(1,j)(1,j)-set problem in graphs

A subset $D \subseteq V$of a graph $G = (V, E)$ is a $(1, j)$-set if every vertex $v \in V \setminus D$ is adjacent to at least $1$ but not more than $j$ vertices in D. The cardinality of a minimum $(1, j)$-set of $G$, denoted as $\gamma_{(1,j)} (G)$, is called the $(1, j)$-domination number of $G$. Given a graph $G = (V, E)$ and an integer $k$, the decision version of the $(1, j)$-set problem is to decide whether $G$ has a $(1, j)$-set of cardinality at most $k$. In this paper, we first obtain an upper bound on $\gamma_{(1,j)} (G)$ using probabilistic methods, for bounded minimum and maximum degree graphs. Our bound is constructive, by the randomized algorithm of Moser and Tardos [MT10], We also show that the $(1, j)$- set problem is NP-complete for chordal graphs. Finally, we design two algorithms for finding $\gamma_{(1,j)} (G)$ of a tree and a split graph, for any fixed $j$, which answers an open question posed in [CHHM13]

Uniformity of point samples in metric spaces using gap ratio

Teramoto et al. defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a bounded subset of $\mathbb{R}^2$. We generalize this definition of measure over all metric spaces by appealing to covering and packing radius. The definition of gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion of a range space and its volume. We also show some interesting connections of gap ratio to Delaunay triangulation and discrepancy in the Euclidean plane. The major focus of this work is on solving optimization related questions about selecting uniform point samples from metric spaces; the uniformity being measured using gap ratio. We consider discrete spaces like graph and set of points in the Euclidean space and continuous spaces like the unit square and path connected spaces. We deduce lower bounds, prove hardness and approximation hardness results. We show that a general approximation algorithm framework gives different approximation ratios for different metric spaces based on the lower bound we deduce. Apart from the above, we show existence of coresets for sampling uniform points from the Euclidean space -- for both the static and the streaming case. This leads to a $\left( 1+\epsilon \right)$-approximation algorithm for uniform sampling from the Euclidean space.Comment: 31 pages, 10 figure

A comparative study of neonatal and maternal outcome between forceps delivery and vacuum extraction

Background: Aims and objectives of the study were to compare maternal and neonatal outcomes of forceps versus vacuum application in assisted vaginal delivery. Methods: This prospective study was conducted in a tertiary care hospital of West Bengal over one year. Women in labor with vertex presentation were delivered by vacuum and forceps. A total of 100 cases were included of which 50 patients selected for forceps delivery and 50 patients for vacuum extraction. The instruments were either silastic cup vacuum extractor or Wrigley`s outlet forceps. Maternal morbidity was studied in terms of cervical tears, vaginal lacerations, episiotomy extension, perineal tears, PPH, and retention of urine. Neonatal morbidity was studied in terms of Apgar score, instrumental injuries, cephalhematoma, NICU admission and the outcome was compared. Chi square test was used to analyze the data. Results: Observations maternal morbidity viz. episiotomy extension, traumatic PPH were significant in the forceps group (p=0.01). With regards to neonatal morbidity, SNCU admission were significantly higher in forceps delivery (p=0.02) and incidence of cephalohematomas were more in ventouse delivery (p=0.02). Conclusions: Vacuum and forceps should remain appropriate tools in the armamentarium of the modern obstetrician. However, ventouse may be chosen first (if there is no fetal distress) as it is significantly less likely to injure the mother.

Cervical lymph node metastasis in squamous cell carcinoma of the buccal mucosa:a retrospective study on pattern of involvement and clinical analysis

The study was performed with an aim to map the pattern of metastasis of squamous cell carcinomas of buccal mucosa to various cervical lymph node levels and analyze its correlation with primary tumor size and histo-pathological grading. 254 patients with squamous cell carcinoma of the buccal mucosa treated with surgery first approach were analyzed retrospectively. The tumor size was noted from pre-operative CT Scans and were divided into early and advanced tumors. The resected specimen was studied to note the histo-pathological grading of the squamous cell carcinoma and the metastatic deposits at various lymph node levels. Out of 254 patients (149 females, 105 males), 145 patients showed histo-pathologically proven metastatic deposits in one or more lymph nodes out of which there were 56 patients showing occult metastasis. 78/145 patients showed metastatic involvement of level IB and/or IA lymph nodes, 31 showed involvement of level II and/or I lymph nodes, 27 showed involvement of level III with or without involvement of level I and II and 9 showed metastasis to level IV and V lymph nodes with or without level I, II or III lymph nodes. Cervical lymph node metastasis had statistically significant association with tumor size with advanced tumors showing worse pattern of metastatic spread beyond level I and II lymph nodes. As the degree of differentiation of squamous cell carcinoma reduced, they were more prone for cervical metastasis with moderately and poorly differentiated squamous cell carcinoma showing higher involvement of level III, IV and V lymph nodes. The majority of buccal mucosa cases showed metastasis to level I, II and III lymph nodes out of which level IB and/or IA was most frequently involved. Metastasis to level IV and V lymph nodes was rare and was seen especially in patients with advanced primary tumor and poor histo-pathologic differentiation