Decomposition Of Vertices Research Articles

Accurate assembly of multiple RNA-seq samples with Aletsch.

High-throughput RNA sequencing has become indispensable for decoding gene activities, yet the challenge of reconstructing full-length transcripts persists. Traditional single-sample assemblers frequently produce fragmented transcripts, especially in single-cell RNA-seq data. While algorithms designed for assembling multiple samples exist, they encounter various limitations. We present Aletsch, a new assembler for multiple bulk or single-cell RNA-seq samples. Aletsch incorporates several algorithmic innovations, including a "bridging" system that can effectively integrate multiple samples to restore missed junctions in individual samples, and a new graph-decomposition algorithm that leverages "supporting" information across multiple samples to guide the decomposition of complex vertices. A standout feature of Aletsch is its application of a random forest model with 50 well-designed features for scoring transcripts. We demonstrate its robust adaptability across different chromosomes, datasets, and species. Our experiments, conducted on RNA-seq data from several protocols, firmly demonstrate Aletsch's significant outperformance over existing meta-assemblers. As an example, when measured with the partial area under the precision-recall curve (pAUC, constrained by precision), Aletsch surpasses the leading assemblers TransMeta by 22.9%-62.1% and PsiCLASS by 23.0%-175.5% on human datasets. Aletsch is freely available at https://github.com/Shao-Group/aletsch. Scripts that reproduce the experimental results of this manuscript is available at https://github.com/Shao-Group/aletsch-test.

Graph theoretic properties of good sets in hypercube

A good set on k vertices is a vertex induced subgraph of the hypercube Q n that has the maximum number of edges. In this paper we discuss graph theoretic properties of good sets including connectivity, vertex decomposition, domination and identifying path covers of good sets.

Geometric vertex decomposition and liaison for toric ideals of graphs

Geometric vertex decomposition and liaison for toric ideals of graphs

Geometric Vertex Decomposition, Gröbner Bases, and Frobenius Splittings for Regular Nilpotent Hessenberg Varieties

Geometric Vertex Decomposition, Gröbner Bases, and Frobenius Splittings for Regular Nilpotent Hessenberg Varieties

On the hat guessing number of a planar graph class

The hat guessing number is a graph invariant based on a hat guessing game introduced by Winkler. Using a new vertex decomposition argument involving an edge density theorem of Erdős for hypergraphs, we show that the hat guessing number of all outerplanar graphs is less than 2125000. We also define the class of layered planar graphs, which contains outerplanar graphs, and we show that every layered planar graph has bounded hat guessing number.

The Normalized Matching Property in Random and Pseudorandom Bipartite Graphs

A bipartite graph $G(X,Y,E)$ with vertex partition $(X,Y)$ is said to have the Normalized Matching Property (NMP) if for any subset $S\subseteq X$ we have $\frac{|N(S)|}{|Y|}\geq\frac{|S|}{|X|}$. In this paper, we prove the following results about the Normalized Matching Property.
 
 The random bipartite graph $\mathbb{G}(k,n,p)$ with $|X|=k,|Y|=n$, and $k\leq n<\exp(k)$, and each pair $(x,y)\in X\times Y$ being an edge in $\mathbb{G}$ independently with probability $p$ has $p=\frac{\log n}{k}$ as the threshold for NMP. This generalizes a classic result of Erdős-Rényi on the $\frac{\log n}{n}$ threshold for the existence of a perfect matching in $\mathbb{G}(n,n,p)$.
 A bipartite graph $G(X,Y)$, with $k=|X|\le |Y|=n$, is said to be Thomason pseudorandom (following A. Thomason (Discrete Math., 1989)) with parameters $(p,\varepsilon)$ if every $x\in X$ has degree at least $pn$ and every pair of distinct $x, x'\in X$ have at most $(1+\varepsilon)p^2n$ common neighbours. We show that Thomason pseudorandom graphs have the following property: Given $\varepsilon>0$ and $n\geq k\gg 0$, there exist functions $f,g$ with $f(x), g(x)\to 0$ as $x\to 0$, and sets $\mathrm{Del}_X\subset X, \ \mathrm{Del}_Y\subset Y$ with $|\mathrm{Del}_X|\leq f(\varepsilon)k,\ |\mathrm{Del}_Y|\leq g(\varepsilon)n$ such that $G(X\setminus \mathrm{Del}_X,Y\setminus \mathrm{Del}_Y)$ has NMP. Enroute, we prove an 'almost' vertex decomposition theorem: Every Thomason pseudorandom bipartite graph $G(X,Y)$ admits - except for a negligible portion of its vertex set - a partition of its vertex set into graphs that are spanned by trees that have NMP, and which arise organically through the Euclidean GCD algorithm. 

Geometric vertex decomposition and liaison

Abstract Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this paper, we establish an explicit connection between these approaches. In particular, we show that each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height $1$ to an ideal of indeterminates and, conversely, that every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes and defining ideals of graded lower bound cluster algebras.

Connected power domination in graphs

The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from which every vertex in the graph can be observed, following a set of rules for power system monitoring. In this paper, we study the problem of finding a minimum power dominating set which is connected; the cardinality of such a set is called the connected power domination number of the graph. We show that the connected power domination number of a graph is NP-hard to compute in general, but can be computed in linear time in cactus graphs and block graphs. We also give various structural results about connected power domination, including a cut vertex decomposition and a characterization of the effects of various vertex and edge operations on the connected power domination number. Finally, we present novel integer programming formulations for power domination, connected power domination, and power propagation time, and give computational results.

Vertex decomposition method for wirelength problem and its applications to enhanced hypercube networks

In this study, the authors discuss the vertex congestion of any embedding from the guest graph into the host graph and outline a rigorous mathematical method to compute the wirelength of that embedding. Further, they show that the computation of the optimal wirelength depends on finding optimal solutions for another graph partition problem such as edge isoperimetric problem in that guest graph. On the other side, they consider an important variant of the popular hypercube network, the enhanced hypercube, and obtain the nested optimal solutions for the edge isoperimetric problem. As a combined output, they illustrate the authors' technique by embedding enhanced hypercube into a caterpillar and from that reducing the linear layout of the enhanced hypercube. As another application of their technique, they embed the hypercube as well as the enhanced hypercube on the two rows extended grid structure with optimal wirelength for the first time and showing that the existing edge congestion technique cannot be used to solve this problem.

Vertex cut method for degree and distance-based topological indices and its applications to silicate networks

We have developed rigorous mathematical and computational techniques to obtain exact analytic expressions for a number of degree and distance-based topological indices of inorganic chemical networks and nanomaterials which are newly emerging areas of reticular chemistry. Derivations of these degree and distance-based topological indices of such chemical structures fall under a larger family of partial cubes for which only limited information is currently available. In the present study this gap is filled by developing a new method based on vertex decomposition and computing the degree and distance-based topological indices for polymeric chains, cyclic and double chain silicates, silicate and oxide networks as a function of n, the order of circumscribing.

Decomposition of Directed Graphs and the Turán Problem

We consider vertex decompositions of (di)graphs appearing in the automata theory and establish some properties of these decompositions. These decompositions are applied to the problem of forbidden subgraphs.

Vertex decompositions of two-dimensional complexes and graphs

We investigate families of two-dimensional simplicial complexes defined in terms of vertex decompositions. They include nonevasive complexes, strongly collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary and Palmer. We investigate the complexity of recognition problems for those families and some of their combinatorial properties. Certain results follow from analogous decomposition techniques for graphs. For example, we prove that it is NP-complete to decide if a graph can be reduced to a discrete graph by a sequence of removals of vertices of degree 3.

Generalized Monogamy of Contextual Inequalities from the No-Disturbance Principle

In this Letter, we demonstrate that the property of monogamy of Bell violations seen for no-signaling correlations in composite systems can be generalized to the monogamy of contextuality in single systems obeying the Gleason property of no disturbance. We show how one can construct monogamies for contextual inequalities by using the graph-theoretic technique of vertex decomposition of a graph representing a set of measurements into subgraphs of suitable independence numbers that themselves admit a joint probability distribution. After establishing that all the subgraphs that are chordal graphs admit a joint probability distribution, we formulate a precise graph-theoretic condition that gives rise to the monogamy of contextuality. We also show how such monogamies arise within quantum theory for a single four-dimensional system and interpret violation of these relations in terms of a violation of causality. These monogamies can be tested with current experimental techniques.

General Framework for Optimization of Distributed Queries

The paper suggests the heuristic approach for selecting the optimal evaluation plan and Semi-join approach for reducing the communication cost. Calculating the cost of each evaluation plan of a query takes lots of computational efforts as well as time. Initially the heuristic approach is used to find best evaluation plan among various plans of a single query. By considering only joins, queries are divided into two: tree and cyclic. Semi-join is a useful tool to reduce the cost of joins. The communication cost in distributed database is mainly depending upon the amount of data transferred from one site to another site. Semi-join approach is sufficient to fully reduce the tree queries while semi-joins are not the full reducers for cyclic queries. Cyclic queries are therefore converted into tree queries using different approaches such as addition of vertices, elimination of vertices, decomposition of vertices and elimination of edges. After conversion of cyclic to tree query, semi-join approach can be used.

Vertex decompositions of sparse graphs into an independent vertex set and a subgraph of maximum degree at most 1

A graph G is (1, 0)-colorable if its vertex set can be partitioned into subsets V1 and V0 so that in G[V1] every vertex has degree at most 1, while G[V0] is edgeless. We prove that every graph with maximum average degree at most \(\tfrac{{12}} {5} \) is (1, 0)-colorable. In particular, every planar graph with girth at least 12 is (1, 0)-colorable. On the other hand, we construct graphs with the maximum average degree arbitrarily close (from above) to \(\tfrac{{12}} {5} \) which are not (1, 0)-colorable.

Connections between Classical and Parametric Network Entropies

This paper explores relationships between classical and parametric measures of graph (or network) complexity. Classical measures are based on vertex decompositions induced by equivalence relations. Parametric measures, on the other hand, are constructed by using information functions to assign probabilities to the vertices. The inequalities established in this paper relating classical and parametric measures lay a foundation for systematic classification of entropy-based measures of graph complexity.

Polygon vertex extremality and decomposition of polygons

In this paper, we show that if we decompose a polygon into two smaller polygons, then by comparing the number of extremal vertices in the original polygon versus the sum of the two smaller polygons, we can gain at most two globally extremal vertices in the smaller polygons, as well as at most two locally extremal vertices. We then will derive two discrete Four-Vertex Theorems from our results.

Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

AbstractA graph G is (k,0)‐colorable if its vertices can be partitioned into subsets V1 and V2 such that in G[V1] every vertex has degree at most k, while G[V2] is edgeless. For every integer k⩾0, we prove that every graph with the maximum average degree smaller than (3k+4)/(k+2) is (k,0)‐colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)‐colorable. On the other hand, we construct planar graphs with girth 6 that are not (k,0)‐colorable for arbitrarily large k. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:83–93, 2010

Gröbner geometry of vertex decompositions and of flagged tableaux

We relate a classic algebro-geometric degeneration technique, dating at least to Hodge 1941 (J. London Math. Soc. 16: 245–255), to the notion of vertex decompositions of simplicial complexes. The good case is when the degeneration is reduced, and we call this a geometric vertex decomposition.

Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3

The best approximation algorithm for Max Cut in graphs of maximum degree 3 uses semidefinite programming, has approximation ratio 0.9326, and its running time is Θ ( n 3.5 log n ) ; but the best combinatorial algorithms have approximation ratio 4/5 only, achieved in O ( n 2 ) time [J.A. Bondy, S.C. Locke, J. Graph Theory 10 (1986) 477–504; E. Halperin, et al., J. Algorithms 53 (2004) 169–185]. Here we present an improved combinatorial approximation, which is a 5/6-approximation algorithm that runs in O ( n 2 ) time, perhaps improvable even to O ( n ) . Our main tool is a new type of vertex decomposition for graphs of maximum degree 3.