Disjoint Vertex Sets Research Articles

Paired coalition in graphs

A paired coalition in a graph G = ( V , E ) consists of two disjoint sets of vertices V 1 and V 2 , neither of which is a paired dominating set but whose union V 1 ∪ V 2 is a paired dominating set. A paired coalition partition (abbreviated pc-partition) in a graph G is a vertex partition π = { V 1 , V 2 , … , V k } such that each set V i of π is not a paired dominating set but forms a paired coalition with another set V j ∈ π . The paired coalition graph PCG ( G , π ) of the graph G with the pc-partition π of G, is the graph whose vertices correspond to the sets of π , and two vertices V i and V j are adjacent in PCG ( G , π ) if and only if their corresponding sets V i and V j form a paired coalition in G. In this paper, we initiate the study of paired coalition partitions and paired coalition graphs. In particular, we determine the paired coalition number of paths and cycles, obtain some results on paired coalition partitions in trees and characterize pair coalition graphs of paths, cycles and trees. We also characterize triangle-free graphs G of order n with PC ( G ) = n and unicyclic graphs G of order n with PC ( G ) = n − 2 .

Efficient High-Quality Clustering for Large Bipartite Graphs

A bipartite graph contains inter-set edges between two disjoint vertex sets, and is widely used to model real-world data, such as user-item purchase records, author-article publications, and biological interactions between drugs and proteins. k-Bipartite Graph Clustering (k-BGC) is to partition the target vertex set in a bipartite graph into k disjoint clusters. The clustering quality is important to the utility of k-BGC in various applications like social network analysis, recommendation systems, text mining, and bioinformatics, to name a few. Existing approaches to k-BGC either output clustering results with compromised quality due to inadequate exploitation of high-order information between vertices, or fail to handle sizable bipartite graphs with billions of edges. Motivated by this, this paper presents two efficient k-BGC solutions, HOPE and HOPE+, which achieve state-of-the-art performance on large-scale bipartite graphs. HOPE obtains high scalability and effectiveness through a new k-BGC problem formulation based on the novel notion of high-order perspective (HOP) vectors and an efficient technique for low-rank approximation of HOP vectors. HOPE+ further elevates the k-BGC performance to another level with a judicious problem transformation and a highly efficient two-stage optimization framework. Two variants, HOPE+ (FNEM) and HOPE+ (SNEM) are designed when either the Frobenius norm or spectral norm is applied in the transformation. Extensive experiments, comparing HOPE and HOPE+ against 13 competitors on 10 real-world datasets, exhibit that our solutions, especially HOPE+, are superior to existing methods in terms of result quality, while being up to orders of magnitude faster. On the largest dataset MAG with 1.1 billion edges, HOPE+ is able to produce clusters with the highest clustering accuracy within 31 minutes, which is unmatched by any existing solution for k-BGC.

Scalable Approximate Butterfly and Bi-triangle Counting for Large Bipartite Networks

A bipartite graph is a graph that consists of two disjoint sets of vertices and only edges between vertices from different vertex sets. In this paper, we study the counting problems of two common types of em motifs in bipartite graphs: (i) butterflies (2x2 bicliques) and (ii) bi-triangles (length-6 cycles). Unlike most of the existing algorithms that aim to obtain exact counts, our goal is to obtain precise enough estimations of these counts in bipartite graphs, as such estimations are already sufficient and of great usefulness in various applications. While there exist approximate algorithms for butterfly counting, these algorithms are mainly based on the techniques designed for general graphs, and hence, they are less effective on bipartite graphs. Not to mention that there is still a lack of study on approximate bi-triangle counting. Motivated by this, we first propose a novel butterfly counting algorithm, called one-sided weighted sampling, which is tailored for bipartite graphs. The basic idea of this algorithm is to estimate the total butterfly count with the number of butterflies containing two randomly sampled vertices from the same side of the two vertex sets. We prove that our estimation is unbiased, and our technique can be further extended (non-trivially) for bi-triangle count estimation. Theoretical analyses under a power-law random bipartite graph model and extensive experiments on multiple large real datasets demonstrate that our proposed approximate counting algorithms can reach high accuracy, yet achieve up to three orders (resp. four orders) of magnitude speed-up over the state-of-the-art exact butterfly (resp. bi-triangle) counting algorithms. Additionally, we present an approximate clustering coefficient estimation framework for bipartite graphs, which shows a similar speed-up over the exact solutions with less than 1% relative error.

Spanning trees in graphs without large bipartite holes

AbstractWe show that for any $\varepsilon \gt 0$ and $\Delta \in \mathbb{N}$ , there exists $\alpha \gt 0$ such that for sufficiently large $n$ , every $n$ -vertex graph $G$ satisfying that $\delta (G)\geq \varepsilon n$ and $e(X, Y)\gt 0$ for every pair of disjoint vertex sets $X, Y\subseteq V(G)$ of size $\alpha n$ contains all spanning trees with maximum degree at most $\Delta$ . This strengthens a result of Böttcher, Han, Kohayakawa, Montgomery, Parczyk, and Person.

A Bipartite Graph Neural Network Approach for Scalable Beamforming Optimization

Deep learning (DL) techniques have been intensively studied for the optimization of multi-user multiple-input single-output (MU-MISO) downlink systems owing to the capability of handling nonconvex formulations. However, the fixed computation structure of existing deep neural networks (DNNs) lacks flexibility with respect to the system size, i.e., the number of antennas or users. This paper develops a bipartite graph neural network (BGNN) framework, a scalable DL solution designed for multi-antenna beamforming optimization. The MU-MISO system is first characterized by a bipartite graph where two disjoint vertex sets, each of which consists of transmit antennas and users, are connected via pairwise edges. These vertex interconnection states are modeled by channel fading coefficients. Thus, a generic beamforming optimization process is interpreted as a computation task over a weighted bipartite graph. This approach partitions the beamforming optimization procedure into multiple suboperations dedicated to individual antenna vertices and user vertices. Separated vertex operations lead to scalable beamforming calculations that are invariant to the system size. The vertex operations are realized by a group of DNN modules that collectively form the BGNN architecture. Identical DNNs are reused at all antennas and users so that the resultant learning structure becomes flexible to the network size. Component DNNs of the BGNN are trained jointly over numerous MU-MISO configurations with randomly varying network sizes. As a result, the trained BGNN can be universally applied to arbitrary MU-MISO systems. Numerical results validate the advantages of the BGNN framework over conventional methods.

Paired 3-Disjoint Path Covers in Bipartite Torus-Like Graphs with Edge Faults

Given two disjoint vertex-sets, [Formula: see text] and [Formula: see text] in a graph, a paired many-to-many [Formula: see text]-disjoint path cover joining [Formula: see text] and [Formula: see text] is a set of pairwise vertex-disjoint paths [Formula: see text] that altogether cover every vertex of the graph, in which each path [Formula: see text] runs from [Formula: see text] to [Formula: see text]. In this paper, we reveal that a bipartite torus-like graph, if built from lower dimensional torus-like graphs that have good disjoint-path-cover properties, retain such good property. As a result, an [Formula: see text]-dimensional bipartite torus, [Formula: see text], with at most [Formula: see text] edge faults has a paired many-to-many [Formula: see text]-disjoint path cover joining arbitrary disjoint sets [Formula: see text] and [Formula: see text] of size [Formula: see text] each such that [Formula: see text] contains the equal numbers of vertices from different parts of the bipartition.

Check if a Graph is Bipartite or not & Bipartite Graph Coloring using Java

Nowadays, graphs including bigraphs are mostly used in various real-world applications such as search engines and social networks. The bigraph or bipartite graph is a graph whose vertex set is split into two disjoint vertex sets such that there is no edge between the same vertex set. The bipartite graphs are colored using only two colors. This article checks if a given graph is bipartite or not and finds the color assignments of the bipartite graph using Java implementation.

Hamilton-connected, vertex-pancyclic and bipartite holes

In 2016, McDiarmid and Yolov gave a tight threshold for the existence of Hamilton cycle in graphs with large minimum degree and without large “bipartite hole”(two disjoint sets of vertices with no edge between them) which extends Dirac's classical Theorem. In detail, an (s,t)-bipartite-hole in a graph G consists of two disjoint vertex sets S and T with |S|=s and |T|=t such that E(G[S,T])=∅. Let α˜(G) be the maximum integer such that G contains an (s,t)-bipartite-hole for every pair of nonnegative integers s and t with s+t=r. Motivated by Bondy's metaconjecture, in this paper, we study the existence of vertex-pancyclicity (every vertex is in a cycle of length i for each i∈[3,n] and Hamilton-connectivity(any two vertices can be connected through a Hamilton path). Our central theorem is that for any given μ>0 and sufficiently large n, if G is an n-vertex graph with α˜(G)=μn and δ(G)≥103μn, then G is Hamilton-connected and vertex-pancyclic.

Providing Fast Reachability Query Services With MGTag: A Multi-Dimensional Graph Labeling Method

Reachability queries ask whether a vertex can reach another vertex on large directed graphs. It is one of the most fundamental graph operators and has attracted researchers in both academics and industry to study it. The main technical challenge is to support fast reachability queries by efficient managing the three main costs: the index construction time, the index size and the query processing time on large/small and sparse/dense graphs. As real world graphs grow bigger in size, these problems remain open challenges that demand high performance solutions. In this paper, we propose a Multi-Dimensional Graph Labeling approach (called MGTag) to supporting fast reachability queries. MGTag is novel in three aspects. First, it recursively partitions a graph into multiple subgraphs with disjoint vertex sets, called non-shared graphs, and several inter-partition edges, called cross-edges. Second, we build a four-dimensional label - one dimension of layer, one dimension of non-shared graph and two dimensions of interval for each vertex in non-shared graphs. Finally, with the four-dimensional labeling scheme, we design algorithms to answer reachability queries efficiently. The extensive experiments on 28 large/small and dense/sparse graphs show that building the high dimensional index is quickly and the index size is also competitive compared with most of the state-of-the-art approaches. The results also show that our approach is more scalable and efficient than the state-of-the-art approaches in answering reachability queries on large/small and sparse/dense graphs.

An average degree condition for independent transversals

In 1994, Erdős, Gyárfás and Łuczak posed the following problem: given disjoint vertex sets V1,…,Vn of size k, with exactly one edge between any pair Vi,Vj, how large can n be such that there will always be an independent transversal? They showed that the maximal n is at most (1+o(1))k2, by providing an explicit construction with these parameters and no independent transversal. They also proved a lower bound which is smaller by a 2e-factor.In this paper, we solve this problem by showing that their upper bound construction is best possible: if n≤(1−o(1))k2, there will always be an independent transversal. In fact, this result is a very special case of a much more general theorem which concerns independent transversals in arbitrary partite graphs that are ‘locally sparse’, meaning that the maximum degree between each pair of parts is relatively small. In this setting, Loh and Sudakov provided a global maximum degree condition for the existence of an independent transversal. We show that this can be relaxed to an average degree condition.We can also use our new theorem to establish tight bounds for a more general version of the Erdős–Gyárfás–Łuczak problem and solve a conjecture of Yuster from 1997. This exploits a connection to the Turán numbers of complete bipartite graphs, which might be of independent interest.

A Bipartite Graph Partition-Based Coclustering Approach With Graph Nonnegative Matrix Factorization for Large Hyperspectral Images

Clustering large hyperspectral images (HSIs) is a very challenging problem because large HSIs have high dimensionality, large spectral variability, and large computational and memory consumption. Recently, sparse subspace clustering (SSC) has achieved remarkable success in HSI clustering. However, most SSC-based methods suffer from the following bottlenecks for large HSIs: 1) high computational consumption and memory space during the construction of the similarity matrix and decomposition of the graph Laplacian matrix and 2) failure to capture the relationships among dictionary atoms, sparse coefficients, and hyperspectral pixels. To address these challenges, we propose a novel algorithm that extends SSC to cocluster large HSIs, called bipartite graph partition with graph nonnegative matrix factorization (BGP-GNMF). Specifically, to fully explore the characteristics of the spectral and spatial contexts in HSIs, we propose a novel superpixel and pixel coclustering framework with bipartite graph partitioning in the joint sparse representation domain, where superpixel-based dictionary atoms are defined as disjoint vertex sets of the bipartite graph and the joint sparsity representation is mapped into the adjacency matrix of the undirected bipartite graph. To overcome the challenges of high computational consumption and large memory space for large HSIs, the bipartite graph partition with orthonormal constrained nonnegative matrix factorization is proposed to simultaneously cluster the structured dictionary atoms and hyperspectral pixels with an indicator matrix. Finally, to exploit the intrinsic geometry of HSIs, we incorporate manifold regularization into the bipartite graph partition to improve final clustering accuracy. The effectiveness and efficiency of the proposed method are verified on three classical HSIs, and the experimental results illustrate the superiority of the proposed method compared with other state-of-the-art HSI clustering methods.

A Graph Theoretic Approach for Multi-Objective Budget Constrained Capsule Wardrobe Recommendation

Traditionally, capsule wardrobes are manually designed by expert fashionistas through their creativity and technical prowess. The goal is to curate minimal fashion items that can be assembled into several compatible and versatile outfits. It is usually a cost and time intensive process, and hence lacks scalability. Although there are a few approaches that attempt to automate the process, they tend to ignore the price of items or shopping budget. In this article, we formulate this task as a multi-objective budget constrained capsule wardrobe recommendation ( MOBCCWR ) problem. It is modeled as a bipartite graph having two disjoint vertex sets corresponding to top-wear and bottom-wear items, respectively. An edge represents compatibility between the corresponding item pairs. The objective is to find a 1-neighbor subset of fashion items as a capsule wardrobe that jointly maximize compatibility and versatility scores by considering corresponding user-specified preference weight coefficients and an overall shopping budget as a means of achieving personalization. We study the complexity class of MOBCCWR , show that it is NP-Complete, and propose a greedy algorithm for finding a near-optimal solution in real time. We also analyze the time complexity and approximation bound for our algorithm. Experimental results show the effectiveness of the proposed approach on both real and synthetic datasets.

An improved linear connectivity bound for tournaments to be highly linked

A digraph is k-linked if for any two disjoint sets of vertices {x1,…,xk} and {y1,…,yk} there are vertex disjoint paths P1,…,Pk such that Pi is directed from xi to yi for i=1,…,k. Pokrovskiy in 2015 proved that every strongly 452k-connected tournament is k-linked. In this paper, we significantly reduce this connectivity bound and show that any (40k−31)-connected tournament is k-linked.

On Triangle Estimation Using Tripartite Independent Set Queries

Estimating the number of triangles in a graph is one of the most fundamental problems in sublinear algorithms. In this work, we provide an algorithm that approximately counts the number of triangles in a graph using only polylogarithmic queries when the number of triangles on any edge in the graph is polylogarithmically bounded. Our query oracle Tripartite Independent Set (TIS) takes three disjoint sets of vertices A, B and C as inputs, and answers whether there exists a triangle having one endpoint in each of these three sets. Our query model generally belongs to the class of group queries (Ron and Tsur ACM Trans. Comput. Theory 8(4), 15, 2016; Dell and Lapinskas 2018) and in particular is inspired by the Bipartite Independent Set (BIS) query oracle of Beame et al. (2018). We extend the algorithmic framework of Beame et al., with TIS replacing BIS, for approximately counting triangles in graphs.

An approximation algorithm for the k-generalized Steiner forest problem

In this paper, we introduce the k-generalized Steiner forest (k-GSF) problem, which is a natural generalization of the k-Steiner forest problem and the generalized Steiner forest problem. In this problem, we are given a connected graph $$G =(V,E)$$ with non-negative costs $$c_{e}$$ for the edges $$e\in E$$ , a set of disjoint vertex sets $${\mathcal {V}}=\{V_{1},V_{2},\ldots ,V_{l}\}$$ and a parameter $$k\le l$$ . The goal is to find a minimum-cost edge set $$F\subseteq E$$ that connects at least k vertex sets in $${\mathcal {V}}$$ . Our main work is to construct an $$O(\sqrt{l})$$ -approximation algorithm for the k-GSF problem based on a greedy approach and an LP-rounding technique.

Efficient bi-triangle counting for large bipartite networks

A bipartite network is a network with two disjoint vertex sets and its edges only exist between vertices from different sets. It has received much interest since it can be used to model the relationship between two different sets of objects in many applications (e.g., the relationship between users and items in E-commerce). In this paper, we study the problem of efficient bi-triangle counting for a large bipartite network, where a bi-triangle is a cycle with three vertices from one vertex set and three vertices from another vertex set. Counting bi-triangles has found many real applications such as computing the transitivity coefficient and clustering coefficient for bipartite networks. To enable efficient bi-triangle counting, we first develop a baseline algorithm relying on the observation that each bi-triangle can be considered as the join of three wedges. Then, we propose a more sophisticated algorithm which regards a bi-triangle as the join of two super-wedges, where a wedge is a path with two edges while a super-wedge is a path with three edges. We further optimize the algorithm by ranking vertices according to their degrees. We have performed extensive experiments on both real and synthetic bipartite networks, where the largest one contains more than one billion edges, and the results show that the proposed solutions are up to five orders of magnitude faster than the baseline method.

Introduction to coalitions in graphs

A coalition in a graph consists of two disjoint sets of vertices V 1 and V 2, neither of which is a dominating set but whose union is a dominating set. A coalition partition in a graph G of order is a vertex partition such that every set Vi of π either is a dominating set consisting of a single vertex of degree n – 1, or is not a dominating set but forms a coalition with another set which is not a dominating set. In this paper we introduce this concept and study its properties.

Torus-like graphs and their paired many-to-many disjoint path covers

Given two disjoint vertex-sets, S={s1,…,sk} and T={t1,…,tk} in a graph, a paired many-to-manyk-disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths {P1,…,Pk} that altogether cover every vertex of the graph, in which each path Pi runs from si to ti. In this paper, we propose a family of graphs, called torus-like graphs,that include torus networks, and reveal that a torus-like graph, if built from lower dimensional torus-like graphs that have good Hamiltonian and disjoint-path-cover properties, retain such good properties. As a result, every m-dimensional nonbipartite torus, m≥2, with at most f vertex and/or edge faults has a paired many-to-many k-disjoint path cover joining arbitrary disjoint sets S and T of size k each, subject to k≥2 and f+2k≤2m. The bound 2m on f+2k is nearly optimal.

Bipartite communities via spectral partitioning

In this paper we are interested in finding communities with bipartite structure. A bipartite community is a pair of disjoint vertex sets S, $$S'$$ such that the number of edges with one endpoint in S and the other endpoint in $$S'$$ is “significantly more than expected.” This additional structure is natural to some applications of community detection. In fact, using other terminology, they have already been used to study correlation networks, social networks, and two distinct biological networks. In 2012 two groups independently [(1) Lee, Oveis Gharan, and Trevisan and (2) Louis, Raghavendra, Tetali, and Vempala] used higher eigenvalues of the normalized Laplacian to find an approximate solution to the k-sparse-cuts problem. In 2015 Liu generalized spectral methods for finding k communities to find k bipartite communities. Our approach improves the bounds on bipartite conductance (measure of strength of a bipartite community) found by Liu and also implies improvements to the original spectral methods by Lee et al. and Louis et al. We also highlight experimental results found when applying a practical algorithm derived from our theoretical results to three distinct real-world networks.

Unpaired Many-to-Many Disjoint Path Covers on Bipartite k-Ary n-Cube Networks with Faulty Elements

The [Formula: see text]-ary [Formula: see text]-cube network is known as one of the most attractive interconnection networks for parallel and distributed systems. A many-to-many [Formula: see text]-disjoint path cover ([Formula: see text]-DPC for short) of a graph is a set of [Formula: see text] vertex-disjoint paths joining two disjoint vertex sets [Formula: see text] and [Formula: see text] of equal size [Formula: see text] that altogether cover every vertex of the graph. The many-to-many [Formula: see text]-DPC is classified as paired if each source in [Formula: see text] is further required to be paired with a specific sink in [Formula: see text], or unpaired otherwise. In this paper, we consider the unpaired many-to-many [Formula: see text]-DPC problem of faulty bipartite [Formula: see text]-ary [Formula: see text]-cube networks [Formula: see text], where the sets [Formula: see text] and [Formula: see text] are chosen in different parts of the bipartition. We show that, every bipartite [Formula: see text], under the condition that [Formula: see text] or less faulty edges are removed, has an unpaired many-to-many [Formula: see text]-DPC for any [Formula: see text] and [Formula: see text] subject to [Formula: see text]. The bound [Formula: see text] is tight here.