Simultaneous Feedback Vertex Set: A Parameterized Perspective

Given a family of graphs $\mathcal{F}$, a graph $G$, and a positive integer $k$, the $\mathcal{F}$-Deletion problem asks whether we can delete at most $k$ vertices from $G$ to obtain a graph in $\mathcal{F}$. $\mathcal{F}$-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A graph $G = (V, \cup_{i=1}^{\alpha} E_{i})$, where the edge set of $G$ is partitioned into $\alpha$ color classes, is called an $\alpha$-edge-colored graph. A natural extension of the $\mathcal{F}$-Deletion problem to edge-colored graphs is the $\alpha$-Simultaneous $\mathcal{F}$-Deletion problem. In the latter problem, we are given an $\alpha$-edge-colored graph $G$ and the goal is to find a set $S$ of at most $k$ vertices such that each graph $G_i \setminus S$, where $G_i = (V, E_i)$ and $1 \leq i \leq \alpha$, is in $\mathcal{F}$. In this work, we study $\alpha$-Simultaneous $\mathcal{F}$-Deletion for $\mathcal{F}$ being the family of forests. In other words, we focus on the $\alpha$-Simultaneous Feedback Vertex Set ($\alpha$-SimFVS) problem. Algorithmically, we show that, like its classical counterpart, $\alpha$-SimFVS parameterized by $k$ is fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed constant $\alpha$. In particular, we give an algorithm running in $2^{O(\alpha k)}n^{O(1)}$ time and a kernel with $O(\alpha k^{3(\alpha + 1)})$ vertices. The running time of our algorithm implies that $\alpha$-SimFVS is FPT even when $\alpha \in o(\log n)$. We complement this positive result by showing that for $\alpha \in O(\log n)$, where $n$ is the number of vertices in the input graph, $\alpha$-SimFVS becomes W[1]-hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014)

Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion

Given a graph $G$ and a parameter $k$, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset $U\subseteq V(G)$ of size at most $k$ that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size $O(k^{161}\log^{58}k)$, and asked whether one can design a kernel of size $O(k^{10})$. While we do not completely resolve this question, we design a significantly smaller kernel of size $O(k^{12}\log^{10}k)$, inspired by the $O(k^2)$-size kernel for Feedback Vertex Set. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution

The opinion of undergraduate medical students on current curriculum and teaching methodology of pharmacology in four medical colleges of India: a questionnaire based study

Background: The objective of current study was to obtain an opinion from 2nd professional year passed medical students on current curriculum, teaching methodology and importance of pharmacology subject and to identify the area of improvement.Methods: A set questionnaire was distributed among randomly distributed to 2nd year passed 100 undergraduate (UG) students to each of four medical colleges. They were instructed to tick out the best possible option of each question on the basis of their own perceptions. They are also asked to give suggestion to improve teaching and learning of pharmacology subject.Results: Out of the 400 students, only 387 responses of students were suitable for data analysis. The majority of students 99.22% (384) were unsatisfied with the practical teaching. Teachings of preparing and dispensing types of exercises were irrelevant in today’s clinical practice according to 87.78% of the students and were in favor of the deletion of such exercises from the curriculum. The analysis showed that 62.27% of the students were the opinion that animals should not be used in experimental pharmacology. More than half of the UGs (63%) supported the use of computer assisted learning. All of the students were interested in the inclusion of case, problem and multiple choice based question discussions in the regular teaching classes followed by quizzes (31.78%) and group discussions (14.47) while small number of students (1.03%) were interested in the conduction of seminars.Conclusion: There is an urgent need to reform the curriculum and practical teaching methods for fulfilling the objective of reading pharmacology

On the Parameterized Complexity of Contraction to Generalization of Trees

For a family of graphs F, the F-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists S subseteq E(G) of size at most k such that G/S belongs to F. Here, G/S is the graph obtained from G by contracting all the edges in S. Heggernes et al.[Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied cal F-Contraction when F is a simple family of graphs such as trees and paths. In this paper, we study the F-Contraction problem, where F generalizes the family of trees. In particular, we define this generalization in a "parameterized way". Let T_ell be the family of graphs such that each graph in T_ell can be made into a tree by deleting at most ell edges. Thus, the problem we study is T_ell-Contraction. We design an FPT algorithm for T_ell-Contraction running in time O((ncol)^{O(k + ell)} * n^{O(1)}). Furthermore, we show that the problem does not admit a polynomial kernel when parameterized by k. Inspired by the negative result for the kernelization, we design a lossy kernel for T_ell-Contraction of size O([k(k + 2ell)] ^{(lceil {frac{alpha}{alpha-1}rceil + 1)}})

A Finite Algorithm for the Realizabilty of a Delaunay Triangulation

The \emph{Delaunay graph} of a point set $P \subseteq \mathbb{R}^2$ is the plane graph with the vertex-set $P$ and the edge-set that contains $\{p,p'\}$ if there exists a disc whose intersection with $P$ is exactly $\{p,p'\}$. Accordingly, a triangulated graph $G$ is \emph{Delaunay realizable} if there exists a triangulation of the Delaunay graph of some $P \subseteq \mathbb{R}^2$, called a \emph{Delaunay triangulation} of $P$, that is isomorphic to $G$. The objective of \textsc{Delaunay Realization} is to compute a point set $P \subseteq \mathbb{R}^2$ that realizes a given graph $G$ (if such a $P$ exists). Known algorithms do not solve \textsc{Delaunay Realization} as they are non-constructive. Obtaining a constructive algorithm for \textsc{Delaunay Realization} was mentioned as an open problem by Hiroshima et al.~\cite{hiroshima2000}. We design an $n^{\mathcal{O}(n)}$-time constructive algorithm for \textsc{Delaunay Realization}. In fact, our algorithm outputs sets of points with {\em integer} coordinates

Carbonization of eucalyptus wood and characterization of the properties of chars for application in metallurgy

In view of the prominent energy & environmental problems associated with the use of fossil fuels, an increasing attention is now being paid by the metallurgists for the alternate energy sources which are renewable and environment friendly in nature. Biomass (i.e. wood) obtained by fast growing trees appears to be highly beneficial and found suitable for plantations under Indian conditions. Charcoal obtained from these energy crops can be used as a reducing agent for iron-ore reduction. The aims of the present project work is i) Characterization of the properties of different components of eucalyptus tree, such as wood, bark, branch and leaves, and ii) Characterization of the physical and chemical properties of charcoals obtained at different carbonization conditions such as, temperature, heating rate and soaking time. It is found from the proximate analysis that the ash content of different components of eucalyptus wood is very low as compared to coal. It is also found that the calorific value of eucalyptus wood is higher than the other components of eucalyptus tree. The results shows that yield of char and their physical & chemical properties depends on the carbonization conditions, viz. temperature, heating rate and soaking time. The char yield decreases gradually with the increasing carbonization temperature and most of the volatilization occurs up to 800°C. The fixed carbon content increases while volatile matter decreases with increase in carbonization temperature. The calorific value of charcoals increases marginally with increase in carbonization temperature. Reactivity of Eucalyptus wood chars towards CO2 was measured and the effects of carbonization conditions were determined. It is found that the reactivity decreases with increasing carbonization temperature. Charcoals obtained under rapid carbonization were found to be more reactive than the chars obtained under slow carbonization