The Complexity of Drawing Graphs on Few Lines and Few Planes

It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the minimum number of lines in $\mathbb{R}^d$ that together can cover all edges of a drawing of $G$. For $d=2$, $G$ must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results. - For $d\in\{2,3\}$, we prove that deciding whether $\rho^1_d(G)\le k$ for a given graph $G$ and integer $k$ is ${\exists\mathbb{R}}$-complete. - Since $\mathrm{NP}\subseteq{\exists\mathbb{R}}$, deciding $\rho^1_d(G)\le k$ is NP-hard for $d\in\{2,3\}$. On the positive side, we show that the problem is fixed-parameter tractable with respect to $k$. - Since ${\exists\mathbb{R}}\subseteq\mathrm{PSPACE}$, both $\rho^1_2(G)$ and $\rho^1_3(G)$ are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to $\rho^1_2$ or $\rho^1_3$ sometimes require irrational coordinates. - Let $\rho^2_3(G)$ be the minimum number of planes in $\mathbb{R}^3$ needed to cover a straight-line drawing of a graph $G$. We prove that deciding whether $\rho^2_3(G)\le k$ is NP-hard for any fixed $k \ge 2$. Hence, the problem is not fixed-parameter tractable with respect to $k$ unless $\mathrm{P}=\mathrm{NP}$

Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study. We show that computing the flip distance between two triangulations of a simple polygon is NP-complete. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.Comment: Accepted versio

Mechanistic Insights into Ring-Opening and Decarboxylation of 2-Pyrones in Liquid Water and Tetrahydrofuran

2-Pyrones, such as triacetic acid lactone, are a promising class of biorenewable platform chemicals that provide access to an array of chemical products and intermediates. We illustrate through the combination of results from experimental studies and first-principle density functional theory calculations that key structural features dictate the mechanisms underlying ring-opening and decarboxylation of 2-pyrones, including the degree of ring saturation, the presence of C═C bonds at the C4═C5 or C5═C6 positions within the ring, as well as the presence of a β-keto group at the C4 position. Our results demonstrate that 2-pyrones undergo a range of reactions unique to their structure, such as retro-Diels–Alder reactions and nucleophilic addition of water. In addition, the reactivity of 2-pyrones and the final products formed is shown to depend on the solvent used and the acidity of the reaction environment. The mechanistic insights obtained here provide guidance for the selective conversion of 2-pyrones to targeted chemicals.Reprinted (adapted) with permission from Journal of American Chemical Society, 135(15); 5699-5708. Doi: 10.1021/ja312075r. Copyright 2013 American Chemical Society. </p

Complex associations between membrane proteins analyzed by analytical ultracentrifugation: Studies on the erythrocyte membrane proteins band 3 and ankyrin

Associations between different water-soluble proteins can be studied by sedimentation equilibrium experiments in the analytical ultracentrifuge and subsequent mathematical analysis of thec(r)-distributions obtained. The analysis can be simplified by labelling one of the proteins with a dye absorbing at wavelengths >300 nm. The method can also be applied to intrinsic membrane proteins in solutions of a nonionic detergent. The present paper both reviews the method and reports application to the associations between two proteins of the human erythrocyte membrane: 1) band 3, the membrane's main intrinsic protein which, in detergent solutions and presumably also in the erythrocyte membrane, is in a monomer/dimer/tetramer association equilibrium, and 2) the cytoskeletal protein ankyrin which links the membrane skeleton to the lipid bilayer by binding to band 3. Ankyrin was labelled with fluorescein isothiocyanate and the detergent used was C12E9. It was found that the only aggregate of ankyrin and band 3 occurring in significant amounts was a complex of one ankyrin molecule and four band 3 molecules. This strongly suggests that, in the erythrocyte membrane, the band 3 tetramer represents the high affinity ankyrin binding site

Total synthesis of 9-dihydroerythronolide B derivatives and of erythronolide B

A convergent total synthesis (22 steps on the longest linear route) of (-)-erythronolide B (5) and two 9-dihydro derivatives (52 and 54) thereof from (R)-2,3-O-isopropylideneglyceraldehyde (20) as the only source of chirality is described. A key step of the synthesis is the regio- and stereocontrolled coupling of the allyl sulfide anion 39 and ketone 26, which can be directed to either α-adduct 40 or 41 by an appropriate choice of the conditions (Scheme V, Table II). From 40 and 41 the seco acids 47 and 49 are prepared, which are smoothly macrolactonized to 50 and 51 according to a modified Yamaguchi procedure. Hydroboration of 50 and 51 proceeds under macrocyclic stereocontrol to afford the 9-dihydroerythronolide B derivatives 52 and 54, of which 54 is converted into 5 by a known oxidation-deketalizaton sequence