Vertex Deletion into Bipartite Permutation Graphs

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines ?? and ??, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP-complete by the classical result of Lewis and Yannakakis [John M. Lewis and Mihalis Yannakakis, 1980]. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time f(k)n^O(1), and also give a polynomial-time 9-approximation algorithm

Vertex deletion into bipartite permutation graphs

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $ℓ_{1}$ and $ℓ_{2}$, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP-complete by the classical result of Lewis and Yannakakis [20]. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time ${\mathcal {O}}(9^k \cdot n^9)$, and also give a polynomial-time 9-approximation algorithm

Vertex deletion into bipartite permutation graphs

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines \u1d4c1₁ and \u1d4c1₂, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP-complete by the classical result of Lewis and Yannakakis [John M. Lewis and Mihalis Yannakakis, 1980]. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time f(k)n^O(1), and also give a polynomial-time 9-approximation algorithm

Vertex deletion into bipartite permutation graphs

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $l_1$ and $l_2$, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP-complete by the classical result of Lewis and Yannakakis. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time $O(9^k\cdot n^9)$, and also give a polynomial-time 9-approximation algorithm.Comment: Extended abstract accepted to International Symposium on Parameterized and Exact Computation (IPEC'20

Beyond circular-arc graphs -- recognizing lollipop graphs and medusa graphs

In 1992 Bir\'{o}, Hujter and Tuza introduced, for every fixed connected graph $H$, the class of $H$-graphs, defined as the intersection graphs of connected subgraphs of some subdivision of $H$. Recently, quite a lot of research has been devoted to understanding the tractability border for various computational problems, such as recognition or isomorphism testing, in classes of $H$-graphs for different graphs $H$. In this work we undertake this research topic, focusing on the recognition problem. Chaplick, T\"{o}pfer, Voborn\'{\i}k, and Zeman showed, for every fixed tree $T$, a polynomial-time algorithm recognizing $T$-graphs. Tucker showed a polynomial time algorithm recognizing $K_3$-graphs (circular-arc graphs). On the other hand, Chaplick at al. showed that recognition of $H$-graphs is $NP$-hard if $H$ contains two different cycles sharing an edge. The main two results of this work narrow the gap between the $NP$-hard and $P$ cases of $H$-graphs recognition. First, we show that recognition of $H$-graphs is $NP$-hard when $H$ contains two different cycles. On the other hand, we show a polynomial-time algorithm recognizing $L$-graphs, where $L$ is a graph containing a cycle and an edge attached to it ($L$-graphs are called lollipop graphs). Our work leaves open the recognition problems of $M$-graphs for every unicyclic graph $M$ different from a cycle and a lollipop. Other results of this work, which shed some light on the cases that remain open, are as follows. Firstly, the recognition of $M$-graphs, where $M$ is a fixed unicyclic graph, admits a polynomial time algorithm if we restrict the input to graphs containing particular holes (hence recognition of $M$-graphs is probably most difficult for chordal graphs). Secondly, the recognition of medusa graphs, which are defined as the union of $M$-graphs, where $M$ runs over all unicyclic graphs, is $NP$-complete

Application of peripheral nerve conduits in clinical practice: A literature review

Understanding the pathomechanisms behind peripheral nerve damage and learning the course of regeneration seem to be crucial for selecting the appropriate methods of treatment. Autografts are currently the gold standard procedure in nerve reconstruction. However, due to the frequency of complications resulting from autografting and a desire to create a better environment for the regeneration of the damaged nerve, artificial conduits have become an approved alternative treatment method. The aim of this mini-review is to present the nerve scaffolds that have been applied in clinical practice to date, and the potential directions of developments in nerve conduit bioengineering. Articles regarding construction and characterization of nerve conduits were used as the theoretical background. All papers, available in PubMed database since 2000, presenting results of application of artificial nerve conduits in clinical trials were included into this mini-review. Fourteen studies including ≤10 patients and 10 trials conducted on >10 patients were analyzed as well as 24 papers focused on artificial nerve conduits per se. Taking into consideration the experiences of the authors investigating nerve conduits in clinical trials, it is essential to point out the emergence of bioresorbable scaffolds, which in the future may significantly change the treatment of peripheral nerve injuries. Also worth mentioning among the advanced conduits are hybrid conduits, which combine several modifications of a synthetic material to provide the optimal regeneration of a damaged nerve

Ray-tracer on CPU and GPU

Projekt Ray-tracer zaimplementowany jest w języku C++. W wersji na GPU jest napisany przy użyciu CUDA. Sceny do generowania są zbudowane z dużej ilości trójkątów (triangle mesh). W celu uzyskania jak najlepszego czasu działania wykorzystywana jest wielowątkowość oraz struktury danych takie jak kd-drzewa.Project Ray-tracer is implemented in C++. GPU version is written in CUDA. Generated scenes are built of many triangles (triangle mesh). In order to improve performance, multithreading and data structures such as kd-trees are used