Parameterized Edge Hamiltonicity

We study the parameterized complexity of the classical Edge Hamiltonian Path problem and give several fixed-parameter tractability results. First, we settle an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT parameterized by vertex cover, and that it also admits a cubic kernel. We then show fixed-parameter tractability even for a generalization of the problem to arbitrary hypergraphs, parameterized by the size of a (supplied) hitting set. We also consider the problem parameterized by treewidth or clique-width. Surprisingly, we show that the problem is FPT for both of these standard parameters, in contrast to its vertex version, which is W-hard for clique-width. Our technique, which may be of independent interest, relies on a structural characterization of clique-width in terms of treewidth and complete bipartite subgraphs due to Gurski and Wanke

Aromatic N versus aromatic F: bioisosterism discovered in RNA base pairing interactions leads to a novel class of universal base analogs

The thermodynamics of base pairing is of fundamental importance. Fluorinated base analogs are valuable tools for investigating pairing interactions. To understand the influence of direct base–base interactions in relation to the role of water, pairing free energies between natural nucleobases and fluorinated analogs are estimated by potential of mean force calculations. Compared to pairing of AU and GC, pairing involving fluorinated analogs is unfavorable by 0.5–1.0 kcal mol−1. Decomposing the pairing free energies into enthalpic and entropic contributions reveals fundamental differences for Watson–Crick pairs compared to pairs involving fluorinated analogs. These differences originate from direct base–base interactions and contributions of water. Pairing free energies of fluorinated base analogs with natural bases are less unfavorable by 0.5–1.0 kcal mol−1 compared to non-fluorinated analogs. This is attributed to stabilizing C–F…H–N dipolar interactions and stronger N…H–C hydrogen bonds, demonstrating direct and indirect influences of fluorine. 7-methyl-7H-purine and its 9-deaza analog (Z) have been suggested as members of a new class of non-fluorinated base analogs. Z is found to be the least destabilizing universal base in the context of RNA known to date. This is the first experimental evidence for nitrogen-containing heterocylces as bioisosteres of aromatic rings bearing fluorine atoms

Factor-criticality and matching extension in DCT-graphs

The class of DCT-graphs is a common generalization of the classes of almost claw-free and quasi claw-free graphs. We prove that every even (2p + 1)-connected DCT-graph G is p-extendable, i.e. every set of p independent edges of G is contained in a perfect matching of G. This result is obtained as a corollary of a stronger result concerning factor-criticality of DCT-graphs

Forbidden Subgraphs, Hamiltonicity and Closure in Claw-Free Graphs

We study the stability of some classes of graphs defined in terms of forbidden subgraphs under the closure operation introduced by the second author. Using these results, we prove that every 2-connected claw-free and P 7 -free, or claw-free and Z 4 - free, or claw-free and eiffel-free graph is either hamiltonian or belongs to a certain class of exceptions (all of them having connectivity 2). 1 Introduction In this paper we consider only finite undirected graphs G = (V (G); E(G)) without loops and multiple edges. For terminology and notation not defined here we refer to [3]. If H 1 ; : : : ; H k (k 1) are graphs, then we say that a graph G is H 1 : : : H k -free if G contains no copy of any of the graphs H 1 ; : : : ; H k as an induced subgraph; the graphs H 1 ; : : : ; H k will be also referred to in this context as forbidden subgraphs. Specifically, the four-vertex star K 1;3 will be also denoted by C and called the claw and in this case we say that G is claw-free. Whenever we..