Given a 3-connected biased graph Ω with three node-disjoint unbalanced circles, at most one of which is a loop, we describe how the bias matroid of Ω is uniquely represented by Ω.

We construct a combinatorial algorithm to find a maximum packing of fully node-disjoint A-paths. In doing this, we discover relations with fractional packings, 2-packings, and b-packings.

This work proposes an efficient disjoint multipath geographic routing algorithm for dense wireless sensor networks (WSN), called Multipath Grid-based Enabled Geographic Routing (MGEGR). The proposed algorithm relies on the construction of a 2-D logical grid in the geographical region of deployment. The objective of the proposed scheme is to determine optimal or near-optimal (within a defined co...

The process of sending a message from one node of a communication network to all other nodes is called broadcasting when the message is the same for all nodes and scattering when each of the nodes receives a different message. In this thesis we prove several upper bounds on the time to broadcast and scatter in the Cube-Connected Cycles (CCCd) and Butterfly (BFd) interconnection networks. We use...

Björklund and Husfeldt developed a randomized polynomial time algorithm to solve the shortest two disjoint paths problem. Their algorithm is based on computation of permanents modulo 4 and the isolation lemma. In this paper, we consider the following generalization of the shortest two disjoint paths problem, and develop a similar algebraic algorithm. The shortest perfect (A + B)-path packing pr...

We study the problem of finding disjoint paths between two nodes in a star interconnection network. Specifically, we show a simple routing algorithm that finds n − 1 disjoint paths between any two nodes s and t in an n-star in optimal O(n) time such that no path has length more than d(s, t) + 4, where d(s, t) is the shortest path length between s and t. This result matches those obtained earlie...

Tightening interatomic distance bounds, obtained from NMR experiments, is a critical step in solving the molecular conformation problem. Tetrangle inequality (resulting from the Cayley-Menger determinant for the 3D embedding) is applied to quadruples of atoms in order to tighten the distance bounds. Complete graph Kn is used to model an n-atom molecule, where the nodes represent the atoms. Only...

A network or graph consists of a set of nodes joined by edges. A set of nodes are form a connected component when every vertex has a path to every other node in the set. The connected components of a network can be computed efficiently using depth-first search or disjoint-sets data structures. The disjoint sets approach has the advantages of easily answering queries whether two nodes are in the...

A phylogenetic network is a generalization of a phylogenetic tree, allowing structural properties that are not tree-like. With the growth of genomic data, much of which does not fit ideal tree models, there is greater need to understand the algorithmics and combinatorics of phylogenetic networks [10, 11]. However, to date, very little has been published on this, with the notable exception of th...