Hardness and approximation for the geodetic set problem in some graph classes

In this paper, we study the computational complexity of finding the \emph{geodetic number} of graphs. A set of vertices $S$ of a graph $G$ is a \emph{geodetic set} if any vertex of $G$ lies in some shortest path between some pair of vertices from $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality. In this paper, we prove that solving the \textsc{MGS} problem is NP-hard on planar graphs with a maximum degree six and line graphs. We also show that unless $P=NP$, there is no polynomial time algorithm to solve the \textsc{MGS} problem with sublogarithmic approximation factor (in terms of the number of vertices) even on graphs with diameter $2$. On the positive side, we give an $O\left(\sqrt[3]{n}\log n\right)$-approximation algorithm for the \textsc{MGS} problem on general graphs of order $n$. We also give a $3$-approximation algorithm for the \textsc{MGS} problem on the family of solid grid graphs which is a subclass of planar graphs

Distribution, abundance and diversity of phytoplankton in the inshore waters of Nizampatnam, South East coast of India

348-356Distribution, abundance and species assemblages of Phytoplankton were studied from inshore waters of Nizampatnam, South East coast of India in March 2007. <span style="mso-fareast-font-family:Calibri;mso-ansi-language:EN-US;mso-bidi-language: GU;mso-bidi-font-weight:bold" lang="EN-US">Significant spatial variations in temperature, pH, salinity, dissolved oxygen (DO), nitrites (NO2) –N)<span style="mso-fareast-font-family:Calibri;mso-ansi-language:EN-US;mso-bidi-language: GU;mso-bidi-font-weight:bold" lang="EN-US">, nitrates (NO3) – N)<span style="mso-fareast-font-family:Calibri;mso-ansi-language: EN-US;mso-bidi-language:GU;mso-bidi-font-weight:bold" lang="EN-US">, phosphate (PO4) –P),<span style="mso-fareast-font-family:Calibri;mso-ansi-language:EN-US;mso-bidi-language: GU;mso-bidi-font-weight:bold" lang="EN-US"> silicate (SiO4) – (Si)<span style="mso-fareast-font-family:Calibri; mso-ansi-language:EN-US;mso-bidi-language:GU;mso-bidi-font-weight:bold" lang="EN-US"> were monitored. A total of 90 species of phytoplankton (net hauls) represented by 5 groups were identified at nine stations, collected along three transects during low tide.<span style="mso-fareast-font-family: Calibri;mso-ansi-language:EN-US;mso-bidi-language:GU;mso-bidi-font-weight:bold" lang="EN-US"> Percentage contribution of each group of phytoplankton was in the order: Bacillariophyceans <span style="mso-fareast-font-family: Calibri;mso-ansi-language:EN-US;mso-bidi-language:GU;mso-bidi-font-weight:bold" lang="EN-US">> Dinophyceans <span style="mso-fareast-font-family: Calibri;mso-ansi-language:EN-US;mso-bidi-language:GU;mso-bidi-font-weight:bold" lang="EN-US">> Cyanophyceans <span style="mso-fareast-font-family: Calibri;mso-ansi-language:EN-US;mso-bidi-language:GU;mso-bidi-font-weight:bold" lang="EN-US">> Euglenophyceans<span style="mso-fareast-font-family:Calibri;mso-ansi-language:EN-US;mso-bidi-language: GU;mso-bidi-font-weight:bold" lang="EN-US">. Pleurosigma angulatum, Navicula sp. were dominant species in the study area. Bray - Curtis similarity and group average clustering, recommended identifying two assemblages of phytoplankton in the study area. High diversity of phytoplankton in the present area suggests stable environmental conditions. </span

Increased human MUC1 protein expression in Kras- and Pten- driven genital tract tumors of MUC1KrasPten triple transgenic mice triggers humoral immunity.

<p>(A) MUC1 immunohistochemistry staining of tumors occurring in the ovary (upper panel), oviduct (middle panel) or endometrium (lower panel). An antibody specific to the human MUC1 extracellular domain (clone HMPV, mouse IgG1) was used at 1∶100. Polarized MUC1 expression throughout the genital tract of healthy female mice at baseline is shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0102409#pone.0102409.s002" target="_blank">Fig. S2</a>. Mouse tumor MUC1 mimics human tumor expression (shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0102409#pone.0102409.s005" target="_blank">Fig. S5</a>). Representative immunohistochemical images shown. Scale bar −50 µm. (B) ELISA measurement of human MUC1 peptide-specific IgG antibodies in sera from MUC1KrasPten mice with tumors (n = 5 ovarian, n = 4 oviductal and n = 4 uterine). Upper panel: presence of antibodies at two different dilutions, using sing as target peptide a 100mer peptide comprising fie 20-aminoacid long peptide from the MUC1 extracellular domain of MUC1. Background levels were detected using sera from KrasPten mice with MUC1 negative tumors (i.e. wild -type for MUC1). Vehicle only was also included as an additional negative control. The assay was run in duplicate and values were plotted as means with standard deviations. Lower panel: box and whisker diagrams (min, Q1, median, Q3, max) of readings at 1∶20 dilution. Antibody levels are significantly higher (compared to control readings) in the ovarian and oviduct tumor group (one way ANOVA p<0.05; *two tail t test; p<0.05). Uterine tumors, p = 0.052.</p

Oviductal and endometrial tumors show endometrioid histology at both primary and satellite locations.

<p>Formalin fixed and paraffin embedded primary and metastatic tumor tissues were analyzed for histo-pathology. Representative images of H&E stained tumor sections are shown. Left column: primary tumors of the genital tract show endometrioid histology in the ovary, oviduct and endometrium. Right column: secondary tumors, including ovarian metastases to the diaphragm (upper), oviduct metastases to the pancreas (middle) and endometrial metastases to the diaphragm (lower) also show endometrioid histology. Scale bar −20 µm.</p

Primary tumors of the ovary, oviduct and uterus have epithelial origin.

<p>Immunohistochemistry staining of tumors occurring in the ovary (upper panels), oviduct (middle panels) or endometrium (lower panels). Antibodies to mouse cytokeratin 8 (an epithelial cell marker, left column) and mouse desmin (right column) were used at 1∶50 dilution. Representative images shown. Scale bar −50 µm.</p

Kras- and Pten- induced tumors differ in nuclear grade and survival based on the anatomical site of mutation activation.

<p>(A) Nuclear grade of primary tumor tissues of the ovary, oviduct and the uterus. Representative H&E images are shown. Scale bars: Main −20 µm, Inset −100 µm. (B) Kaplan Meyer curve shows that mice with uterine tumors survive significantly more than those with ovarian tumors (* p = 0.0015) or those with ductal tumors (# p = 0.0016). Individual group comparison after post ANOVA Bonferroni correction (p<0.016). The numbers of mice in each tumor group and median survival time for each tumor type are listed <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0102409#pone.0102409.s006" target="_blank">Table S1</a>. Mice with premalignant lesions in the uterine tumor category were excluded from analyses. (C) Splenic Treg/CD8 T cell ratios in mice with ovarian, oviduct or uterine tumors (n = 5 mice/group), represented as box and whisker diagrams (min, Q1, median, Q3, max). CD4 and CD8 T lymphocytes were gated under the CD3 population. Foxp3 cells were gated under the CD4 population. One way ANOVA for comparison of all means (p<0.03) and two tail t tests between any two groups show significant differences between the ratios in uterine tumors and any of the other two tumor types, ovarian and oviduct (p<0.02 and p<0.01, respectively).</p

Induction of floxed Kras^G12D and Pten mutations in the female genital tract results in large tumors at the primary site, accompanied at late stages by numerous peritoneal implants.

<p>(A) Left panel: female genital tract anatomy of a healthy mouse (Ov, ovaries; OvB, ovarian bursa; OvD, oviduct; UT, uterus). Right panel: schematic representation of the murine female genital tract showing the ovaries, oviducts and uterine horns. Arrows indicate AdCre delivery routes: under the ovarian bursa, inside the oviduct and inside the uterine horn. All injections were unilateral, keeping the contralateral site as control. (B) Mice carrying conditional mutations in oncogenic Kras^G12D and tumor suppressor Pten pathways were injected with AdCre either under the left ovarian bursa (n = 12), left oviduct (n = 9), or left uterine horn (n = 12). Representative gross images of primary tumors (left column) are shown for ovarian, oviductal and uterine injections. Right column shows numerous loco- regional metastatic small tumor deposits (arrows) that accompany the corresponding primary tumor. The nodules were often located on the diaphragm (upper and lower panels) and liver (middle panel).</p