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# Finding cliques in a graph

### Clique problem - Wikipedi

1. Common formulations of the clique problem include finding a maximum clique (a clique with the largest possible number of vertices), finding a maximum weight clique in a weighted graph, listing all maximal cliques (cliques that cannot be enlarged), and solving the decision problem of testing whether a graph contains a clique larger than a given size
2. A clique is a subset of vertices of an undirected graph G such that every two distinct vertices in the clique are adjacent; that is, its induced subgraph is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs
3. •problem: finding all cliques of a graph efficiently •hard task (in terms of memory and runtime) •Bron-Kerbosch algorithm is one efficient solution •several applications - perhaps some more could be invented (operating on ontologies e.g.) Michaela Regneri Finding Cliques 30. Literature • Coen Bron and Joep Kerbosch (1973): Algorithm 457: Finding All Cliques of an Undirected Graph.
4. The cliques were defined as non-extendable groups such that each pair of persons within the group had a good relationship. The problem of finding cliques is the same as finding maximum complete subgraphs in a graph. The same sociological method was used by Schaay  . In a computer-program he listed all complete subgraphs of 3 vertices and constructe
5. # We can create a new graph where the nodes are neighborhoods and two neighborhoods are connected if # they are adjacent: I = nx.Graph() for i, n in enumerate(cycles): for j, m in enumerate(cycles[i + 1:], start=i + 1): if not n.isdisjoint(m): I.add_edge(i, j
6. Count the number of maximal cliques present in the graph and print it. Use the nx.find_cliques () function of G to find the maximal cliques. The nx.find_cliques () function returns a generator object. To count the number of maximal cliques, you need to first convert it to a list with list () and then use the len () function

A clique of size k in a graph G is a clique of graph G containing k vertices, i.e. the degree of each vertex is k-1 in that clique. So particularly, if there is a subset of k vertices that are connected to each other in the graph G, we say that graph contains a k-clique Your problem (assuming I understand your English correctly) is counting the number of cliques in a graph of size n = 50. Counting the number of cliques in a graph is #P-complete (see this paper, which shows that counting the number of independent sets in a graph is #P-complete even for bipartite graphs) ### Clique in Graphs - OpenGenus IQ: Learn Computer Scienc

Note however, that for an input graph which is itself a clique, there are (n 17) cliques of comprised of 17 vertices, and hence this trivial algorithm is optimal if we consider only worst case scaling, since simply reading a list of all such cliques takes (n 17) steps. In order to do better than this, you need a graph with some structure All the vertices whose degree is greater than or equal to (K-1) are found and checked which subset of K vertices form a clique. When another edge is added to the present list, it is checked if by adding that edge, the list still forms a clique or not. The following steps can be followed to compute the result

The optimal solution is like this because in a complete graph there are 2^n cliques. Consider all subsets of nodes using a recursive function. And for each subset if all the edges are present between nodes of the subset, add 1 to your counter: (This is almost a pseudocode in C++ graphs, an approximation of the k-clique densest subgraph , as well as, for finding quasi-cliques. Finally, we are able to estimate the accuracy of the approach proposed in  in approximating the number of k-cliques, for those graphs whose exact count of k-cliques was not known before our work. The rest of the paper is organized as. What is a clique? A clique in graph theory is an interesting concept with a lot of depth to explore. We define the term and give some examples in today's mat..

However, graph data tends to be uncertain in practice due to noise, incompleteness and inaccuracy. This paper investigates the problem of finding top-k maximal cliques in an uncertain graph. A new model of uncertain graphs is presented, and an intuitive measure is introduced to evaluate the significance of vertex sets. An optimized branch-and. Finding the Maximum Clique in Massive Graphs Can Lu, Jeffrey Xu Yu, Hao Wei, Yikai Zhang The Chinese University of Hong Kong, Hong Kong, China flucan,yu,hwei,ykzhangg@se.cuhk.edu.hk 1. ABSTRACT Cliques refer to subgraphs in an undirected graph such that ver-tices in each subgraph are pairwise adjacent. The maximum clique problem, to ﬁnd the clique with most vertices in a given graph, has. For the number of maximal cliques, take the complement of a disjoint union of triangles. Since the number of maximal independent sets is exactly the same (in the complement), you can count the number of maximal independent set in a graph that is a disjoint union of triangles. This number is 3 n / 3 (Moon and Moser, 1965) What are maximum cliques and maximal cliques in graph theory? We'll be defining both terms in today's video graph theory lesson, as well as going over an exa.. maximal clique problem is the problem of finding in a given graph the clique with the largest number of vertices. For finding a maximal clique in a graph is an NP-hard problem, and itifficultis d to obtain the exact solution efficiently. It is also difficult to obtain even a satisfactory approximate solution. Nevertheless, many practical problems can b

cliques find all complete subgraphs in the input graph, obeying the size limitations given in the min and max arguments. largest_cliques finds all largest cliques in the input graph. A clique is largest if there is no other clique including more vertices. max_cliques finds all maximal cliques in the input graph Abstract. The problem of finding a maximum clique in a graph is prototypical for many clustering and similarity problems; however, in many real-world scenarios, the classical problem of finding a complete subgraph needs to be relaxed to finding an almost complete subgraph, a so-called quasi-clique.In this work, we demonstrate how two previously existing definitions of quasi-cliques can be.

Details. cliques find all complete subgraphs in the input graph, obeying the size limitations given in the min and max arguments.. largest_cliques finds all largest cliques in the input graph. A clique is largest if there is no other clique including more vertices. max_cliques finds all maximal cliques in the input graph. A clique in maximal if it cannot be extended to a larger clique To obtain a list of cliques, use list (find_cliques (G)). Based on the algorithm published by Bron & Kerbosch (1973) [R198] as adapated by Tomita, Tanaka and Takahashi (2006) [R199] and discussed in Cazals and Karande (2008) [R200]. The method essentially unrolls the recursion used in the references to avoid issues of recursion stack depth Finding cliques in a given graph is an important procedure in discrete mathematical modeling. The paper will show how concepts such as splitting partitions, quasi coloring, node and edge dominance. This is a graph: This is a clique (a set of vertices where everything is connected to everything): These are the maximal cliques (cliques that cannot be expanded more): The problem asks to find the maximal cliques in an undirected graph. The interest seems to be mainly scientific, but there could be useful real-world applications as well

This paper develops a family of algorithms that are variations on the Bron-Kerbosch algorithm for finding all the cliques of a simple undirected graph. The algorithms are developed in a stepwise manner, from a recursive algorithm for generating all combinations of zero or more objects chosen from N objects. Experimental results are given So below are two steps to find if graph can be divided in two Cliques or not. Find complement of Graph. Below is complement graph is above shown graph. In complement, all original edges are removed. And the vertices which did not have an edge between them, now have an edge connecting them. Return true if complement is Bipartite, else false. Above shown graph is Bipartite. Checking whether a.

In this thesis, the problem of finding cliques has addressed in large graphs such as social networks. The problem of finding all cliques in a graph is known to be NP problem. A new heuristic approach for finding cliques has provided. The new approach represents a graph coloring technique which is based on the using of Largest Degree Coloring algorithm, which colors a graph starting with the. Maximum Clique Problem: For a given undirected graph G find a maximum clique of G Finding all cliques of an undirected graph. Comm. ACM, 16 (9) (1973), pp. 575-577. CrossRef View Record in Scopus Google Scholar. 11. R Garraghan, P.M Pardalos. An exact algorithm for the maximum clique problem. Oper. Res. Lett., 9 (1990), pp. 375-382. Google Scholar. 12. M.R Garey, D.S Johnson. Computers and. For a graph , a clique is a set such that all pairs of vertices in are adjacent in .Determining the maximum size of a clique in a graph is a classic NP-Complete problem, but sometimes you just need to find or count cliques anyway.. Today, we discuss the cliquer algorithm, as designed by Niskanen and Östergård. Their very efficient implementation is available on their cliquer homepage Eine Clique bezeichnet in der Graphentheorie eine Teilmenge von Knoten in einem ungerichteten Graphen, bei der jedes Knotenpaar durch eine Kante verbunden ist. Zu entscheiden, ob ein Graph eine Clique einer bestimmten Mindestgröße enthält, wird Cliquenproblem genannt und gilt, wie das Finden von größten Cliquen, als algorithmisch schwierig (NP-vollständig) Keyword Search in Graphs: Finding r-cliques Mehdi Kargar and Aijun An Department of Computer Science and Engineering York University, Toronto, Canada fkargar,aang@cse.yorku.ca ABSTRACT Keyword search over a graph nds a substructure of the graph containing all or some of the input keywords. Most of previous methods in this area nd connected minimal trees that cover all the query keywords. Description bttroductian. A maximal complete subgraph (clique) is a complete subgraph that is not contained in any other complete subgraph. A recent paper  describes a number of techniques to find maximal complete subgraphs of a given undirected graph. In this paper, we present two backtracking algorithms, using a branchand-bound technique  to cut off branches that cannot lead to a clique Finding Maximal Cliques in Massive Networks by H*-graph James Cheng School of Computer Engineering Nanyang Technological University, Singapore j.cheng@acm.org Yiping Ke Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong ypke@se.cuhk.edu.hk Ada Wai-Chee Fu Department of Computer Science and Engineering The Chinese University of Hong Kong adafu@cse.

of maximal cliques within a graph; on a graph with n vertices, the largest possible number of maximal cliques is 3n 3 1. For the case of uncertain graphs, we present the ﬁrst matching upper and lower bounds for the largest number of a-maximal cliques in a graph on n vertices. We show that for any 0 < a < 1, the maximum number of a-maximal cliques possible in an uncertain graph is n bn=2c, i. Finding cliques. Hello Tiago! Thanks for the great library. I use a function triadic_census to find all combinations of subgraphs based on the clique of size 3: K3. What can you propose, if I.. Finding Cliques in a Graph Description of the problem. Today's post is about finding cliques in a graph: A clique in an undirected graph G is a subgraph which is complete. One specific algorithm on finding cliques is a family of algorithms called Bron-Kerbosch Algrithm(s).They were simple enough to implement

Maximal cliques are cliques that cannot be extended by adding an adjacent edge, and are a useful property of the graph when finding communities. NetworkX provides a function that allows you to identify the nodes involved in each maximal clique in a graph: nx.find_cliques(G) Various algorithms for finding cliques of graphs were developed by Bierstone  and Mulligan  and presented in an analysis of clustering techniques by Augustson and Minker . The Bierstone  algorithm has errors and it was corrected by Mulligan and Corneil . In  Coen Bron and Joep Kerboscht presented two versions of backtracking algorithms, using a branch- and-bound technique to. The classical problem of finding a clique of largest cardinality in an arbitrary graph is NP-complete. For that reason earlier work diverges into two directions. The first concerns algorithms solving the problem for arbitrary graphs in reasonable (but exponential) time, the other restricts to special classes of graphs where polynomial methods can be found. Here, the two directions are combined. Find all the k-cliques in an undirected graph. astarSearch: Compute astarSearch for a graph bandwidth: Compute bandwidth for an undirected graph bccluster: Graph clustering based on edge betweenness centrality bellman.ford.sp: Bellman-Ford shortest paths using boost C++ betweenness: Compute betweenness centrality for an undirected graph bfs: Breadth and Depth-first searc Cliques in a graph correspond to independent sets in its complement. And the largest clique, a maximum clique, would correspond to the largest, the maximum independent set in its complement. So the clique number of a graph equals the independence number of its complement. By now a new notation, little omega of G equals alpha of the complement of G. Explore our Catalog Join for free and get.

### Finding cliques (I) Pytho

We recently discussed cliquer, a fast algorithm for finding and counting maximum cliques in a graph. Today, I would like to discuss a modification of this algorithm when applied to graphs that have a very restrictive form of symmetry. Specifically, we will discuss circulant graphs and interval subgraphs of distance graphs. These graphs are particularl cliques within a graph; on a graph with n vertices, the larg-est possible number of maximal cliques is 3n3.1 For the case of uncertain graphs, we present the ﬁrst matching upper and lower bounds for the largest number of a-maximal cli-ques in a graph on n vertices. We show that for any 0 < a < 1, the maximum number of a-maximal cliques possible in an uncertain graph is ðn bn=2cÞ, i.e. The maximum clique problem (MCP) is to find a maximum clique in a given graph G. all cliques in a graph. Some of these methods are called vertex sequence methods, which produce the cliques of G from the cliques of G\{v}. Other algorithms are based on backtracking method, for example the algorithm proposed by Bron and Kerbosch. Branch and Bound Algorithms Branch and Bound Algorithms have. In more technical terms we are looking for cliques in the graph. A clique is a set of nodes which are all connected to each other. Actually to be more precise we are looking for the maximum clique in the graph to get the lower bound. What we later do is to actually search for all maximal cliques. A clique is maximal if we can't add another node to it to make it bigger i.e 1,2 and 8 form a. Finding cliques in a given graph is an important procedure in discrete mathematical modeling. The paper will show how concepts such as splitting partitions, quasi coloring, node and edge dominance are related to clique search problems. In particular we will discuss the connection with parallel clique search algorithms. These concepts also suggest practical guide lines to inspect a given graph.

find_cliques¶ find_cliques(G) [source] ¶. Search for all maximal cliques in a graph. Maximal cliques are the largest complete subgraph containing a given node. The largest maximal clique is sometimes called the maximum clique How can I list all cliques of an Undirected Graph ? (Not all maximal cliques, like the Bron-Kerbosch algorithm) Stack Exchange Network. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange. Loading 0 +0; Tour Start here for a. We base it on the classic backtracking method for finding all maximal cliques in non-partite graphs, and show that it runs in O(3 n/3) time. In Section 3 , we employ a series of innovative combinatorial constructions to prove an asymptotically tight bound on the maximum number of maximal k -partite cliques in a k -partite graph, thereby establishing MMCE's asymptotic optimality Finding Maximal Cliques in Massive Networks by H*-graph James Cheng Yiping Ke School of Computer Department of Systems Engineering Engineering and Engineering Nanyang Technological Management University, Singapore The Chinese University of j.cheng@acm.org Hong Kong ypke@se.cuhk.edu.hk Ada Wai-Chee Fu Jeffrey Xu Yu Linhong Zhu Department of Computer Department of Systems School of Computer. Maximum clique algorithms differ from maximal clique algorithms (e.g., Bron-Kerbosch algorithm). The maximal search is for all maximal cliques in a graph (cliques that cannot be enlarged), while the maximum clique algorithms find a maximum clique (a clique with the largest number of vertices). This makes maximum clique algorithms about an order. A graph clique is a set of nodes \$\mathcal{C}\$ in which each node is connected to all other nodes in \$\mathcal{C}\$. I have this small program for finding largest cliques from undirected graphs. I have two algorithms: SparseGraphLargestCliqueFinder: it begins with trivial clique candidates of size 1. It moves towards clique candidates of.

### Algorithm to find cliques of a given size k in O(n^k) time

• You might miss many maximal cliques with this approach; in fact, you might very well find none of them. Consider a cycle graph on at least $4$ vertices. In this graph, the maximal cliques are just the edges (there is no clique on $3$ or more vertices). But for each vertex, your approach will try to take both neighbors of that vertex and see if they form a clique; they don't, so your approach.
• cliques in a graph, they focus on a different problem, that of enumerating the kcliques with the highest probability of existence. We present bounds on the number of such cliques that could exist, while by deﬁnition, their problem requires them to output no more than kcliques. We provide a runtime complexity analysis of our algorithm and show that it is near optimal. No runtime complexity.
• Given a graph, in the maximum clique problem, one desires to find the largest number of vertices, any two of which are adjacent. A branch-and-bound algorithm for the maximum clique problem—which is computationally equivalent to the maximum independent (stable) set problem—is presented with the vertex order taken from a coloring of the vertices and with a new pruning strategy

In this paper, we present two efficient algorithms for finding maximum cliques of an overlap graph when it is given in the form of a family of n intervals. The first algorithm finds a maximum clique in O ( n . log n + Min { m, n ‐ ω}) time, where m is the number of edges and ω is the size of a maximum clique, respectively, of the graph Abstract We present a depth-first search algorithm for generating all maximal cliques of an undirected graph, in which pruning methods are employed as in the Bron-Kerbosch algorithm. All the maximal cliques generated are output in a tree-like form. Subsequently, we prove that its worst-case time complexity is O(3 n/3 ) for an n-vertex graph. This is optimal as a function of n, since there. Details. cliques find all complete subgraphs in the input graph, obeying the size limitations given in the min and max arguments.. largest.cliques finds all largest cliques in the input graph. A clique is largest if there is no other clique including more vertices. maximal.cliques finds all maximal cliques in the input graph. A clique in maximal if it cannot be extended to a larger clique A Novel Approach to Finding Near-Cliques: The Triangle-Densest Subgraph Problem Charalampos E. Tsourakakis ICERM, Brown University charalampos tsourakakis@brown.edu May 20, 2014 Abstract Many graph mining applications rely on detecting subgraphs which are large near-cliques. There exists a dichotomy between the results in the existing work related to this problem: on the one hand formulations. Problem 2647. Find the maximal cliques in an undirected graph. Created by Matthew Eicholtz; Like (0 ### algorithms - Finding all cliques of a graph - Computer

Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result many algorithms for finding. The smallest cliques are composed of two actors: the dyad. But dyads can be extended to become more and more inclusive -- forming strong or closely connected regions in graphs. A number of approaches to finding groups in graphs can be developed by extending the close-coupling of dyads to larger structures What is the best way to identify quasi-cliques in a graph in Mathematica? I don't think it has been implemented, so are there any solutions to do this? Or do I have to export the graph data somehow and do the hard work myself? EDIT: A quasi-clique is an almost complete subgraph. More formally (from this paper): Given an undirected graph $(V, E)$, and two parameters $\lambda$ and $\gamma$ with.

### ds.algorithms - Finding cliques in a big graph ..

Finding all maximal cliques in dynamic graphs Volker Stix Vienna University of Economics Department of Information Business Augasse 2{6 A-1090 Vienna / Austri In many practical cases one has to choose an arrangement of different objects so that they are compatible. Whenever the compatibility of the objects can be checked by a pair-wise comparison the problem can be modelled using the graph-theoretic notion of cliques. We consider a special case of the problem where the objects can be grouped so that exactly one object in every group has to be chosen The functions find cliques, ie. complete subgraphs in a graph: cliques: The functions find cliques, ie. complete subgraphs in a graph: clique_num: The functions find cliques, ie. complete subgraphs in a graph: closeness: Closeness centrality of vertices: closeness.estimate: Closeness centrality of vertices: cluster.distribution: Connected.

### Find all cliques of size K in an undirected graph

• The resulting data pool held all possible cliques of the graph and the string with the largest number of ones would be the maximum clique. To create the complete data pool, the scientists utilized DNA pairing (each nucleic acid has a complement, so if two strands contain a series of complementary DNA, they will stick to eachother). The scientists arranged the cliques so each vertex is.
• Cliques in Multipartite Graphs Cliques Maximum Clique Maximal Clique Enumeration Bipartite and multipartite graphs Maximal Biclique Enumeration Maximum k-partite cliques Maximal k-partite clique enumeration. Applications Molecular Biology Telecommunications Natural Language Processing Social Network Analysis Transportation Operations Research Chemistry Textile Manufacturing Drug Discovery.
• Finding maximal cliques in massive networks by James Cheng, Yiping Ke - ACM Trans. Database Syst Maximal clique enumeration is a fundamental problem in graph theory and has important applications in many areas such as social network analysis and bioinformatics
• iKanren and core.logic when looking for solutions to relational systems
• Find all cliques of size K in an undirected graph, Count the number of Prime Cliques in an undirected graph, Two Clique Problem (Check if Graph can be divided in two Cliques), Find if an undirected graph contains an independent set of a given size, Convert the undirected graph into directed graph such that there is no path of length greater than 1, Convert undirected connected graph to.

### Finding All Cliques of an Undirected Graph - Stack Overflo

• imum.
• k-cliques are used in project selection, pattern matching, finance, and network analysis. FindKClique can be used to find k-cliques of different sizes, from 1 to the largest possible size (in general n for a graph on n vertices). FindKClique can be used to find a single k-clique of specified size, a specified number of cliques, or all
• Finding cliques in random graphs by adaptive probing Based on joint works with U. Feige, D. Gamarnik, J. Neeman, B. Schiffer, and P. Tetali Miklos Z. Racz The 19th International Conference on Random Structures and Algorithms July 16, 2019. Finding cliques Erdős-Rényi random graph ( ,1/2) Finding cliques Erdős-Rényi random graph ( ,1/2) Largest clique ������ ≈2log Finding cliques Erdős.
• imum vertex degree Î´ must have a maximum clique of size at least âŒˆ n/(nâˆ'Î´) âŒ‰ and that this condition is the best possible in terms of nand Î´. As a corollary, we obtain new bounds on the famous Ramsey numbers in terms of the maximum and.
• and my attempt at finding cliques with different number of nodes and degrees. def plot_cliques(graph, number_of_nodes, degree): ug = ox.save_load.get_undirected(graph) cliques = nx.find_cliques(ug) cliques_nodes = [clq for clq in cliques if len(clq) >= number_of_nodes] print({} cliques with more than {} nodes..format(len(cliques_nodes), number_of_nodes)) nodes = set(n for clq in cliques.

Finding a large hidden clique in a random graph Noga Alon y Michael Krivelevich z Benny Sudakov x Abstract We consider the following probabilistic model of a graph on nlabeled vertices. First choose a random graph G(n;1=2) and then choose randomly a subset Qof vertices of size kand force it to be a clique by joining every pair of vertices of Qby an edge. The problem is to give a polynomial. DAA - Max Cliques. Advertisements. Previous Page. Next Page . In an undirected graph, a clique is a complete sub-graph of the given graph. Complete sub-graph means, all the vertices of this sub-graph is connected to all other vertices of this sub-graph. The Max-Clique problem is the computational problem of finding maximum clique of the graph. Max clique is used in many real-world problems. Finding large cliques in sparse semi-random graphs by simple randomized search heuristics Tobias Storch∗,1 Remote Sensing Technology Institute, German Aerospace Center - DLR, 82234 Wessling, Germany Received 16 June 2006; received in revised form 14 March 2007; accepted 21 June 2007 Communicated by T. Baeck Abstract Surprisingly, general heuristics often solve some instances of hard.

### Listing k-cliques in Sparse Real-World Graphs

• or is at most 2ctn for c. Key words. AMS subject classiﬁcations. 05C35, 05C69.
• ate the algorithm once you find 1000 cliques. I already got accepted on this problem with the above algorithm, believe me, it's fine. If you're getting TLE you're doing something wrong, my implementation runs in 0.08s on.
• imum vertex.
• finding all the cliques in a graph . Revision en2, by surajkvm007, 2016-03-13 15:03:17 can anyone please provide me with a hint to solve this problem . basically we need to find all the cliques in the graph but in short time. graph, clique. Compare Revisions. History Revisions.
• This paper develops a family of algorithms that are variations on the Bron-Kerbosch algorithm for finding all the cliques of a simple undirected graph. The algorithms are developed in a stepwise manner, from a recursive algorithm for generating all combinations of zero or more objects chosen fromN objects. Experimental results are given

Finding maximum cliques in circle graphs. D. Rotem. Department of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L361. Search for more papers by this author. J. Urrutia. Department of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L361. Department of Combinatorics and Optimization. Search for more papers by this author. D. Rotem. Department of. Independent sets and cliques S V is independent if no edge of G has both of its endpoints in S. (G)=maximum size of an independent set of G. Lemma 1 S is independent iff V n S is a cover. Corollary 1 (G)+ (G)= : 1. L E is an edge covering if every v 2 V is contained in an edge of L. 0(G)=minimum size of an edge cover 0(G)=maximum size of a matching. Theorem 1 If there are no isolated vertices. IntroductionA clique of an undirected graph is a complete subgraph which is not contained in any other complete subgraph. The problem of determining all cliques or the maximum clique of a graph is NP-complete. Clique-finding algorithms are used in many application areas [1,2,3,4,5,9,11,12,16]. Some applications involve the comparison of small.    Note that such graphs involve at most two cliques and that, if K 4 has a crossing, combining it with any other clique would violate 1-planarity (see Figure 4a,b). The next lemma accounts for cliques with five or six vertices. Lemma 2. There exists no 1-plane simple topological graph that contains two cliques, one of which with at least five vertices, whose edges cross each other. Proof. The question of finding maximal groups of sequences that pairwise tolerate each other now reduces to the question of finding all maximal cliques, in the respective c-max-tolerance graph. For general graphs, the computation of all maximal cliques is an NP-hard problem, since it can be reduced to the maximum clique problem which is again a classical NP-complete graph problem [ 7 ] Algorithm 457: finding all cliques of an undirected graph. Comm. ACM 16, 9, 575--577. Google Scholar; Byskov, J. M. 2003. Algorithms for k-colouring and finding maximal independent sets. In Proceedings of the Symposium on Discrete Algorithms (SODA). 456--457. Google Scholar; Cazals, F. and Karande, C. 2008. A note on the problem of reporting.

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