Maximum Independent Sets in Subcubic Graphs: New Results

The maximum independent set problem is known to be NP-hard in the class of subcubic graphs, i.e. graphs of vertex degree at most 3. We present a polynomial-time solution in a subclass of subcubic graphs generalizing several previously known results

Precise Localization of the Soft Gamma Repeater SGR 1627-41 and the Anomalous X-ray Pulsar AXP 1E1841-045 with Chandra

We present precise localizations of AXP 1E1841-045 and SGR 1627-41 with Chandra. We obtained new infrared observations of SGR 1627-41 and reanalyzed archival observations of AXP 1E1841-045 in order to refine their positions and search for infrared counterparts. A faint source is detected inside the error circle of AXP 1E1841-045. In the case of SGR 1627-41, several sources are located within the error radius of the X-ray position and we discuss the likelihood of one of them being the counterpart. We compare the properties of our candidates to those of other known AXP and SGR counterparts. We find that the counterpart candidates for SGR 1627-41 and SGR 1806-20 would have to be intrinsically much brighter than AXPs to have detectable counterparts with the observational limits currently available for these sources. To confirm the reported counterpart of SGR 1806-20, we obtained new IR observations during the July 2003 burst activation of the source. No brightening of the suggested counterpart is detected, implying that the counterpart of SGR 1806-20 remains yet to be identified.Comment: 29 pages, 4 figures, accepted for publication in Ap

Tree decompositions with small cost

The f-cost of a tree decomposition ({Xi | i e I}, T = (I;F)) for a function f : N -> R+ is defined as EieI f(|Xi|). This measure associates with the running time or memory use of some algorithms that use the tree decomposition. In this paper we investigate the problem to find tree decompositions of minimum f-cost. A function f : N -> R+ is fast, if for every i e N: f(i+1) => 2*f(i). We show that for fast functions f, every graph G has a tree decomposition of minimum f-cost that corresponds to a minimal triangulation of G; if f is not fast, this does not hold. We give polynomial time algorithms for the problem, assuming f is a fast function, for graphs that has a polynomial number of minimal separators, for graphs of treewidth at most two, and for cographs, and show that the problem is NP-hard for bipartite graphs and for cobipartite graphs. We also discuss results for a weighted variant of the problem derived of an application from probabilistic networks

Independent Set Reconfiguration in Cographs

We study the following independent set reconfiguration problem, called TAR-Reachability: given two independent sets $I$ and $J$ of a graph $G$, both of size at least $k$, is it possible to transform $I$ into $J$ by adding and removing vertices one-by-one, while maintaining an independent set of size at least $k$ throughout? This problem is known to be PSPACE-hard in general. For the case that $G$ is a cograph (i.e. $P_4$-free graph) on $n$ vertices, we show that it can be solved in time $O(n^2)$, and that the length of a shortest reconfiguration sequence from $I$ to $J$ is bounded by $4n-2k$, if such a sequence exists. More generally, we show that if $X$ is a graph class for which (i) TAR-Reachability can be solved efficiently, (ii) maximum independent sets can be computed efficiently, and which satisfies a certain additional property, then the problem can be solved efficiently for any graph that can be obtained from a collection of graphs in $X$ using disjoint union and complete join operations. Chordal graphs are given as an example of such a class $X$

Synthesis and Pharmacology of Halogenated δ-Opiod Selective [\u3csub\u3eD\u3c/sub\u3eAla\u3csup\u3e2\u3c/sup\u3e] Deltorphin II Peptide Analogs

Deltorphins are naturally occurring peptides produced by the skin of the giant monkey frog (Phyllomedusa bicolor). They are δ-opioid receptor-selective agonists. Herein, we report the design and synthesis of a peptide, Tyr-d-Ala-(pI)Phe-Glu-Ile-Ile-Gly-NH2 3 (GATE3-8), based on the [d-Ala2]deltorphin II template, which is δ-selective in in vitro radioligand binding assays over the μ- and κ-opioid receptors. It is a full agonist in [35S]GTPγS functional assays and analgesic when administered supraspinally to mice. Analgesia of 3 (GATE3-8) is blocked by the selective δ receptor antagonist naltrindole, indicating that the analgesic action of 3 is mediated by the δ-opioid receptor. We have established a radioligand in which 125I is incorporated into 3 (GATE3-8). The radioligand has a KD of 0.1 nM in Chinese hamster ovary (CHO) cells expressing the δ receptor. Additionally, a series of peptides based on 3 (GATE3-8) was synthesized by incorporating various halogens in the para position on the aromatic ring of Phe3. The peptides were characterized for binding affinity at the μ-, δ-, and κ-opioid receptors, which showed a linear correlation between binding affinity and the size of the halogen substituent. These peptides may be interesting tools for probing δ-opioid receptor pharmacology

On the complexity of color-avoiding site and bond percolation

The mathematical analysis of robustness and error-tolerance of complex networks has been in the center of research interest. On the other hand, little work has been done when the attack-tolerance of the vertices or edges are not independent but certain classes of vertices or edges share a mutual vulnerability. In this study, we consider a graph and we assign colors to the vertices or edges, where the color-classes correspond to the shared vulnerabilities. An important problem is to find robustly connected vertex sets: nodes that remain connected to each other by paths providing any type of error (i.e. erasing any vertices or edges of the given color). This is also known as color-avoiding percolation. In this paper, we study various possible modeling approaches of shared vulnerabilities, we analyze the computational complexity of finding the robustly (color-avoiding) connected components. We find that the presented approaches differ significantly regarding their complexity.Comment: 14 page

Retention of Supraspinal Delta-like Analgesia and Loss of Morphine Tolerance in δ Opioid Receptor Knockout Mice

AbstractGene targeting was used to delete exon 2 of mouse DOR-1, which encodes the δ opioid receptor. Essentially all 3H-[D-Pen2,D-Pen5]enkephalin (3H-DPDPE) and 3H-[D-Ala2,D-Glu4]deltorphin (3H-deltorphin-2) binding is absent from mutant mice, demonstrating that DOR-1 encodes both δ1 and δ2 receptor subtypes. Homozygous mutant mice display markedly reduced spinal δ analgesia, but peptide δ agonists retain supraspinal analgesic potency that is only partially antagonized by naltrindole. Retained DPDPE analgesia is also demonstrated upon formalin testing, while the nonpeptide δ agonist BW373U69 exhibits enhanced activity in DOR-1 mutant mice. Together, these findings suggest the existence of a second delta-like analgesic system. FinallyDOR-1 mutant mice do not develop analgesic tolerance to morphine, genetically demonstrating a central role for DOR-1 in this process

Syzygies of torsion bundles and the geometry of the level l modular variety over M_g

We formulate, and in some cases prove, three statements concerning the purity or, more generally the naturality of the resolution of various rings one can attach to a generic curve of genus g and a torsion point of order l in its Jacobian. These statements can be viewed an analogues of Green's Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding non-vanishing locus in the moduli space R_{g,l} of twisted level l curves of genus g and use this to derive results about the birational geometry of R_{g, l}. For instance, we prove that R_{g,3} is a variety of general type when g>11 and the Kodaira dimension of R_{11,3} is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the Prym-Green conjecture in genus 8 and level 2.Comment: 35 pages, appeared in Invent Math. We correct an inaccuracy in the statement of Prop 2.

Maximum Independent Sets in Subcubic Graphs: New Results

International audienceWe consider the complexity of the classical Independent Set problem on classes of subcubic graphs characterized by a finite set of forbidden induced subgraphs. It is well-known that a necessary condition for Independent Set to be tractable in such a class (unless P=NP) is that the set of forbidden induced subgraphs includes a subdivided star S k,k,k , for some k. Here, S k,k,k is the graph obtained by taking three paths of length k and identifying one of their endpoints. It is an interesting open question whether this condition is also sufficient: is Independent Set tractable on all hereditary classes of subcu-bic graphs that exclude some S k,k,k ? A positive answer to this question would provide a complete classification of the complexity of Independent Set on all classes of subcubic graphs characterized by a finite set of forbidden induced subgraphs. The best currently known result of this type is tractability for S2,2,2-free graphs. In this paper we generalize this result by showing that the problem remains tractable on S 2,k,k-free graphs, for any fixed k. Along the way, we show that subcubic Independent Set is tractable for graphs excluding a type of graph we call an "apple with a long stem", generalizing known results for apple-free graphs

Register Allocation After Classical SSA Elimination is NP-Complete

Abstract. Chaitin proved that register allocation is equivalent to graph coloring and hence NP-complete. Recently, Bouchez, Brisk, and Hack have proved independently that the interference graph of a program in static single assignment (SSA) form is chordal and therefore colorable in linear time. Can we use the result of Bouchez et al. to do register allocation in polynomial time by first transforming the program to SSA form, then performing register allocation, and finally doing the classical SSA elimination that replaces φ-functions with copy instructions? In this paper we show that the answer is no, unless P = NP: register allocation after classical SSA elimination is NP-complete. Chaitin’s proof technique does not work for programs after classical SSA elimination; instead we use a reduction from the graph coloring problem for circular arc graphs.