Complete Subgraphs Research Articles

Edge erasures and chordal graphs

We prove several results about chordal graphs and weighted chordal graphs by focusing on exposed edges. These are edges that are properly contained in a single maximal complete subgraph. This leads to a characterization of chordal graphs via deletions of a sequence of exposed edges from a complete graph. Most interesting is that in this context the connected components of the edge-induced subgraph of exposed edges are 2 -edge connected. We use this latter fact in the weighted case to give a modified version of Kruskal's second algorithm for finding a minimum spanning tree in a weighted chordal graph. This modified algorithm benefits from being local in an important sense.

Optimization of data allocation in hierarchical memory for blocked shortest paths algorithms

Thispaper is devoted to the reduction of data transfer between the main memory and direct mapped cache for blocked shortest paths algorithms (BSPA), which represent data by a D[M×M] matrix of blocks. For large graphs, the cache size S = δ×M2, δ < 1 is smaller than the matrix size. The cache assigns a group of main memory blocks to a single cache block. BSPA performs multiple recalculations of a block over one or two other blocks and may access up to three blocks simultaneously. If the blocks are assigned to the same cache block, conflicts occur among the blocks, which imply active transfer of data between memory levels. The distribution of blocks on groups and the block conflict count strongly depends on the allocation and ordering of the matrix blocks in main memory. To solve the problem of optimal block allocation, the paper introduces a block conflict weighted graph and recognizes two cases of block mapping: non-conflict and minimum-conflict. In first case, it formulates an equitable color-class-size constrained coloring problem on the conflict graph and solves it by developing deterministic and random algorithms. In second case, the paper formulates a problem of weighted defective color-count constrained coloring of the conflict graph and solves it by developing a random algorithm. Experimental results show that the equitable random algorithm provides an upper bound of the cache size that is very close to the lower bound estimated over the size of a complete subgraph, and show that a non-conflict matrix allocation is possible at δ = 0.5 for M = 4 and at δ = 0.1 for M = 20. For a low cache size, the weighted defective algorithm gives the number of remaining conflicts that is up to 8.8 times less than the original BSPA gives. The proposed model and algorithms are applicable to set-associative cache as well.

(p,q)-biclique counting and enumeration for large sparse bipartite graphs

In this paper, we study the problem of ( p , q)-biclique counting and enumeration for large sparse bipartite graphs. Given a bipartite G = ( U, V , E), and two integer parameters p and q, we aim to efficiently count and enumerate all (p, q)-bicliques in G , where a (p, q)-biclique B ( L, R ) is a complete subgraph of G with L ⊆ U, R ⊆ V , |L| = p, and |R| = q. The problem of (p, q)-biclique counting and enumeration has many applications, such as graph neural network information aggregation, densest subgraph detection, and cohesive subgroup analysis, etc. Despite the wide range of applications, to the best of our knowledge, we note that there is no efficient and scalable solution to this problem in the literature. This problem is computationally challenging, due to the worst-case exponential number of (p, q)-bicliques. In this paper, we propose a competitive branch-and-bound baseline method, namely BCList, which explores the search space in a depth-first manner, together with a variety of pruning techniques. Although BCList offers a useful computation framework to our problem, its worst-case time complexity is exponential to p + q. To alleviate this, we propose an advanced approach, called BCList++. Particularly, BCList++ applies a layer based exploring strategy to enumerate ( p, q )-bicliques by anchoring the search on either U or V only, which has a worst-case time complexity exponential to either p or q only. Consequently, a vital task is to choose a layer with the least computation cost. To this end, we develop a cost model, which is built upon an unbiased estimator for the density of 2-hop graph induced by U or V. To improve computation efficiency, BCList++ exploits pre-allocated arrays and vertex labeling techniques such that the frequent subgraph creating operations can be substituted by array element switching operations. We conduct extensive experiments on 16 real-life datasets, and the experimental results demonstrate that BCList++ significantly outperforms the baseline methods by up to 3 orders of magnitude. We show via a case study that (p, q)-bicliques optimize the efficiency of graph neural networks.

On EPO-groups with a given property on their conjugacy classes

Let G be a finite group and V(G) be the set of all non-central conjugacy classes of G. Consider the conjugacy class graph of G which is defined as follows: its vertex set is the set V(G) and two distinct vertices xG and yG are connected with an edge if o(x) and o(y) are not coprime. In this article, we determine the structure of groups G whose non-trivial elements have prime orders and whose graph does not have any complete subgraph with six vertices.

A Note on Stability for Maximal F-Free Graphs

Popielarz, Sahasrabuddhe and Snyder in 2018 proved that maximal K_{r+1}-free graphs with (1-frac{1}{r})frac{n^2}{2}-o(n^{frac{r+1}{r}}) edges contain a complete r-partite subgraph on n-o(n) vertices. This was very recently extended to odd cycles in place of K_3 by Wang, Wang, Yang and Yuan. We further extend it to some other 3-chromatic graphs, and obtain some other stability results along the way.

Three-way conflict analysis based on hesitant fuzzy information systems

Conflict occurs naturally in all walks of life at any time. When agents face a conflict, they usually have multiple alternatives and hesitate to make a decision. This paper studies conflict analysis in the case of hesitant fuzzy setting, i.e. the attitude of an agent on an issue has multiple choices. We first propose a conflict analysis model based on hesitant fuzzy information systems. Subsequently, we present the concepts of conflict, neutrality and alliance sets on one issue and multiple issues respectively, and introduce a method to compute the three sets via Bayesian decision theory. Moreover, the degree of conflict, neutrality and alliance among agents with respect to an issue are defined respectively. According to the conflict and alliance degrees, the set of issues is divided into three parts: the main reason, secondary reason and irrelevant reason. Finally, two methods of computing coalitions are discussed. One is based on Bayesian decision theory. The other is to find a complete subgraph of the coalition graph.

Unified Formulation of Multiagent Coordination With Relative Measurements

This article provides a unified solution to a general coordination problem of multiagent systems with relative measurements, including uncertain transformations in translation, rotation, reflection, and scale. First, we introduce relative measurements described by a matrix group expressing such transformations in a unified way. Then, we derive a necessary, and sufficient condition to achieve a coordination task by relative, distributed control according to a network topology, and a class of measurement information. Especially, a strict class of all realizable coordination tasks is characterized with an orbit associated with the transformation matrix group on measurement. Next, we show that the network topology required to coordination is the clique rigidity. Then, the clique rigidity for concrete coordination tasks is associated with conventional connectivity, e.g., connectedness and rigidity. Moreover, an intuitive condition is derived as a connectivity condition of the intersection graph of the maximal cliques (i.e., complete subgraphs). Finally, the new method is applied to formation control with unknown, heterogeneous scale factors and its effectiveness is demonstrated through simulations for both 2-D and 3-D spaces.

A local core number based algorithm for the maximum clique problem

‎The maximum clique problem (MCP) is to determine a complete subgraph of maximum cardinality in a graph‎. ‎MCP is a fundamental problem in combinatorial optimization and is noticeable for its wide range of applications‎. ‎In this paper‎, ‎we present two branch-and-bound exact algorithms for finding a maximum clique in an undirected graph‎. ‎Many efficient exact branch and bound maximum clique algorithms use approximate coloring to compute an upper bound on the clique number but‎, ‎as a new pruning strategy‎, ‎we show that local core number is more efficient‎. ‎Moreover‎, ‎instead of neighbors set of a vertex‎, ‎our search area is restricted to a subset of the set in each subproblem which speeds up clique finding process‎. ‎This subset is based on the core of the vertices of a given graph‎. ‎We improved the MCQ and MaxCliqueDyn algorithms with respect to the new pruning strategy and search area restriction‎. ‎Experimental results demonstrate that the improved algorithms outperform the previous well-known algorithms for many instances when applied to DIMACS benchmark and random graphs‎.

The Action of the Weyl Group on the E_8 Root System

Let varGamma be the graph on the roots of the E_8 root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let varGamma _c be the subgraph of varGamma consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of varGamma that are either monochromatic, or of size at most 3, or a maximal clique in varGamma _c for some color set c, or whose vertices are the vertices of a face of the E_8 root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of varGamma if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of varGamma , in terms of the restrictions of f to certain special subgraphs of K of size at most 7.

Unextendible Sets of Mutually Unbiased Basis Obtained from Complete Subgraphs

We propose a method, based on the search and identification of complete subgraphs of a regular graph, to obtain sets of Pauli operators whose eigenstates form unextendible complete sets of mutually unbiased bases of n-qubit systems. With this method we can obtain results for complete and inextensible sets of mubs for 2, 3, 4 and 5 qubits.

The biclique partitioning polytope

Given a bipartite graph G=(V1,V2,E), a biclique of G is a complete bipartite subgraph of G, and a biclique partitioning of G is a set A⊆E such that the bipartite graph G′=(V1,V2,A) is a vertex-disjoint union of bicliques. The biclique partitioning problem (BPP) consists of, given a complete bipartite graph G=(V1,V2,E) with edge weights we∈R for all e∈E (thus, negative weights are allowed), finding a biclique partitioning A⊆E of minimum weight. The bicluster editing problem (BEP) is a variant of the BPP and consists of editing a minimum number of edges of an input bipartite graph G in order to transform its edge set into a biclique partitioning. Editing an edge consists of either adding it to the graph or deleting it from the graph. In addition to the BPP and the BEP, other problems in the literature aim at finding biclique partitionings, and this motivates the study of the biclique partitioning polytope Pnm of the complete bipartite graph Knm (i.e., Pnm is the convex hull of the incidence vectors of the biclique partitionings of Knm). In this paper we develop such a polyhedral study and show that ladder, bellows, and grid inequalities induce facets of Pnm. Our computational results show that these inequalities are very effective in solving the BEP. In particular, they are able to improve the value of the relaxed solution by up to 20%.

Self-Adaptive Skeleton Approaches to Detect Self-Organized Coalitions From Brain Functional Networks Through Probabilistic Mixture Models

Detecting self-organized coalitions from functional networks is one of the most important ways to uncover functional mechanisms in the brain. Determining these raises well-known technical challenges in terms of scale imbalance, outliers and hard-examples. In this article, we propose a novel self-adaptive skeleton approach to detect coalitions through an approximation method based on probabilistic mixture models. The nodes in the networks are characterized in terms of robust k -order complete subgraphs ( k -clique ) as essential substructures. The k -clique enumeration algorithm quickly enumerates all k -cliques in a parallel manner for a given network. Then, the cliques, from max -clique down to min -clique, of each order k , are hierarchically embedded into a probabilistic mixture model. They are self-adapted to the corresponding structure density of coalitions in the brain functional networks through different order k . All the cliques are merged and evolved into robust skeletons to sustain each unbalanced coalition by eliminating outliers and separating overlaps. We call this the k -CLIque Merging Evolution (CLIME) algorithm. The experimental results illustrate that the proposed approaches are robust to density variation and coalition mixture and can enable the effective detection of coalitions from real brain functional networks. There exist potential cognitive functional relations between the regions of interest in the coalitions revealed by our methods, which suggests the approach can be usefully applied in neuroscientific studies.

Accelerating the Bron-Kerbosch Algorithm for Maximal Clique Enumeration Using GPUs

Maximal clique enumeration (MCE) is a classic problem in graph theory to identify all complete subgraphs in a graph. In prior MCE work, the Bron-Kerbosch algorithm is one of the most popular solutions, and there are several improved algorithms proposed on CPU platforms. However, while few studies have focused on the related issue of parallel implementation, recently, there have been numerous explorations of the acceleration of general purpose applications using a graphics processing unit (GPU) to reduce the computing power consumption. In this article, we develop a GPU-based Bron-Kerbosch algorithm that efficiently solves the MCE problem in parallel by optimizing the process of subproblem decomposition and computing resource usage. To speed up the computations, we use coalesced memory accesses and warp reductions to increase bandwidth and reduce memory latency. Our experimental results show that the proposed algorithm can fully exploit the resources of GPU architectures, allowing for the vast acceleration of operations to solve the MCE problem.

Generalized Paley graphs and their complete subgraphs of orders three and four

Let \(k \ge 2\) be an integer. Let q be a prime power such that \(q \equiv 1 ({\mathrm{mod}}\,\,k)\) if q is even, or, \(q \equiv 1 ({\mathrm{mod}}\,\,2k)\) if q is odd. The generalized Paley graph of order q, \(G_k(q)\), is the graph with vertex set \(\mathbb {F}_q\) where ab is an edge if and only if \({a-b}\) is a kth power residue. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in \(G_k(q)\), \(\mathcal {K}_4(G_k(q))\), which holds for all k. This generalizes the results of Evans, Pulham and Sheehan on the original (\(k=2\)) Paley graph. We also provide a formula, in terms of Jacobi sums, for the number of complete subgraphs of order three contained in \(G_k(q)\), \(\mathcal {K}_3(G_k(q))\). In both cases, we give explicit determinations of these formulae for small k. We show that zero values of \(\mathcal {K}_4(G_k(q))\) (resp. \(\mathcal {K}_3(G_k(q))\)) yield lower bounds for the multicolor diagonal Ramsey numbers \(R_k(4)=R(4,4,\ldots ,4)\) (resp. \(R_k(3)\)). We state explicitly these lower bounds for small k and compare to known bounds. We also examine the relationship between both \(\mathcal {K}_4(G_k(q))\) and \(\mathcal {K}_3(G_k(q))\), when q is prime, and Fourier coefficients of modular forms.

Research on Game Strategy of Crossing Desert

This paper analyzes the players in the game of "crossing the desert", and puts forward the optimal customs clearance strategy to help the game players reach the destination within the specified time under the condition that the remaining funds are retained as much as possible. Firstly, the redundant paths are eliminated and the complete subgraph of the game map is drawn to improve the speed of computer optimization. Then, the multiple rules in the game are transformed into the constraints in the planning model, such as load constraint, remaining fund constraint, resource margin constraint and so on. By solving the model based on adjacency matrix and mixed integer programming, the rationality of the optimization strategy is verified by an example. The waiting model is used to decompose the number of players and consider different space and time to maximize the revenue of a mine as much as possible, while reducing the basic consumption of latecomers in a certain area and the consumption of resources purchased in the same village.

Discovering the maximum k-clique on social networks using bat optimization algorithm

The k-clique problem is identifying the largest complete subgraph of size k on a network, and it has many applications in Social Network Analysis (SNA), coding theory, geometry, etc. Due to the NP-Complete nature of the problem, the meta-heuristic approaches have raised the interest of the researchers and some algorithms are developed. In this paper, a new algorithm based on the Bat optimization approach is developed for finding the maximum k-clique on a social network to increase the convergence speed and evaluation criteria such as Precision, Recall, and F1-score. The proposed algorithm is simulated in Matlab® software over Dolphin social network and DIMACS dataset for k = 3, 4, 5. The computational results show that the convergence speed on the former dataset is increased in comparison with the Genetic Algorithm (GA) and Ant Colony Optimization (ACO) approaches. Besides, the evaluation criteria are also modified on the latter dataset and the F1-score is obtained as 100% for k = 5.

Edge clique covers in graphs with independence number two

AbstractThe edge clique cover number of a graph is the size of the smallest collection of complete subgraphs whose union covers all edges of . Chen, Jacobson, Kézdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if is claw‐free, then is bounded above by its order (denoted ). Recently, Javadi and Hajebi verified this conjecture for claw‐free graphs with an independence number at least three. We study the edge clique cover number of graphs with independence number two, which are necessarily claw‐free. We give the first known proof of a linear bound in for for such graphs, improving upon the bou nd of due to Javadi, Maleki, and Omoomi. More precisely we prove that is at most the minimum of and , where is the minimum degree of . In the fractional version of the problem, we improve these upper bounds to . We also verify the conjecture for some specific subfamilies, for example, when the edge packing number with respect to cliques (a lower bound for ) equals , and when contains no induced subgraph isomorphic to where is any fixed graph of order 4.

On Numerical Invariant of Graph

Let G = (V (G), E(G)) be a finite graph with n vertices, where V (G) = {v i, … , v n} denote vertex set of G and E(G) denote an edge set G. In this paper, we have introduced the graph G p , q for G. The motivation for introducing the graph G p , q are as follows: Many mathematicians studied the properties of graph G by using boundary operator and co-boundary operator which consider only the relation between vertices and an edges of a graph G. There is no place for complete subgraph (whose vertices greater than 2) of a graph G in boundary operator and co-boundary operator of G. Hence to study the relation between complete subgraph of G and also to study the properties of G, we have introduced an edge-boundary operator and an edge-co-boundary operator on G p,q .

Recognizing Generating Subgraphs Revisited

A graph [Formula: see text] is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function [Formula: see text] is defined on its vertices. Then [Formula: see text] is [Formula: see text]well-covered if all maximal independent sets are of the same weight. For every graph [Formula: see text], the set of weight functions [Formula: see text] such that [Formula: see text] is [Formula: see text]-well-covered is a vector space, denoted as WCW(G). Deciding whether an input graph [Formula: see text] is well-covered is co-NP-complete. Therefore, finding WCW(G) is co-NP-hard. A generating subgraph of a graph [Formula: see text] is an induced complete bipartite subgraph [Formula: see text] of [Formula: see text] on vertex sets of bipartition [Formula: see text] and [Formula: see text], such that each of [Formula: see text] and [Formula: see text] is a maximal independent set of [Formula: see text], for some independent set [Formula: see text]. If [Formula: see text] is generating, then [Formula: see text] for every weight function [Formula: see text]. Therefore, generating subgraphs play an important role in finding WCW(G). The decision problem whether a subgraph of an input graph is generating is known to be NP-complete. In this article we prove NP- completeness of the problem for graphs without cycles of length 3 and 5, and for bipartite graphs with girth at least 6. On the other hand, we supply polynomial algorithms for recognizing generating subgraphs and finding WCW(G), when the input graph is bipartite without cycles of length 6. We also present a polynomial algorithm which finds WCW(G) when [Formula: see text] does not contain cycles of lengths 3, 4, 5, and 7.

Biclique Graphs of K3-free Graphs and Bipartite Graphs

A biclique of a graph is a maximal complete bipartite subgraph. The biclique graph of a graph G, KB(G), defined as the intersection graph of the bicliques of G, was introduced and characterized in 2010 by Groshaus and Szwarcfiter. However, this characterization does not lead to polynomial time recognition algorithms, and the time complexity of its recognition problem remains open. There are some works on this problem when restricted to some classes. In this work we give a characterization of the biclique graph of a K3-free graph G. We prove that KB(G) is the square graph of a particular graph which we call Mutually Included Biclique Graph of G, KBm(G). Although it does not lead to a polynomial time recognition algorithm, it gives a new tool to prove properties of biclique graphs (restricted to K3-free graphs) using known properties of square graphs. For instance we generalize a property about induced P3's in biclique graphs to a property about stars and proved a conjecture posted by Groshaus and Montero, when restricted to K3-free graphs. Also we give another characterization of the class of biclique graphs of bipartite graphs. We prove that KB(bipartite) = (IIC-comparability)2, where IIC-comparability is a subclass of comparability graphs that we call Interval Intersection Closed Comparability.