Cliques In Graphs Research Articles

Theoretically and Practically Efficient Maximum Defective Clique Search

The study of k -defective cliques, defined as induced subgraphs that differ from cliques by at most k missing edges, has attracted much attention in graph analysis due to their relevance in various applications, including social network analysis and implicit interaction predictions. However, determining the maximum k -defective clique in graphs has been proven to be an NP-hard problem, presenting significant challenges in finding an efficient solution. To address this problem, we develop a theoretically and practically efficient algorithm that leverages newly-designed branch reduction rules and a pivot-based branching technique. Our analysis establishes that the time complexity of the proposed algorithm is bounded by O(mγ k n ), where γ k is a real value strictly less than 2 (e.g., when k= 1, 2, and 3, γ k = 1.466, 1.755, and 1.889, respectively). To our knowledge, this algorithm achieves the best worst-case time complexity to date compared to state-of-the-art solutions. Moreover, to further reduce unnecessary branches, we propose a time-efficient upper bound-based pruning technique, which is obtained by manipulating information such as the number of distinct colors assigned to vertices and the presence of non-neighbors among them. Additionally, we employ an ordering-based heuristic approach as a preprocessing step to improve computational efficiency. Finally, we conduct extensive experiments on a diverse set of over 300 graphs to evaluate the efficiency of the proposed solutions. The results demonstrate that our algorithm achieves a speedup of 3 orders of magnitude over state-of-the-art solutions in processing most of real-world graphs.

Ant algorithms for finding weighted and unweighted maximum cliques in d-division graphs

This article deals with the problem of finding the maximum number of maximum cliques in a weighted graph with all edges between vertices from different d-division of a graph with the minimum total weight of all these cliques, and the problem of finding the maximum number of maximum cliques in a nonweighted graph with not all edges between vertices from different d-division of the graph. This article presents new ant algorithms with new desire functions for these problems. These algorithms were tested for their purpose with different changing input parameters, the test results were tabulated and discussed, the best algorithms were indicated.

Characterising clique convergence for locally cyclic graphs of minimum degree δ ≥ 6

The clique graph kG of a graph G has as its vertices the cliques (maximal complete subgraphs) of G, two of which are adjacent in kG if they have non-empty intersection in G. We say that G is clique convergent if knG≅kmG for some n≠m, and that G is clique divergent otherwise. We completely characterise the clique convergent graphs in the class of (not necessarily finite) locally cyclic graphs of minimum degree δ≥6, showing that for such graphs clique divergence is a global phenomenon, dependent on the existence of large substructures. More precisely, we establish that such a graph is clique divergent if and only if its universal triangular cover contains arbitrarily large members from the family of so-called “triangular-shaped graphs”.

Reduced Clique Graphs: A Correction to “Chordal Graphs and Their Clique Graphs”

Galinier, Habib, and Paul introduced the reduced clique graph of a chordal graph G. The nodes of the reduced clique graph are the maximal cliques of G, and two nodes are joined by an edge if and only if they form a non-disjoint separating pair of cliques in G. In this case the weight of the edge is the size of the intersection of the two cliques. A clique tree of G is a tree with the maximal cliques of G as its nodes, where for any v∈V(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v\\in V(G)$$\\end{document}, the subgraph induced by the nodes containing v is connected. Galinier et al. prove that a spanning tree of the reduced clique graph is a clique tree if and only if it has maximum weight, but their proof contains an error. We explain and correct this error. In addition, we initiate a study of the structure of reduced clique graphs by proving that they cannot contain any induced cycle of length five (although they may contain induced cycles of length three or any even integer greater than two). We show that no cycle of length four or more is isomorphic to a reduced clique graph. We prove that the class of clique graphs of chordal graphs is not comparable to the class of reduced clique graphs of chordal graphs by providing examples that are in each of these classes without being in the other.

Concatenated Nanopore DNA Codes.

In nanopore sequencers, single-stranded DNA molecules (or k-mers) enter a small opening in a membrane called a nanopore and modulate the ionic current through the pore, producing a channel output in the form of a noisy piecewise constant signal. An important problem in DNA-based data storage is finding a set of k-mers, i.e. a DNA code, that is robust against noisy sample duplication introduced by nanopore sequencers. Good DNA codes should contain as many k-mers as possible that produce distinguishable current signals (squiggles) as measured by the sequencer. The dissimilarity between squiggles can be estimated using a bound on their pairwise error probability, which is used as a metric for code design. Unfortunately, code construction using the union bound is limited to small k's due to the difficulty of finding maximum cliques in large graphs. In this paper, we construct large codes by concatenating codewords from a base code, thereby packing more information in a single strand while retaining the storage efficiency of the base code. To facilitate decoding, we include a circumfix in the base code to reduce the effect of the nanopore channel memory. We show that the decoding complexity scales as [Formula: see text], where m is the number of concatenated k-mers. Simulations show that the base code error rate is stable as m increases.

Cliques in derangement graphs for innately transitive groups

Abstract Given a permutation group 𝐺, the derangement graph of 𝐺 is the Cayley graph with connection set the derangements of 𝐺. The group 𝐺 is said to be innately transitive if 𝐺 has a transitive minimal normal subgroup. Clearly, every primitive group is innately transitive. We show that, besides an infinite family of explicit exceptions, there exists a function f : N → N f\colon\mathbb{N}\to\mathbb{N} such that, if 𝐺 is innately transitive of degree 𝑛 and the derangement graph of 𝐺 has no clique of size 𝑘, then n ≤ f ⁢ ( k ) n\leq f(k) . Motivation for this work arises from investigations on Erdős–Ko–Rado type theorems for permutation groups.

Approximation algorithms for job scheduling with block-type conflict graphs

The problem of scheduling jobs on parallel machines (identical, uniform, or unrelated), under incompatibility relation modeled as a block graph, under the makespan optimality criterion, is considered in this paper. No two jobs that are in the relation (equivalently in the same block) may be scheduled on the same machine in this model.The presented model stems from a well-established line of research combining scheduling theory with methods relevant to graph coloring. Recently, cluster graphs and their extensions like block graphs were given additional attention. We complement hardness results provided by other researchers for block graphs by providing approximation algorithms. In particular, we provide a 2-approximation algorithm for P|G=block graph|Cmax and a PTAS for the case when the jobs are unit time in addition. In the case of uniform machines, we analyze two cases. The first one is when the number of blocks is bounded, i.e. Q|G=k − block graph|Cmax. For this case, we provide a PTAS, improving upon results presented by D. Page and R. Solis-Oba. The improvement is two-fold: we allow richer graph structure, and we allow the number of machine speeds to be part of the input. Due to strong NP-hardness of Q|G=2 − clique graph|Cmax, the result establishes the approximation status of Q|G=k − block graph|Cmax. The PTAS might be of independent interest because the problem is tightly related to the NUMERICAL k-DIMENSIONAL MATCHING WITH TARGET SUMS problem. The second case that we analyze is when the number of blocks is arbitrary, but the number of cut-vertices is bounded and jobs are of unit time. In this case, we present an exact algorithm. In addition, we present an FPTAS for graphs with bounded treewidth and a bounded number of unrelated machines.The paper ends with extensive tests of the selected algorithms.

Recursion Algorithm for Highlighting the Maximum Clique of the Graph

The work considers the polynomial algorithm for calculating the maximum clique of the non orientable graph. The algorithm is based on the properties of the subspace of sections and the subspace of cycles belonging to the the space sugrapha of a non — unoriented graph. It is shown that the algorithm for the allocation of the set of maximum clicks of the graph has an exponential computational complexity. Examples of highlighting the maximum clique of the graph are considered.

The Maximum Number of Cliques in Graphs with Bounded Odd Circumference

AbstractIn this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Turán-type result is an extension of the celebrated Erdős and Gallai theorem and a strengthening of Luo’s recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.

Faster maximal clique enumeration in large real-world link streams

Link streams offer a good model for representing interactions over time. They consist of links $(b,e,u,v)$, where $u$ and $v$ are vertices interacting during the whole time interval $[b,e]$. In this paper, we deal with the problem of enumerating maximal cliques in link streams. A clique is a pair $(C,[t_0,t_1])$, where $C$ is a set of vertices that all interact pairwise during the full interval $[t_0,t_1]$. It is maximal when neither its set of vertices nor its time interval can be increased. Some main works solving this problem are based on the famous Bron-Kerbosch algorithm for enumerating maximal cliques in graphs. We take this idea as a starting point to propose a new algorithm which matches the cliques of the instantaneous graphs formed by links existing at a given time $t$ to the maximal cliques of the link stream. We prove its correctness and compute its complexity, which is better than the state-of-the art ones in many cases of interest. We also study the output-sensitive complexity, which is close to the output size, thereby showing that our algorithm is efficient. To confirm this, we perform experiments on link streams used in the state of the art, and on massive link streams, up to 100 million links. In all cases our algorithm is faster, mostly by a factor of at least 10 and up to a factor of $10^4$. Moreover, it scales to massive link streams for which the existing algorithms are not able to provide the solution.

Number of cliques of Paley-type graphs over finite commutative local rings

Number of cliques of Paley-type graphs over finite commutative local rings

A family of \(2\)-groups and an associated family of semisymmetric, locally \(2\)-arc-transitive graphs

A mixed dihedral group is a group \(H\) with two disjoint subgroups \(X\) and \(Y\), each elementary abelian of order \(2^n\), such that \(H\) is generated by \(X\cup Y\), and \(H/H'\cong X\times Y\). In this paper, for each \(n\geq 2\), we construct a mixed dihedral \(2\)-group \(H\) of nilpotency class \(3\) and order \(2^a\) where \(a=(n^3+n^2+4n)/2\), and a corresponding graph \(\Sigma\), which is the clique graph of a Cayley graph of \(H\). We prove that \(\Sigma\) is semisymmetric, that is, \({\mathop{\rm Aut}}(\Sigma)\) acts transitively on the edges but intransitively on the vertices of \(\Sigma\). These graphs are the first known semisymmetric graphs constructed from groups that are not \(2\)-generated (indeed \(H\) requires \(2n\) generators). Additionally, we prove that \(\Sigma\) is locally \(2\)-arc-transitive, and is a normal cover of the `basic' locally \(2\)-arc-transitive graph \({\rm\bf K}_{2^n,2^n}\). As such, the construction of this family of graphs contributes to the investigation of normal covers of prime-power order of basic locally \(2\)-arc-transitive graphs – the `local' analogue of a question posed by C. H. Li.

A Dynamically Programmable Quantum Photonic Microprocessor for Graph Computation

AbstractQuantum computing has grown extensively, especially in system design and development, and the current research focus has gradually evolved from validating quantum advantage to practical applications. In particular, nondeterministic‐polynomial‐time (NP)‐complete problems are central in numerous important application areas. Still, in practice, it is difficult to solved efficiently with conventional computers, limited by the exponential jump in hardness. Here, a quantum photonic microprocessor based on Gaussian boson sampling (GBS) that offers dynamic programmability to solve various graph‐related NP‐complete problems is demonstrated. The system with optical, electrical, and thermal packaging implements a GBS with 16 modes of single‐mode squeezed vacuum states, a universal programmable 16‐mode interferometer, and a single photon readout on all outputs with high accuracy, generality, and controllability. The developed system is applied to demonstrate applications in solving NP‐complete problems, manifesting the ability of photonic quantum computing to realize practical applications for conventionally intractable computations. The GBS‐based quantum photonic microprocessor is applied to solve task assignment, Boolean satisfiability, graph clique, max cut, and vertex cover. These demonstrations suggest an excellent benchmarking platform, paving the way toward large‐scale combinatorial optimization.

Some New Properties of Cyclotomic Cliques Arrangements

In this paper, we study the arrangement of lines in the euclidean plane constructed from the geometric clique graph generated by the regular -gon. The vertices of this clique arrangement are located on finite concentric circles that we call orbits. We focus especially on the number of orbits and the number of vertices inside and outside the regular -gon. Combinatorics in finite sets of quadruplets of integers provide information on the way the orbits are distributed. Next, using cyclotomic fields, we give galoisian properties of the radii of the orbits and their cardinalities. Keywords: arrangement of lines in the plane, cyclotomic fields, geometric graphs, Galois theory

Fairness-Aware Maximal Clique in Large Graphs: Concepts and Algorithms

Cohesive subgraph mining on attributed graphs is a fundamental problem in graph data analysis. Existing cohesive subgraph mining algorithms on attributed graphs do not consider the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fairness</i> of attributes in the subgraph. In this paper, we, for the first time, introduce fairness into the widely-used clique model to mine fairness-aware cohesive subgraphs. In particular, we propose three novel fairness-aware maximal clique models on attributed graphs, called weak fair clique, strong fair clique and relative fair clique, respectively. To enumerate all weak fair cliques, we develop an efficient backtracking algorithm called WFCEnum equipped with a novel colorful <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -core based pruning technique. We also propose an efficient enumeration algorithm called SFCEnum to find all strong fair cliques based on a new attribute-alternatively-selection search technique. To further improve the efficiency, we also present several non-trivial ordering techniques for both weak and strong fair clique enumerations. To enumerate all relative fair cliques, we design an enhanced colorful <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -core based pruning technique for 2D attributes, and develop two efficient search algorithms: RFCRefineEnum and RFCAlterEnum for arbitrary dimension attributes. The results of extensive experiments on four real-world graphs demonstrate the efficiency, scalability and effectiveness of the proposed algorithms.

On the Homotopy Type of the Iterated Clique Graphs of Low Degree

On the Homotopy Type of the Iterated Clique Graphs of Low Degree

Iterated Clique Reductions in Vertex Weighted Coloring for Large Sparse Graphs.

The Minimum Vertex Weighted Coloring (MinVWC) problem is an important generalization of the classic Minimum Vertex Coloring (MinVC) problem which is NP-hard. Given a simple undirected graph G=(V,E), the MinVC problem is to find a coloring s.t. any pair of adjacent vertices are assigned different colors and the number of colors used is minimized. The MinVWC problem associates each vertex with a positive weight and defines the weight of a color to be the weight of its heaviest vertices, then the goal is the find a coloring that minimizes the sum of weights over all colors. Among various approaches, reduction is an effective one. It tries to obtain a subgraph whose optimal solutions can conveniently be extended into optimal ones for the whole graph, without costly branching. In this paper, we propose a reduction algorithm based on maximal clique enumeration. More specifically our algorithm utilizes a certain proportion of maximal cliques and obtains lower bounds in order to perform reductions. It alternates between clique sampling and graph reductions and consists of three successive procedures: promising clique reductions, better bound reductions and post reductions. Experimental results show that our algorithm returns considerably smaller subgraphs for numerous large benchmark graphs, compared to the most recent method named RedLS. Also, we evaluate individual impacts and some practical properties of our algorithm. Furthermore, we have a theorem which indicates that the reduction effects of our algorithm are equivalent to that of a counterpart which enumerates all maximal cliques in the whole graph if the run time is sufficiently long.

Simplicial network analysis on EEG signals

It is well known that the epileptic signal analysis helps with automating the diagnosis and detection of a seizure, instead of relying on the traditional process of expert - visual inspection, which is both time-consuming and tedious. Recently, application of network analysis has grown as a surmounting approach for the interpretation of signals. In general, network-based signal analysis approach involves the conversion of time series procured at different physiological conditions into networks like hyper graphs, visibility graphs etc. followed by the derivation of network topological properties. In this paper, we make use of newly developed simplicial approach, where the cliques of visibility graphs are considered as simplices and are analyzed to obtain maximal cliques. Employing simplices would help in securing not only global and significant details but also localized and subtle details of dynamical behavior, for a given time series. The maximal cliques are mathematically evaluated to calculate three independent simplicial characterizers and maximum dimensionality, that define the structural anatomy and connectivity of the entire network at different topological levels. The maximum values of all the measures acquired are processed to differentiate normal signals against pathological EEG signals, using support vector machine through 10-fold cross validation. The classification analysis is performed on all the possible normal versus epileptic (ictal and inter-ictal) combinations, using two different EEG databases. In the case of University of Bonn database, the results are compared to that of conventional network parameters namely average degree, Global efficiency, Average path length and Assortativity. The classification results of both databases achieved very good accuracies, indicating that the proposed algorithm is an efficient and reliable approach for detecting epileptic EEG signals.

Supersolvable saturated matroids and chordal graphs

A matroid is supersolvable if it has a maximal chain of flats, each of which is modular. A matroid is saturated if every round flat is modular. In this article we present supersolvable saturated matroids as analogues to chordal graphs, and we show that several results for chordal graphs hold in this matroidal context. In particular, we consider matroid analogues of the reduced clique graph and clique trees for chordal graphs. The latter is a maximum-weight spanning tree of the former. We also show that the matroid analogue of a clique tree is an optimal decomposition for the matroid parameter of tree-width.

Maximality of subfields as cliques in Cayley graphs over finite fields

Maximality of subfields as cliques in Cayley graphs over finite fields