Clique Size Research Articles

Hitting forbidden induced subgraphs on bounded treewidth graphs

For a fixed graph H, the H-IS-Deletion problem asks, given a graph G, for the minimum size of a set S⊆V(G) such that G∖S excludes H as an induced subgraph. We are interested in determining, for a fixed H, the smallest function fH(t) such that H-IS-Deletion can be solved in time fH(t)⋅nO(1) assuming the Exponential Time Hypothesis, where t and n denote the treewidth and the number of vertices of G, respectively. We show that fH(t)=2O(th−2) for every H on h≥3 vertices, and that fH(t)=2O(t) if H is a clique or an independent set. When H deviates slightly from a clique, the function fH(t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then fH(t)=2Θ(th−2). Moreover, fH(t)=2Ω(th) when H=Kh,h, answering a question of Pilipczuk [MFCS 2011].

Splits with forbidden subgraphs

In this paper, we fix a graph H and ask into how many vertices each vertex of a clique of size n can be “split” such that the resulting graph is H-free. Formally: A graph is an (n,k)-graph if its vertex set is a pairwise disjoint union of n parts of size at most k, such that there is an edge between any two distinct parts. Letf(n,H)=min⁡{k∈N:there is an (n,k)-graph G such that H⊈G}. Barbanera and Ueckerdt [4] observed that f(n,H)=2 for any graph H that is not bipartite. If a graph H is bipartite and has a well-defined Turán exponent, i.e., ex(n,H)=Θ(nr) for some r, we show that Ω(n2/r−1)=f(n,H)=O(n2/r−1log1/r⁡n). We extend this result to all bipartite graphs for which upper and a lower Turán exponents do not differ by much. In addition, we prove that f(n,K2,t)=Θ(n1/3) for any fixed integer t≥2.

Stakeholder theory and management: Understanding longitudinal collaboration networks.

This paper explores the evolution of research collaboration networks in the 'stakeholder theory and management' (STM) discipline and identifies the longitudinal effect of co-authorship networks on research performance, i.e., research productivity and citation counts. Research articles totaling 6,127 records from 1989 to 2020 were harvested from the Web of Science Database and transformed into bibliometric data using Bibexcel, followed by applying social network analysis to compare and analyze scientific collaboration networks at the author, institution and country levels. This work maps the structure of these networks across three consecutive sub-periods (t1: 1989-1999; t2: 2000-2010; t3: 2011-2020) and explores the association between authors' social network properties and their research performance. The results show that authors collaboration network was fragmented all through the periods, however, with an increase in the number and size of cliques. Similar results were observed in the institutional collaboration network but with less fragmentation between institutions reflected by the increase in network density as time passed. The international collaboration had evolved from an uncondensed, fragmented and highly centralized network, to a highly dense and less fragmented network in t3. Moreover, a positive association was reported between authors' research performance and centrality and structural hole measures in t3 as opposed to ego-density, constraint and tie strength in t1. The findings can be used by policy makers to improve collaboration and develop research programs that can enhance several scientific fields. Central authors identified in the networks are better positioned to receive government funding, maximize research outputs and improve research community reputation. Viewed from a network's perspective, scientists can understand how collaborative relationships influence research performance and consider where to invest their decision and choices.

Imbalance parameterized by twin cover revisited

Imbalance is a classical graph layout problem with several applications, particularly in graph drawing. The imbalance of a layout σ is determined by how well-balanced the neighbors of the vertices are in σ. In the present work, we study the problem of Imbalance parameterized by the twin cover of a graph. We show that Imbalance is XP parameterized by twin cover, and FPT when parameterized by the twin cover and the size of the largest clique outside the twin cover. In contrast, we introduce a notion of succinct representations of graphs in terms of their twin cover and demonstrate that Imbalance is NP-hard in the setting of succinct representations, even for graphs that have a twin cover of size one.

Numerical experiments with LP formulations of the maximum clique problem

The maximum clique problems calls for determining the size of the largest clique in a given graph. This graph problem affords a number of zero-one linear programming formulations. In this case study we deal with some of these formulations. We consider ways for tightening the formulations. We carry out numerical experiments to see the improvements the tightened formulations provide.

An Analysis Framework for Examination Timetabling

An examination timetabling problem taken from real world universities was proposed at the International Timetabling Competition (ITC2007). The aim was to establish a common base for comparing different solution approaches. This paper presents new preprocessing methods that disclose hidden constraints and significantly increase the number of new edges that can be added to the conflict graph. Results show that the size of the maximum clique of the obtained conflict graph has been more than doubled for two instances as a result of our preprocessing. These larger cliques mean that instances can be analyzed in advance of a solution and end users gain useful information for making decisions. In addition, we have looked at the different criteria that compose the objective function, in order to provide more useful insights into the difficulty of problems in practice. We propose new integer programming formulations using clique inequalities to compute optimal solutions for 4 criteria and to obtain lower bounds for the 3 others. Results are presented and discussed for all the benchmark instances.

Perfect Nilpotent Graphs

Let R be a commutative ring with identity. The nilpotent graph of R, denoted by ΓN(R), is a graph with vertex set ZN(R)∗, and two vertices x and y are adjacent if and only if xy is nilpotent, where ZN(R) = {x ∈ R∣xy is nilpotent, for some y ∈ R∗}. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. In this paper, we characterize all rings whose ΓN(R) is perfect. In addition, it is shown that for a ring R, if R is Artinian, then ω(ΓN(R)) = χ(ΓN(R)) = |Nil(R)∗| + |Max(R)|.

Strong Cliques in Claw-Free Graphs

For a graph G, L(G)^2 is the square of the line graph of G – that is, vertices of L(G)^2 are edges of G and two edges e,fin E(G) are adjacent in L(G)^2 if at least one vertex of e is adjacent to a vertex of f and ene f. The strong chromatic index of G, denoted by s'(G), is the chromatic number of L(G)^2. A strong clique in G is a clique in L(G)^2. Finding a bound for the maximum size of a strong clique in a graph with given maximum degree is a problem connected to a famous conjecture of Erdős and Nešetřil concerning strong chromatic index of graphs. In this note we prove that a size of a strong clique in a claw-free graph with maximum degree varDelta is at most varDelta ^2 + frac{1}{2}varDelta . This result improves the only known result 1.125varDelta ^2+varDelta , which is a bound for the strong chromatic index of claw-free graphs.

ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH

In this paper, we give a criterion of the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ is Gorenstein if and only if (1) sizes of maximal cliques are constant (say $n$) and (2) (a) $n=1$, (b) $n=2$ and there is no odd cycle without chord and length at least 7 or (c) $n\geq 3$ and there is no odd cycle without chord and length at least 5.

Modeling Higher-Order Interactions in Complex Networks by Edge Product of Graphs

Abstract Many graph products have been applied to generate complex networks with striking properties observed in real-world systems. In this paper, we propose a simple generative model for simplicial networks by iteratively using edge corona product. We present a comprehensive analysis of the structural properties of the network model, including degree distribution, diameter, clustering coefficient, as well as distribution of clique sizes, obtaining explicit expressions for these relevant quantities, which agree with the behaviors found in diverse real networks. Moreover, we obtain exact expressions for all the eigenvalues and their associated multiplicities of the normalized Laplacian matrix, based on which we derive explicit formulas for mixing time, mean hitting time and the number of spanning trees. Thus, as previous models generated by other graph products, our model is also an exactly solvable one, whose structural properties can be analytically treated. More interestingly, the expressions for the spectra of our model are also exactly determined, which is sharp contrast to previous models whose spectra can only be given recursively at most. This advantage makes our model a good test bed and an ideal substrate network for studying dynamical processes, especially those closely related to the spectra of normalized Laplacian matrix, in order to uncover the influences of simplicial structure on these processes.

Using Machine Learning for Quantum Annealing Accuracy Prediction

Quantum annealers, such as the device built by D-Wave Systems, Inc., offer a way to compute solutions of NP-hard problems that can be expressed in Ising or quadratic unconstrained binary optimization (QUBO) form. Although such solutions are typically of very high quality, problem instances are usually not solved to optimality due to imperfections of the current generations quantum annealers. In this contribution, we aim to understand some of the factors contributing to the hardness of a problem instance, and to use machine learning models to predict the accuracy of the D-Wave 2000Q annealer for solving specific problems. We focus on the maximum clique problem, a classic NP-hard problem with important applications in network analysis, bioinformatics, and computational chemistry. By training a machine learning classification model on basic problem characteristics such as the number of edges in the graph, or annealing parameters, such as the D-Wave’s chain strength, we are able to rank certain features in the order of their contribution to the solution hardness, and present a simple decision tree which allows to predict whether a problem will be solvable to optimality with the D-Wave 2000Q. We extend these results by training a machine learning regression model that predicts the clique size found by D-Wave.

Social complexity and the fractal structure of group size in primate social evolution.

Compared to most other mammals and birds, anthropoid primates have unusually complex societies characterised by bonded social groups. Among primates, this effect is encapsulated in the social brain hypothesis: the robust correlation between various indices of social complexity (social group size, grooming clique size, tactical behaviour, coalition formation) and brain size. Hitherto, this has always been interpreted as a simple, unitary relationship. Using data for five different indices of brain volume from four independent brain databases, we show that the distribution of group size plotted against brain size is best described as a set of four distinct, very narrowly defined grades which are unrelated to phylogeny. The allocation of genera to these grades is highly consistent across the different data sets and brain indices. We show that these grades correspond to the progressive evolution of bonded social groups. In addition, we show, for those species that live in multilevel social systems, that the typical sizes of the different grouping levels in each case coincide with different grades. This suggests that the grades correspond to demographic attractors that are especially stable. Using five different cognitive indices, we show that the grades correlate with increasing social cognitive skills, suggesting that the cognitive demands of managing group cohesion increase progressively across grades. We argue that the grades themselves represent glass ceilings on animals' capacity to maintain social and spatial coherence during foraging and that, in order to evolve more highly bonded groups, species have to be able to invest in costly forms of cognition.

CoCoS: Fast and Accurate Distributed Triangle Counting in Graph Streams

Given a graph stream, how can we estimate the number of triangles in it using multiple machines with limited storage? Specifically, how should edges be processed and sampled across the machines for rapid and accurate estimation? The count of triangles (i.e., cliques of size three) has proven useful in numerous applications, including anomaly detection, community detection, and link recommendation. For triangle counting in large and dynamic graphs, recent work has focused largely on streaming algorithms and distributed algorithms but little on their combinations for “the best of both worlds.” In this work, we propose CoCoS , a fast and accurate distributed streaming algorithm for estimating the counts of global triangles (i.e., all triangles) and local triangles incident to each node. Making one pass over the input stream, CoCoS carefully processes and stores the edges across multiple machines so that the redundant use of computational and storage resources is minimized. Compared to baselines, CoCoS is: (a) accurate: giving up to smaller estimation error; (b) fast : up to faster, scaling linearly with the size of the input stream; and (c) theoretically sound : yielding unbiased estimates.

How do network attributes impact information virality in social networks?

PurposeSocial media platforms play a key role in information propagation and there is a need to study the same. This study aims to explore the impact of the number of close communities (represented by cliques), the size of these close communities and its impact on information virality.Design/methodology/approachThis study identified 6,786 users from over 11 million tweets for analysis using sentiment mining and network science methods. Inferential analysis has also been established by introducing multiple regression analysis and path analysis.FindingsSentiments of content did not have a significant impact on the information virality. However, there exists a stagewise development relationship between communities of close friends, user reputation and information propagation through virality.Research limitations/implicationsThis paper contributes to the theory by introducing a stagewise progression model for influencers to manage and develop their social networks.Originality/valueThere is a gap in the existing literature on the role of the number and size of cliques on information propagation and virality. This study attempts to address this gap.

The complexity of subtree intersection representation of chordal graphs and linear time chordal graph generation

It is known that any chordal graph on n vertices can be represented as the intersection of n subtrees in a tree on n nodes (Gavril in J Comb Theory 16:47–56, 1974). This characterization has been recently used to generate random chordal graphs on n vertices by generating n subtrees of a tree on n nodes. The space (and thus time) complexity of an algorithm generating n subtrees of a tree on n nodes is at least the sum of the sizes of the generated subtrees. The determination of this sum was left as an open question in Şeker et al. (Generation of random chordal graphs using subtrees of a tree. arXiv preprint arXiv:1810.13326, 2018). In this paper, we show that the sum of the sizes of n subtrees in a tree on n nodes is \(\varTheta (m\sqrt{n})\). We also show that we can confine ourselves to contraction-minimal subtree intersection representations because they are sufficient to generate every chordal graph with strictly positive probability. Moreover, the sum of the sizes of the subtrees in a contraction-minimal representation is at most \(2m+n\). We use this result to derive the first linear-time random chordal graph generator. Based on contraction-minimal representations, we also derive connectivity-related structural properties of chordal graphs. Besides these theoretical results, we also conduct experiments to study the quality of the chordal graphs generated by our algorithm and compare them to those generated by existing methods from the literature. Our algorithm does not generate chordal graphs uniformly at random, which is a quite challenging open question, irrespective of the time complexity of the generator. However, our experimental study suggests that the generated graphs have a fairly varied structure as indicated by the sizes of maximal cliques. Furthermore, our algorithm is simple to implement and produces graphs with 10,000 vertices and \(4 \times 10^7\) edges in less than one second on a laptop computer.

Improved hardness and approximation results for single allocation hub location problems

Given a metric graph G=(V,E,w) and an integer k, we aim to find a single allocation k-hub location, which is a spanning subgraph consisting of a clique of size k such that every node outside of the clique is adjacent to exactly one node inside the clique. For various objective functions studied in the literature, we present improved hardness and approximation results.

Many Cliques with Few Edges

Recently Cutler and Radcliffe proved that the graph on $n$ vertices with maximum degree at most $r$ having the most cliques is a disjoint union of $\lfloor n/(r+1)\rfloor$ cliques of size $r+1$ together with a clique on the remainder of the vertices. It is very natural also to consider this question when the limiting resource is edges rather than vertices. In this paper we prove that among graphs with $m$ edges and maximum degree at most $r$, the graph that has the most cliques of size at least two is the disjoint union of $\bigl\lfloor m \bigm/\binom{r+1}{2} \bigr\rfloor$ cliques of size $r+1$ together with the colex graph using the remainder of the edges.

Graph classes with linear Ramsey numbers

The Ramsey number RX(p,q) for a class of graphs X is the minimum n such that every graph in X with at least n vertices has either a clique of size p or an independent set of size q. We say that Ramsey numbers are linear in X if there is a constant k such that RX(p,q)≤k(p+q) for all p,q. In the present paper we conjecture that if X is a hereditary class defined by finitely many forbidden induced subgraphs, then Ramsey numbers are linear in X if and only if X excludes a forest, a disjoint union of cliques and their complements. We prove the “only if” part of this conjecture and verify the “if” part for a variety of classes. We also apply the notion of linearity to bipartite Ramsey numbers and reveal a number of similarities and differences between the bipartite and non-bipartite case.

A branch-and-cut algorithm for the Edge Interdiction Clique Problem

Given a graph G and an interdiction budget k∈N, the Edge Interdiction Clique Problem (EICP) asks to find a subset of at most k edges to remove from G so that the size of the maximum clique, in the interdicted graph, is minimized. The EICP belongs to the family of interdiction problems with the aim of reducing the clique number of the graph. The EICP optimal solutions, called optimal interdiction policies, determine the subset of most vital edges of a graph which are crucial for preserving its clique number. We propose a new set-covering-based Integer Linear Programming (ILP) formulation for the EICP with an exponential number of constraints, called the clique-covering inequalities. We design a new branch-and-cut algorithm which is enhanced by a tailored separation procedure and by an effective heuristic initialization phase. Thanks to the new exact algorithm, we manage to solve the EICP in several sets of instances from the literature. Extensive tests show that the new exact algorithm greatly outperforms the state-of-the-art approaches for the EICP.

On the complexity of coloring ‐graphs

AbstractAn ‐graph is a graph that can be partitioned into r independent sets and ℓ cliques. In the Graph Coloring problem, we are given as input a graph G, and the objective is to determine the minimum integer k such that G admits a proper vertex k‐coloring. In this work, we describe a Poly versus NP‐hard dichotomy of this problem regarding the parameters r and ℓ of ‐graphs, which determine the boundaries of the ‐hardness of Graph Coloring for such classes. We also analyze the complexity of the problem on ()‐graphs under the parameterized complexity perspective. We show that given a (2, 1)‐partition of a graph G, finding an optimal coloring of G is ‐complete even when K1 is a maximal clique of size 3; XP but W[1]‐hard when parameterized by ; fixed‐parameter tractable (FPT) and admits a polynomial kernel when parameterized by . Besides, concerning the case where K1 is a maximal clique of size 3, a P versus NPh dichotomy regarding the neighborhood of K1 is provided; furthermore, an FPT algorithm parameterized by the number of vertices having no neighbor in K1 is presented.