Complexity Results for the Spanning Tree Congestion Problem

We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the complexity of this problem. First, we show that for every fixed k and d the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k ≥ 10. For very small values of k however, the problem becomes polynomially solvable. We also show that it is NP-hard to approximate the spanning tree congestion within a factor better than 11/10. On planar graphs, we prove the problem is NP-hard in general, but solvable in linear time for fixed k

Growth Hormone Promotes Hair Cell Regeneration in the Zebrafish (Danio rerio) Inner Ear following Acoustic Trauma

BACKGROUND: Previous microarray analysis showed that growth hormone (GH) was significantly upregulated following acoustic trauma in the zebrafish (Danio rerio) ear suggesting that GH may play an important role in the process of auditory hair cell regeneration. Our objective was to examine the effects of exogenous and endogenous GH on zebrafish inner ear epithelia following acoustic trauma. METHODOLOGY/PRINCIPAL FINDINGS: We induced auditory hair cell damage by exposing zebrafish to acoustic overstimulation. Fish were then injected intraperitoneally with either carp GH or buffer, and placed in a recovery tank for either one or two days. Phalloidin-, bromodeoxyuridine (BrdU)-, and TUNEL-labeling were used to examine hair cell densities, cell proliferation, and apoptosis, respectively. Two days post-trauma, saccular hair cell densities in GH-treated fish were similar to that of baseline controls, whereas buffer-injected fish showed significantly reduced densities of hair cell bundles. Cell proliferation was greater and apoptosis reduced in the saccules, lagenae, and utricles of GH-treated fish one day following trauma compared to controls. Fluorescent in situ hybridization (FISH) was used to examine the localization of GH mRNA in the zebrafish ear. At one day post-trauma, GH mRNA expression appeared to be localized perinuclearly around erythrocytes in the blood vessels of the inner ear epithelia. In order to examine the effects of endogenous GH on the process of cell proliferation in the ear, a GH antagonist was injected into zebrafish immediately following acoustic trauma, resulting in significantly decreased cell proliferation one day post-trauma in all three zebrafish inner ear end organs. CONCLUSIONS/SIGNIFICANCE: Our results show that exogenous GH promotes post-trauma auditory hair cell regeneration in the zebrafish ear through stimulating proliferation and suppressing apoptosis, and that endogenous GH signals are present in the zebrafish ear during the process of auditory hair cell regeneration

Ueber Immunisierung mit atoxischen Toxinen 1 )

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