Induced disjoint paths in claw-free graphs.

Paths P1,…,Pk in a graph G = (V,E) are said to be mutually induced if for any 1 ≤ i < j ≤ k, Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to test whether a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that Pi connects si and ti for i = 1,…,k. This problem is known to be NP-complete already for k = 2, but for n-vertex claw-free graphs, Fiala et al.gave an nO(k)-time algorithm. We improve the latter result by showing that the problem is fixed-parameter tractable for claw-free graphs when parameterized by k. Several related problems, such as the k-in-a-Path problem, are shown to be fixed-parameter tractable for claw-free graphs as well. We prove that an improvement of these results in certain directions is unlikely, for example by noting that the Induced Disjoint Paths problem cannot have a polynomial kernel for line graphs (a type of claw-free graphs), unless NP ⊆ coNP/poly. Moreover, the problem becomes NP-complete, even when k = 2, for the more general class of K1,4-free graphs. Finally, we show that the nO(k)-time algorithm of Fiala et al.for testing whether a claw-free graph contains some k-vertex graph H as a topological induced minor is essentially optimal by proving that this problem is W[1]-hard even if G and H are line graphs

Electron Correlated Ab Initio Study of Amino Group Flexibility for Improvement of Molecular Mechanics Simulations on Nucleic Acid Conformations and Interactions

High level ab initio studies demonstrate substantial conformational flexibility of amino groups of nucleic acid bases. This flexibility is important for biological functions of DNA. Existing force field models of molecular mechanics do not describe this phenomenon due to a lack of quantitative experimental data necessary for an adjustment of empirical parameters. We have performed extensive calculations of nucleic acid bases at the MP2/6-31G(d,p) level of ab initio theory for broad set of amino group configurations. Two-dimensional maps of energy and geometrical characteristics as functions of two amino hydrogen torsions have been constructed. We approximate the maps by polynomial expressions, which can be used in molecular mechanics calculations. Detailed considerations of these maps enable us to propose a method for determination of numerical coefficients in the developed formulae using restricted sets of points obtained via higher-level calculations