On Structural Parameterizations of the Edge Disjoint Paths Problem

In this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain -hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the [1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

Heterostemon amoris (Leguminosae, Detarioideae), a new species from Colombia and a key to the species of Heterostemon

Heterostemon amoris a new species from Colombia is described, illustrated, and a key to the species of the genus is presented. Heterostemon amoris is characterized by its leaves with 5-7 pairs of leaflets, basal pair of leaflets 0.6-0.8 × 0.4-0.6 cm, considerably smaller and falcate, intermediate leaflets 3.8-5.5 × 1.1-2.0 cm, ovate to obovate, and terminal pair of leaflets 2.0-2.3 × 0.7-0.8 cm, ovate, shorter than the intermediate ones. This new species is only known from four localities in the departments of Guainía and Vaupés.

Calmodulin-Cork Model of Gap Junction Channel Gating—One Molecule, Two Mechanisms

The Calmodulin-Cork gating model is based on evidence for the direct role of calmodulin (CaM) in channel gating. Indeed, chemical gating of cell-to-cell channels is sensitive to nanomolar cytosolic calcium concentrations [Ca2+]i. Calmodulin inhibitors and inhibition of CaM expression prevent chemical gating. CaMCC, a CaM mutant with higher Ca2+-sensitivity greatly increases chemical gating sensitivity (in CaMCC the NH2-terminal EF-hand pair (res. 9–76) is replaced by the COOH-terminal pair (res. 82–148). Calmodulin colocalizes with connexins. Connexins have high-affinity CaM binding sites. Several connexin mutants paired to wild-type connexins have a high gating sensitivity that is eliminated by inhibition of CaM expression. Repeated transjunctional voltage (Vj) pulses slowly and progressively close a large number of channels by the chemical/slow gate (CaM lobe). At the single-channel level, the chemical/slow gate closes and opens slowly with on-off fluctuations. The model proposes two types of CaM-driven gating: “Ca-CaM-Cork” and “CaM-Cork”. In the first, gating involves Ca2+-induced CaM-activation. In the second, gating takes place without [Ca2+]i rise. The Ca-CaM-Cork gating is only reversed by a return of [Ca2+]i to resting values, while the CaM-Cork gating is reversed by Vj positive at the gated side.