Disjoint Cliques Research Articles

Pairwise disjoint maximal cliques in random graphs and sequential motion planning on random right angled Artin groups

The clique number of a random graph in the Erdös–Rényi model [Formula: see text] yields a random variable which takes values asymptotically almost surely (as [Formula: see text]) within one of an explicit logarithmic function [Formula: see text]. We show that random graphs have, asymptotically almost surely, arbitrarily many pairwise disjoint cliques with [Formula: see text] vertices. Such a result is motivated by, and applied to, the multi-tasking version of Farber’s topological model to study the motion planning problem in robotics. Indeed, we study the behavior of all the higher topological complexities of Eilenberg–MacLane spaces of type [Formula: see text], where [Formula: see text] is a random right angled Artin group.

Unique function words characterize genomic proteins

Between 2009 and 2016 the number of protein sequences from known species increased 10-fold from 8 million to 85 million. About 80% of these sequences contain at least one region recognized by the conserved domain architecture retrieval tool (CDART) as a sequence motif. Motifs provide clues to biological function but CDART often matches the same region of a protein by two or more profiles. Such synonyms complicate estimates of functional complexity. We do full-linkage clustering of redundant profiles by finding maximum disjoint cliques: Each cluster is replaced by a single representative profile to give what we term a unique function word (UFW). From 2009 to 2016, the number of sequence profiles used by CDART increased by 80%; the number of UFWs increased more slowly by 30%, indicating that the number of UFWs may be saturating. The number of sequences matched by a single UFW (sequences with single domain architectures) increased as slowly as the number of different words, whereas the number of sequences matched by a combination of two or more UFWs in sequences with multiple domain architectures (MDAs) increased at the same rate as the total number of sequences. This combinatorial arrangement of a limited number of UFWs in MDAs accounts for the genomic diversity of protein sequences. Although eukaryotes and prokaryotes use very similar sets of "words" or UFWs (57% shared), the "sentences" (MDAs) are different (1.3% shared).

Disjoint cliques in claw-free graphs

A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K1,3. Let s and k be two integers with 0 ≤ s ≤ k and let G be a claw-free graph of order n. In this paper, we investigate clique partition problems in claw-free graphs. It is proved that if n ≥ 3s+4(k−s) and d(x)+d(y) ≥ n−2s+2k+1 for any pair of non-adjacent vertices x, y of G, then G contains s disjoint K3s and k − s disjoint K4s such that all of them are disjoint. Moreover, the degree condition is sharp in some cases.

One more remark on the adjoint polynomial

The adjoint polynomial of G is h(G,x)=∑k=1n(−1)n−kak(G)xk,where ak(G) denotes the number of ways one can cover all vertices of the graph G by exactly k disjoint cliques of G. In this paper we show the adjoint polynomial of a graph G is a simple transformation of the independence polynomial of another graph Ĝ. This enables us to use the rich theory of independence polynomials to study the adjoint polynomials. In particular we give new proofs of several theorems of R. Liu and P. Csikvári.

Engineering Agreement: The Naming Game with Asymmetric and Heterogeneous Agents

Being popular in language evolution, cognitive science, and culture dynamics, the Naming Game has been widely used to analyze how agents reach global consensus via communications in multi-agent systems. Most prior work considered networks that are symmetric and homogeneous (e.g., vertex transitive). In this paper we consider asymmetric or heterogeneous settings that complement the current literature: 1) we show that increasing asymmetry in network topology can improve convergence rates. The star graph empirically converges faster than all previously studied graphs; 2) we consider graph topologies that are particularly challenging for naming game such as disjoint cliques or multi-level trees and ask how much extra homogeneity (random edges) is required to allow convergence or fast convergence. We provided theoretical analysis which was confirmed by simulations; 3) we analyze how consensus can be manipulated when stubborn nodes are introduced at different points of the process. Early introduction of stubborn nodes can easily influence the outcome in certain family of networks while late introduction of stubborn nodes has much less power.

The complexity of partitioning into disjoint cliques and a triangle-free graph

Motivated by Chudnovsky’s structure theorem of bull-free graphs, Abu-Khzam, Feghali, and Müller have recently proved that deciding if a graph has a vertex partition into disjoint cliques and a triangle-free graph is NP-complete for five graph classes. The problem is trivial for the intersection of these five classes. We prove that the problem is NP-complete for the intersection of two subsets of size four among the five classes. We also show NP-completeness for other small classes, such as graphs with maximum degree 4 and line graphs.

Parameterized algorithms for min–max 2-cluster editing

For a given graph and an integer t, the Min---Max 2-Clustering problem asks if there exists a modification of a given graph into two maximal disjoint cliques by inserting or deleting edges such that the number of the editing edges incident to each vertex is at most t. It has been shown that the problem can be solved in polynomial time for $$t

Community detection with the weighted parsimony criterion

Community detection in networks has been studied extensively in the last decade. Many criteria, expressing the quality of the partitions obtained, as well as a few exact algorithms and a large number of heuristics have been proposed. The parsimony criterion consists in minimizing the number of edges added or removed from the given network in order to transform it into a set of disjoint cliques. Recently Zhang, Qiu and Zhang have proposed a weighted parsimony model in which a weight coefficient is introduced to balance the numbers of inserted and deleted edges. These authors propose rules to select a good value of the coefficient, use simulated annealing to find optimal or near-optimal solutions and solve a series of real and artificial instances. In the present paper, an algorithm is proposed for solving exactly the weighted parsimony problem for all values of the parameter. This algorithm is based on iteratively solving the problem for a set of given values of the parameter using a row generation algorithm. This procedure is combined with a search procedure to find all lowest breakpoints of the value curve (i.e., the weighted sum of inserted and deleted edges). Computational results on a series of artificial and real world networks from the literature are reported. It appears that several partitions for the same network may be informative and that the set of solutions usually contains at least one intuitively appealing partition.

Partitioning a graph into disjoint cliques and a triangle-free graph

A graph G=(V,E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques (i.e., G[A] is P3-free) and B induces a triangle-free graph (i.e., G[B] is K3-free). In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be NP-complete on arbitrary graphs. Here it is proved that if a graph G is bull-free, planar, perfect, K4-free or does not contain certain holes then deciding whether G is partitionable is NP-complete. This answers an open question posed by Thomassé, Trotignon and Vušković. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs.

Asymptotic Degree Distribution of a Duplication–Deletion Random Graph Model

We study a discrete-time duplication-deletion random graph model and analyse its asymptotic degree distribution. The random graphs consists of disjoint cliques. In each time step either a new vertex is brought in with probability $0<p<1$ and attached to an existing clique, chosen with probability proportional to the clique size, or all the edges of a random vertex are deleted with probability $1-p$. We prove almost sure convergence of the asymptotic degree distribution and find its exact values in terms of a hypergeometric integral, expressed in terms of the parameter $p$. In the regime $0<p<\frac{1}{2}$ we show that the degree sequence decays exponentially at rate $\frac{p}{1-p}$, whereas it satisfies a power-law with exponent $\frac{p}{2p-1}$ if $\frac{1}{2}<p<1$. At the threshold $p=\frac{1}{2}$ the degree sequence lies between a power-law and exponential decay.

Unavoidable vertex-minors in large prime graphs

A graph is prime (with respect to the split decomposition) if its vertex set does not admit a partition (A,B) (called a split) with |A|, |B|⩾2 such that the set of edges joining A and B induces a complete bipartite graph.We prove that for each n, there exists N such that every prime graph on at least N vertices contains a vertex-minor isomorphic to either a cycle of length n or a graph consisting of two disjoint cliques of size n joined by a matching.

Convex optimization for the planted k-disjoint-clique problem

We consider the \(k\) -disjoint-clique problem. The input is an undirected graph \(G\) in which the nodes represent data items, and edges indicate a similarity between the corresponding items. The problem is to find within the graph \(k\) disjoint cliques that cover the maximum number of nodes of \(G\). This problem may be understood as a general way to pose the classical ‘clustering’ problem. In clustering, one is given data items and a distance function, and one wishes to partition the data into disjoint clusters of data items, such that the items in each cluster are close to each other. Our formulation additionally allows ‘noise’ nodes to be present in the input data that are not part of any of the cliques. The \(k\)-disjoint-clique problem is NP-hard, but we show that a convex relaxation can solve it in polynomial time for input instances constructed in a certain way. The input instances for which our algorithm finds the optimal solution consist of \(k\) disjoint large cliques (called ‘planted cliques’) that are then obscured by noise edges inserted either at random or by an adversary, as well as additional nodes not belonging to any of the \(k\) planted cliques.

Guaranteed clustering and biclustering via semidefinite programming

Identifying clusters of similar objects in data plays a significant role in a wide range of applications. As a model problem for clustering, we consider the densest $$k$$ -disjoint-clique problem, whose goal is to identify the collection of $$k$$ disjoint cliques of a given weighted complete graph maximizing the sum of the densities of the complete subgraphs induced by these cliques. In this paper, we establish conditions ensuring exact recovery of the densest $$k$$ cliques of a given graph from the optimal solution of a particular semidefinite program. In particular, the semidefinite relaxation is exact for input graphs corresponding to data consisting of $$k$$ large, distinct clusters and a smaller number of outliers. This approach also yields a semidefinite relaxation with similar recovery guarantees for the biclustering problem. Given a set of objects and a set of features exhibited by these objects, biclustering seeks to simultaneously group the objects and features according to their expression levels. This problem may be posed as that of partitioning the nodes of a weighted bipartite complete graph such that the sum of the densities of the resulting bipartite complete subgraphs is maximized. As in our analysis of the densest $$k$$ -disjoint-clique problem, we show that the correct partition of the objects and features can be recovered from the optimal solution of a semidefinite program in the case that the given data consists of several disjoint sets of objects exhibiting similar features. Empirical evidence from numerical experiments supporting these theoretical guarantees is also provided.

Graph partitions for the multidimensional assignment problem

In this paper, we consider two decomposition schemes for the graph theoretical description of the axial Multidimensional Assignment Problem (MAP). The problem is defined as finding n disjoint cliques of size m with minimum total cost in Km×n, which is an m-partite graph with n elements per dimension. Even though the 2-dimensional assignment problem is solvable in polynomial time, extending the problem to include n≥3 dimensions renders it \(\mathcal{NP}\)-hard. We propose two novel decomposition schemes for partitioning a MAP into disjoint subproblems, that can then be recombined to provide both upper and lower bounds to the original problem. For each of the partitioning schemes, we investigate and compare the efficiency of distinct exact and heuristic methodologies, namely augmentation and partitioning. Computational results for the methods, along with a hybrid one that consists of both partitioning schemes, are presented to depict the success of our approaches on large-scale instances.

ON THE NUMBER OF CLIQUES AND CYCLES IN GRAPHS

We give a new recursive method to compute the number of cliques and cycles of a graph‎. ‎This method is related‎, ‎respectively to the number of disjoint cliques in the complement graph and to the sum of permanent function over all principal minors of the adjacency matrix of the graph‎. ‎In particular‎, ‎let $G$ be a graph and let $overline {G}$ be its complement‎, ‎then given the chromatic polynomial of $overline {G}$‎, ‎we give a recursive method to compute the number of cliques of $G$‎. ‎Also given the adjacency matrix $A$ of $G$ we give a recursive method to compute the number of cycles by computing the sum of permanent function of the principal minors of $A$‎. ‎In both cases we confront to a new computable parameter which is defined as the number of disjoint cliques in $G$‎.

Hadwiger’s Conjecture and inflations of the Petersen graph

An inflation of a graph G is obtained by replacing vertices in G by disjoint cliques and adding all possible edges between any pair of cliques corresponding to adjacent vertices in G. We prove that the chromatic number of an arbitrary inflation F of the Petersen graph is equal to the chromatic number of some inflated 5-cycle contained in F. As a corollary, we find that Hadwiger’s Conjecture holds for any inflation of the Petersen graph. This solves a problem posed by Bjarne Toft.

Two remarks on the adjoint polynomial

One can define the adjoint polynomial of the graph G as follows. Let ak(G) denote the number of ways one can cover all vertices of the graph G by exactly k disjoint cliques of G. Then the adjoint polynomial of G is h(G,x)=∑k=1n(−1)n−kak(G)xk, where n denotes the number of vertices of the graph G. In this paper we show that the largest real root γ(G) of h(G,x) has the largest absolute value among the roots. We also prove that γ(G)≤4(Δ−1), where Δ denotes the largest degree of the graph G. This bound is sharp.

Approximation algorithms for clique-transversal sets and clique-independent sets in cubic graphs

A clique-transversal set S of a graph G is a subset of vertices intersecting all the cliques of G, where a clique is a complete subgraph maximal under inclusion and having at least two vertices. A clique-independent set of the graph G is a set of pairwise disjoint cliques of G. The clique-transversal number τ C ( G ) of G is the cardinality of the smallest clique-transversal set in G and the clique-independence number α C ( G ) of G is the cardinality of the largest clique-independent set in G. This paper proves that determining τ C ( G ) and α C ( G ) is NP-complete for a cubic planar graph G of girth 3. Further we propose two approximation algorithms for determining τ C ( G ) and α C ( G ) in a cubic graph G.

A 2 k kernel for the cluster editing problem

The cluster editing problem is a decision problem that, for a graph G and a parameter k, determines if one can apply at most k edge insertion/deletion operations on G so that the resulting graph becomes a union of disjoint cliques. The problem has attracted much attention because of its applications in a variety of areas. In this paper, we present a polynomial-time kernelization algorithm for the problem that produces a kernel of size bounded by 2 k. More precisely, we develop an O ( m n ) -time algorithm that, on a graph G of n vertices and m edges and a parameter k, produces a graph G ′ and a parameter k ′ such that k ′ ⩽ k , that G ′ has at most 2 k ′ vertices, and that ( G , k ) is a yes-instance if and only if ( G ′ , k ′ ) is a yes-instance of the cluster editing problem. This improves the previously best kernel of size 4 k for the problem.

The minimum spanning tree problem with conflict constraints and its variations

We consider the minimum spanning tree problem with conflict constraints (MSTC). The problem is known to be strongly NP-hard and computing even a feasible solution is NP-hard. When the underlying graph is a cactus, we show that the feasibility problem is polynomially bounded whereas the optimization version is still NP-hard. When the conflict graph is a collection of disjoint cliques, (equivalently, when the conflict relation is transitive) we observe that MSTC can be solved in polynomial time. We also identify other special cases of MSTC that can be solved in polynomial time. Exploiting these polynomially solvable special cases we derive strong lower bounds. Also, various heuristic algorithms and feasibility tests are discussed along with preliminary experimental results. As a byproduct of this investigation, we show that if an ϵ -optimal solution to the maximum clique problem can be obtained in polynomial time, then a ( 3 ϵ − 1 ) -optimal solution to the maximum edge clique partitioning (Max-ECP) problem can be obtained in polynomial time. As a consequence, we have a polynomial time approximation algorithm for the Max-ECP with performance ratio O ( n ( log log n ) 2 log 3 n ) , improving the best previously known bound of O ( n ) .