The Complexity of Contracting Bipartite Graphs into Small Cycles

For a positive integer $\ell \geq 3$, the $C_\ell$-Contractibility problemtakes as input an undirected simple graph $G$ and determines whether $G$ can betransformed into a graph isomorphic to $C_\ell$ (the induced cycle on $\ell$vertices) using only edge contractions. Brouwer and Veldman [JGT 1987] showedthat $C_4$-Contractibility is NP-complete in general graphs. It is easy toverify that $C_3$-Contractibility is polynomial-time solvable. Dabrowski andPaulusma [IPL 2017] showed that $C_{\ell}$-Contractibility is \NP-complete\ onbipartite graphs for $\ell = 6$ and posed as open problems the status of theproblem when $\ell$ is 4 or 5. In this paper, we show that both$C_5$-Contractibility and $C_4$-Contractibility are NP-complete on bipartitegraphs.<br

The efficacy of doppler indices in predicting the neontal outcome in term preeclamptic women with intrauterine growth restriction: an observational study in a tertiary care centre

Background: Doppler flow velocimetry of the umbilical and fetal cerebral circulation is a non-invasive modality used to access the fetal well-being. Doppler is comparatively more specific and is potentially a useful tool in predicting adverse perinatal outcome in high risk cases. Objectives of this study were to evaluate the efficacy of Middle cerebral artery pulsatility index (MCA-PI), umbilical artery pulsatility index (UA-PI) and cerebroplacental ratio (CPR) doppler indices in assessment of fetal well-being. To document neonatal outcome in preeclamptic women with doppler changes.Methods: A retrospective observational study of term preeclamptic women with clinical IUGR admitting in labour room of RLJ Hospital from January 2019 to December 2019. All these women underwent Doppler study and were followed up till delivery.Results: A total 89 term preeclamptic women, 47.19% women had normal delivery, 52.81% lower segment caesarean section. 74.16 % delivered babies required NICU (neonatal ICU) care, 51.69 % babies had a longer duration of NICU care (more than 5 days). The perinatal complications like respiratory distress 8.99% low birth weight 39.33%, meconium stained 10.11%, still born 4.49% and perinatal asphyxia (6.06%). Women with abnormal MCA-PI 46.07% of cases, UA-PI in 40.45% and CPR 57.30%.Conclusions: It was observed that all three parameters CPR, MCA-PI and UA-PI when taken into account together are good utilities in predicting perinatal outcome

Dynamic Parameterized Problems

In this work, we study the parameterized complexity of various classical graph-theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. Vertex subset problems on graphs typically deal with finding a subset of vertices having certain properties that are of interest to us. In real-world applications, the graph under consideration often changes over time and due to this dynamics, the solution at hand might lose the desired properties. The goal in the area of dynamic graph algorithms is to efficiently maintain a solution under these changes. Recomputing a new solution on the new graph is an expensive task especially when the number of modifications made to the graph is significantly smaller than the size of the graph. In the context of parameterized algorithms, two natural parameters are the size k of the symmetric difference of the edge sets of the two graphs (on n vertices) and the size r of the symmetric difference of the two solutions. We study the Dynamic Pi-Deletion problem which is the dynamic variant of the Pi-Deletion problem and show NP-hardness, fixed-parameter tractability and kernelization results. For specific cases of Dynamic Pi-Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved FPT algorithms and give linear kernels. Specifically, we show that Dynamic Vertex Cover admits algorithms with running times 1.1740^k*n^{O(1)} (polynomial space) and 1.1277^k*n^{O(1)} (exponential space). Then, we show that Dynamic Feedback Vertex Set admits a randomized algorithm with 1.6667^k*n^{O(1)} running time. Finally, we consider Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set and describe algorithms with 2^k*n^{O(1)} running time improving over the known running time bounds for these problems. Additionally, for Dynamic Dominating Set and Dynamic Connected Dominating Set, we show that this is the optimal running time (up to polynomial factors) assuming the Set Cover Conjecture

Packing Arc-Disjoint 4-Cycles in Oriented Graphs

Given a directed graph G and a positive integer k, the Arc Disjoint r-Cycle Packing problem asks whether G has k arc-disjoint r-cycles. We show that, for each integer r ? 3, Arc Disjoint r-Cycle Packing is NP-complete on oriented graphs with girth r. When r is even, the same result holds even when the input class is further restricted to be bipartite. On the positive side, focusing on r = 4 in oriented graphs, we study the complexity of the problem with respect to two parameterizations: solution size and vertex cover size. For the former, we give a cubic kernel with quadratic number of vertices. This is smaller than the compression size guaranteed by a reduction to the well-known 4-Set Packing. For the latter, we show fixed-parameter tractability using an unapparent integer linear programming formulation of an equivalent problem