A paired many-to-many -disjoint path cover (-DPC for short) of a graph is a set of  disjoint paths joining  distinct source-sink pairs in which each vertex of the graph is covered by a path. A two-dimensional × torus is a graph defined as the product of two cycles  and  of length  and , respectively. In this paper, we deal with an × bipartite torus, even ≥ , with a single faul...

We show that the notoriously difficult problem of finding and reporting the smallest number of vertex-disjoint paths that cover the vertices of a graph can be solved timeand work-optimally for cographs. Our algorithm solves this problem in O(log n) time using n log n processors on the EREW-PRAM for an n-vertex cograph G represented by its cotree.

In this paper, we introduce the identifying path cover problem: an identifying path cover of a graph G is a set P of paths such that each vertex belongs to a path of P , and for each pair u, v of vertices, there is a path of P which includes exactly one of u, v. This problem is related to a large variety of identification problems. We investigate the identifying path cover problem in some famil...

A paired many-to-many k-disjoint path cover (k-DPC for short) of a graph is a set of k disjoint paths joining k distinct source-sink pairs that cover all the vertices of the graph. Extending the notion of DPC, we define a paired many-to-many bipartite k-DPC of a bipartite graph G to be a set of k disjoint paths joining k distinct source-sink pairs that altogether cover the same number of vertic...

9 Adleman wrote the first paper in which it is shown that deoxyribonucleic acid (DNA) strands could be employed towards calculating solutions to an instance of the NP-complete Hamiltonian path problem (HPP). Lipton also demonstrated that Adleman’s techniques could be used to solve the NP-complete satisfiability (SAT) problem (the first NP-complete problem). In this paper, it is proved how the D...

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 cycle cover of a graph G is a collection of disjoint cycles that spans G. Generally, a (possibly disconnected) cycle cover is easier to construct than a connected (Hamiltonian) cycle cover. One might expect this since the cycle cover property is local whereas connectivity is a global constraint. We compare the hardness of CONNECTED CYCLE COVER and CYCLE COVER under various constraints (both l...

Over the past decade, sequencing read length has increased from tens to hundreds and then to thousands of bases. Current cDNA synthesis methods prevent RNA-seq reads from being long enough to entirely capture all the RNA transcripts, but long reads can still provide connectivity information on chains of multiple exons that are included in transcripts. We demonstrate that exploiting full connect...

A k-disjoint path cover (k-DPC for short) of a graph is a set of k internally vertex-disjoint paths from given sources to sinks that collectively cover every vertex in the graph. In this paper, we establish a necessary and sufficient condition for the cube of a connected graph to have a 3-DPC joining a single source to three sinks. We also show that the cube of a connected graph always has a 3-...

A k-disjoint path cover of a graph is defined as a set of k internally vertexdisjoint paths connecting given sources and sinks in such a way that every vertex of the graph is covered by a path in the set. In this paper, we analyze the k-disjoint path cover of recursive circulant G(2m, 4) under the condition that at most f faulty vertices and/or edges are removed. It is shown that whenm ≥ 3, G(2...