Disjoint Cliques Research Articles

Estimating the parameters of epidemic spread on two-layer random graphs: a classical and a neural network approach

In this paper, we propose and analyse various methods for estimating the infection parameter τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} in an SIR (susceptible-infected-recovered) epidemic spread model simulated on two-layer random graphs with a preferential attachment component. The statistics that can be used for the estimates is the number of susceptible and infected vertices. Our classical approach is based on the maximum likelihood method, while the machine learning approach uses a graph neural network (GNN). The underlying graph has a layer consisting of disjoint cliques (representing e.g. households) and a random second layer with a preferential attachment structure. Our simulation study reveals that the estimation is poorer at the beginning of the epidemic, for larger preferential attachment parameter of the graph, and for larger τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}. As for the neural networks, we find that dense networks offer better training datasets. Moreover, GNN perfomance is measured better using the l2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_2$$\\end{document} loss function rather than cross-entropy.

Graphs with degree sequence [formula omitted] and [formula omitted]

In this paper we study the class of graphs Gm,n that have the same degree sequence as two disjoint cliques Km and Kn, as well as the class G¯m,n of the complements of such graphs. We establish various properties of Gm,n and G¯m,n related to recognition, connectivity, diameter, bipartiteness, Hamiltonicity, and pancyclicity. We also show that several classical optimization problems on these graphs are NP-hard, while others can be solved efficiently.

Generalized Turán results for disjoint cliques

The generalized Turán number ex(n,H,F) is the largest number of copies of H in n-vertex F-free graphs. We denote by tF the vertex-disjoint union of t copies of F. Gerbner, Methuku and Vizer in 2019 determined the order of magnitude of ex(n,Ks,tKr). We extend this result in three directions. First, we determine ex(n,Ks,tKr) exactly for sufficiently large n. Second, we determine the asymptotics of the analogous number for p-uniform hypergraphs. Third, we determine the order of magnitude of ex(n,H,tKr) for every graph H, and also of the analogous number for p-uniform hypergraphs.

Bounds on the Clique and the Independence Number for Certain Classes of Graphs

In this paper, we study the class of graphs Gm,n that have the same degree sequence as two disjoint cliques Km and Kn, as well as the class G¯m,n of the complements of such graphs. The problems of finding a maximum clique and a maximum independent set are NP-hard on Gm,n. Therefore, looking for upper and lower bounds for the clique and independence numbers of such graphs is a challenging task. In this article, we obtain such bounds, as well as other related results. In particular, we consider the class of regular graphs, which are degree-equivalent to arbitrarily many identical cliques, as well as such graphs of bounded degree.

Spectral Extremal Graphs for Disjoint Cliques

Let $kK_{r+1}$ be the graph consisting of $k$ vertex-disjoint copies of the complete graph $K_{r+1}$. Moon [Canad. J. Math. 20 (1968) 95--102] and Simonovits [Theory of Graphs (Proc. colloq., Tihany, 1996)] independently showed that if $n$ is sufficiently large, then the join of a complete graph $K_{k-1}$ and an $r$-partite Turán graph $T_{n-k+1,r}$ is the unique extremal graph for $kK_{r+1}$. In this paper we consider the graph which has the maximum spectral radius among all graphs without $k$ disjoint cliques. We show that if $G$ attains the maximum spectral radius over all $n$-vertex $kK_{r+1}$-free graphs for sufficiently large $n$, then $G$ is isomorphic to the join of a complete graph $K_{k-1}$ and an $r$-partite Turán graph $T_{n-k+1,r}$.

Total tessellation cover: Bounds, hardness, and applications

The concept of graph tessellation cover was defined in the context of quantum walk models, and is a current research area in graph theory. In this work, we propose a generalization called total tessellation cover. A tessellation of a graph G is a partition of its vertex set V ( G ) into vertex disjoint cliques. A tessellation cover of G is a set of tessellations that covers its edge set E ( G ) . A total tessellation cover of G consists of a tessellation cover together with a compatible vertex coloring, such that the color of each vertex is different from the tessellation labels of the edges incident to the vertex. The total tessellation cover number T t ( G ) is the size of a minimum total tessellation cover of G . We present lower bounds T t ( G ) ≥ ω ( G ) and T t ( G ) ≥ s ( G ) + 1 , where ω ( G ) is the size of a maximum clique, and s ( G ) is the number of edges of a maximum induced star subgraph. A graph G is called a good total tessellable if T t ( G ) = ω ( G ) or T t ( G ) = s ( G ) + 1 . We study the complexity of the k - total tessellability problem, which aims to decide whether a given graph G has T t ( G ) ≤ k . We prove that k - total tessellability is in P for good total tessellable graphs. We establish the NP -completeness of the k - total tessellability when restricted to the following graph classes: bipartite graphs, line graphs of triangle-free graphs, universal graphs, ( 2 , 1 ) -chordal graphs and planar graphs.

Relationship externalities

We propose a model of network formation where agent's payoffs depend on the connected component they belong to in a way that is specific enough to be tractable yet general enough to accommodate a number of economically relevant settings. Among them are formation in the presence of contagion via links and collaboration with spillovers. A key feature of this setting is that the externalities stem from links rather than nodes. We characterize stable and efficient networks. Under negative externalities, disjoint cliques are stable and efficient. Under positive externalities complete networks and star networks are stable. Efficient networks feature a mix: pineapple networks which consist of one large clique and a star network appended to each other.

On Generalized Turán Number of Two Disjoint Cliques

On Generalized Turán Number of Two Disjoint Cliques

Cover by disjoint cliques cuts for the knapsack problem with conflicting items

The knapsack problem with conflicting items arises in several real-world applications. We propose a family of strong cutting planes and prove that a subfamily of these cuts can be facet-defining. Computational experiments show that the proposed cuts are very effective in reducing integrality gaps, providing dual bounds up to 78% tighter than formulations strengthened with traditional combinatorial cuts. We also show that it is possible to adapt a recently proposed lifting procedure to further strengthen the proposed cuts.

Total Completion Time Minimization for Scheduling with Incompatibility Cliques

This paper considers parallel machine scheduling with incompatibilities between jobs. The jobs form a graph equivalent to a collection of disjoint cliques. No two jobs in a clique are allowed to be assigned to the same machine. Scheduling with incompatibilities between jobs represents a well-established line of research in scheduling theory and the case of disjoint cliques has received increasing attention in recent years. While the research up to this point has been focused on the makespan objective, we broaden the scope and study the classical total completion time criterion. In the setting without incompatibilities, this objective is well-known to admit polynomial time algorithms even for unrelated machines via matching techniques. We show that the introduction of incompatibility cliques results in a richer, more interesting picture. We prove that scheduling on identical machines remains solvable in polynomial time, while scheduling on unrelated machines becomes APX-hard. Next, we study the problem under the paradigm of fixed-parameter tractable algorithms (FPT). In particular, we consider a problem variant with assignment restrictions for the cliques rather than the jobs. We prove that, despite still being APX-hard, it can be solved in FPT time with respect to the number of cliques. Moreover, we show that the problem on unrelated machines can be solved in FPT time for reasonable parameters, in particular, the parameter combination: maximum processing time, number of job kinds, and number of machines or maximum processing time, number of job kinds, and number of cliques. The latter results are extensions of known results for the case without incompatibilities, and can even be further extended to the case of total weighted completion time. All of the FPT results make use of n-fold Integer Programs that recently received great attention by proving their usefulness for scheduling problems.

Understanding the conformational change and inhibition of hyperthermophilic GH10 xylanase in ionic liquid

Xylan is an abundant hemicellulosic feedstock of pentose sugars present in the Lignocellulose biomass, offering massive potential for the generation of biofuel and numerous value-added by-products. The extraction of xylan is a major challenge due to the typical organization of cellulose, hemicellulose and lignin. Ionic Liquid (IL) is a promising solvent for the deconstruction of the complex lignocellulosic structure. However, the residual IL present in the pretreated xylan significantly affects xylanase activity during the hydrolysis. Thus, a molecular understanding focusing on the IL-xylanase interaction needs to be explored for developing IL tolerant xylanases for efficient xylan hydrolysis. In this study, atomistic molecular dynamics (MD) simulations were performed to investigate the IL tolerance of hyperthermophilic GH10B xylanase from Thermotoga maritima (TmXYN10B) in both aqueous and water/1-Ethyl-3-methylimidazolium acetate (EmimAc) binary solvent systems. The enzyme was stable in all solvents at high temperature (363 K). However, the opening and closing motions of the xylanase were lost in high EmimAc concentration. Besides, the formation of a considerable number of hydrogen bonds between acetate anion and amino acids at the enzyme surface might change its native motion in high EmimAc system. The IL induced conformational change was charecterized by constructing Protein Structure Network (PSN) on each snapshot from the MD simulation trajectory. Here we captured the enzyme conformational change by identifying disjoint cliques and communities at the active site in high EmimAc concentration. Furthermore, Emim cations populated the active site with higher binding occupancy and suggested to inhibit the enzymatic hydrolysis. Finally, this study provides a theoretical understanding of the reduced activity of the TmXYN10B xylanase in IL, which can pave the path for engineering xylanases for biofuel production.

The maximum number of triangles in a graph of given maximum degree

Maximizing or minimizing the number of copies of a fixed graph in a large host graph is one of the most classical topics in extremal graph theory. Indeed, one of the most famous problems in extremal graph theory, the Erdős-Rademacher problem, which can be traced back to the 1940s, asks to determine the minimum number of triangles in a graph with a given number of vertices and edges. It was conjectured that the mnimum is attained by complete multipartite graphs with all parts but one of the same size whilst the remaining part may be smaller. The problem was widely open in the regime of four or more parts until Razborov resolved the problem asymptotically in 2008 as one of the first applications of his newly developed flag algebra method. This catalyzed a line of research on the structure of extremal graphs and extensions. In particular, Reiher asymptotically solved in 2016 the conjecture of Lovász and Simonovits from the 1970s that the same graphs are also minimizers for cliques of arbitrary size. This paper deals with a problem concerning the opposite direction: _What is the maximum number of triangles in a graph with a given number $n$ of vertices and a given maximum degree $D$?_ Gan, Loh and Sudakov in 2015 conjectured that the graph maximizing the number of triangles is always a union of disjoint cliques of size $D+1$ and another clique that may be smaller, and showed that if such a graph maximizes the number of triangles, it also maximizes the number of cliques of any size $r\ge 4$. The author presents a remarkably simple and elegant argument that proves the conjecture exactly for all $n$ and $D$.

Turán problems for vertex-disjoint cliques in multi-partite hypergraphs

For two s-uniform hypergraphs H and F, the Turán number exs(H,F) is the maximum number of edges in an F-free subgraph of H. Let s,r,k,n1,…,nr be integers satisfying 2≤s≤r and n1≤n2≤⋯≤nr. De Silva, Heysse and Young determined ex2(Kn1,…,nr,kK2) and De Silva, Heysse, Kapilow, Schenfisch and Young determined ex2(Kn1,…,nr,kKr). In this paper, as a generalization of these results, we consider three Turán-type problems for k disjoint cliques in r-partite s-uniform hypergraphs. First, we consider a multi-partite version of the Erdős matching conjecture and determine exs(Kn1,…,nr(s),kKs(s)) for n1≥s3k2+sr. Then, using a probabilistic argument, we determine exs(Kn1,…,nr(s),kKr(s)) for all n1≥k. Recently, Alon and Shikhelman determined asymptotically, for all F, the generalized Turán number ex2(Kn,Ks,F), which is the maximum number of copies of Ks in an F-free graph on n vertices. Here we determine ex2(Kn1,…,nr,Ks,kKr) with n1≥k and n3=⋯=nr. Utilizing a result on rainbow matchings due to Glebov, Sudakov and Szabó, we determine ex2(Kn1,…,nr,Ks,kKr) for all n1,…,nr with n4≥rr(k−1)k2r−2.

Discrete-Time Quantum Walks on Oriented Graphs

The interest in quantum walks has been steadily increasing during the last two decades. It is still worth to present new forms of quantum walks that might find practical applications and new physical behaviors. In this work, we define discrete-time quantum walks on arbitrary oriented graphs by partitioning a graph into tessellations, which is a collection of disjoint cliques that cover the vertex set. By using the adjacency matrices associated with the tessellations, we define local unitary operators, whose product is the evolution operator of our quantum walk model. We introduce a parameter, called alpha, that quantifies the amount of orientation. We show that the parameter alpha can be tuned in order to increase the amount of quantum walk-based transport on oriented graphs.

Formal Modeling and Verification of a Distributed Algorithm for Constructing Maximal Cliques in Static Networks

Constructing maximal cliques is a well-known problem in the graph theory. Despite this problemhas been well studied for decades, few efforts have been presented in distributed computing. The main contribution of this paper is to propose a distributed algorithm, modeled by local computations, for constructing all maximal and disjoint cliques in astatic network. In order to prove the correctness of this algorithm, we use the formal Event-B method, which is based on the refinement technique. The latter consistsin enriching a model in a step by step fashion. It is the foundation of the correct by construction approach which provides an easy way to prove algorithms.

There Does Not Exist a Distance-Regular Graph with Intersection Array $$\{80, 54,12; 1, 6, 60\}$$

In this paper, we will show that there does not exist a distance-regular graph $$\Gamma $$ with intersection array $$\{80, 54,12; 1, 6, 60\}$$. We first show that a local graph $$\Delta $$ of $$\Gamma $$ does not contain a coclique with 5 vertices, and then we prove that the graph $$\Gamma $$ is geometric by showing that $$\Delta $$ consists of 4 disjoint cliques with 20 vertices. Then we apply a result of Koolen and Bang to the graph $$\Gamma $$, and we could obtain that there is no such a distance-regular graph.

The graph tessellation cover number: Chromatic bounds, efficient algorithms and hardness

A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges, and the tessellation cover number is the size of the smallest tessellation cover. These concepts are motivated by their application to quantum walk models, in special, the evolution operator of the staggered model is obtained from a graph tessellation cover. We show that the minimum between the chromatic index of the graph and the chromatic number of its clique graph, which we call chromatic upper bound, is tight with respect to the tessellation cover number for star-octahedral and windmill graphs; whereas for (3,p)-extended wheel graphs, the tessellation cover number is 3 and the chromatic upper bound is 3p. The t-tessellability problem aims to decide whether there is a tessellation cover of the graph with t tessellations. Using graph classes whose tessellation cover numbers achieve the chromatic upper bound, we obtain that t-tessellability is polynomial-time solvable for bipartite, {triangle, proper major}-free, threshold, and diamond-free K-perfect graphs; whereas is NP-complete for triangle-free for t≥3, unichord-free for t≥3, planar for t=3, biplanar for t≥3, chordal (2,1)-graphs for t≥4, (1,2)-graphs for t≥4, and diamond-free with diameter at most five for t=3. We improve the complexity of 2-tessellability problem to linear time.

SimUrb – software for identifying similar municipalities by comparing Urban indices using a graph algorithm

The paper describes the SimUrb tool, which was used to calculate the similarity between municipalities (or other territorial units) according to several characteristics (attributes). The SimUrb tool allows similar groups between ordered sequences to be found. The tool was designed to work with hundreds of records. In order to find similar groups, the tool employs graph theory, in which similar groups are represented as cliques on a simple graph. The Bron-Kerbosch algorithm was used to search for them. The degree of similarity was determined from a metric based on the Euclidean distance between strings. Finding all non-trivial cliques would have been difficult or impossible for such data to be feasible in this study. Therefore, the SimUrb tool was used to find similar groups as the largest disjoint cliques in the respective graph. A case study is introduced in the second part of the paper to illustrate SimUrb’s functionality. The results of standard grouping methods (two methods offered by ArcGIS software) and groups defined under official planning documents were compared to the results from the SimUrb software. We concluded that SimUrb can be used in many applications where the user needs to define groups of objects with the same degree of similarity.

Application of the “descent with mutations” metaheuristic to a clique partitioning problem

We study here the application of the “descent with mutations” metaheuristic to a problem arising from the field of classification and cluster analysis (dealing more precisely with the aggregation of symmetric relations) and which can be represented as a clique partitioning of a weighted graph. In this problem, we deai with a complete undirected graphe G; the edges of G have weights which can be positive, negative or equal to 0; the aim is to partition the vertices of G into disjoint cliques (whose number depends on G in order to minimize the sum of the weights of the edges with their two extremities in a same clique; this problem is NP-hard. The “descent with mutations” is a local search metaheuristic, of which the design is very simple and is based on local transformation. It consists in randomly performing random elementary transformations, irrespective improvement or worsening with respect to the objective function. We compare it with another very efficient metaheuristic, which is a simulated annealing method improved by the addition of some ingredients coming from the noising methods. Experiments show that the descent with mutations is at least as efficient for the studied problem as this improved simulated annealing, usually a little better, while it is much easier to design and to tune.

The Branch and Cut Method for the Clique Partitioning Problem

A numerical study is carried out of the branch and cut method adapted for solving the clique partitioning problem (CPP). The problem is to find a family of pairwise disjoint cliques with minimum total weight in a complete edge-weighted graph. The two particular cases of the CPP are considered: The first is known as the aggregating binary relations problem (ABRP), and the second is the graph approximation problem (GAP). For the previously known class of facet inequalities of the polytope of the problem, the cutting-plane algorithm is developed. This algorithm includes the two new basic elements: finding a solution with given guaranteed accuracy and a local search procedure to solve the problem of inequality identification. The proposed cutting-plane algorithm is used to construct lower bounds in the branch and cut method. Some special heuristics are used to search upper bounds for the exact solution. We perform a numerical experiment on randomly generated graphs. Our method makes it possible to find an optimal solution for the previously studied cases of the ABRP and for new problems of large dimension. The GAP turns out to be a more complicated case of the CPP in the computational aspect. Moreover, some simple and difficult classes of the GAPs are identified for our algorithm.