of disjoint paths connects a set of sources to a set of sinks (no fixed pairing between the sources and sinks) on the boundary of a rectangle where either the longest path length or the total path length is minimized. One algorithm finds the set of disjoint paths with the longest path length minimized in O(n log n...

Uncovering problem is one of three main problems of hidden Markov model (HMM), which aims to find out optimal state sequence that is most likely to produce a given observation sequence. Although Viterbi is the best algorithm to solve uncovering problem, I introduce a new viewpoint of how to solve HMM uncovering problem. The proposed algorithm is called longest-path algorithm in which the uncove...