# Questions tagged [clique]

A clique is a subset of the vertices of a graph such that every pair of vertices in the subset is connected by an edge.

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### The equivalence relations cover problem (in graph theory)

An equivalence relation on a finite vertex set can be represented by an undirected graph that is a disjoint union of cliques. The vertex set represents the elements and an edge represents that two ...
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### Having trouble in understanding the definition of a clique

My definition says A clique is a graph that has an edge connecting every pair of vertices but as I understand, an edge connects only two vertices. Like $A-B$. If we want to connect three vertices,...
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### Number of maximal cliques in a ($2C_4$, $C_5$, $P_5$)-free graph [closed]

So far, I have found out that chordal graphs have linear number of maximal cliques with respect to the number of vertices. In general case, it is exponential. I am trying to determine whether the ...
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### Modified Clique Problem

We know CLIQUE and HALF-CLIQUE problems are NP-complete. Now consider the class of graphs (let's call it $\mathcal{G}_{2K}$) where a graph $G=(V,E)$ is a member of $\mathcal{G}_{2K}$ iff $G$ has two ...
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### Expected number of maximal cliques in $G(n,p)$

The $G(n,p)$ random graph model creates graphs with $n$ vertices and each possible edge exists independently with probability $p\in (0,1)$. Much is known about the (expected) size of a largest ...
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### Understanding CLIQUE structure

I am working on the following problem: Recall the definition of a complete graph Kn is a graph with n vertices such that every vertex is connected to every other vertex. Recall also that a clique is ...
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### An exact solution for biclique vertex-cover problem on a bipartite graph

The biclique vertex-cover problem asks whether the vertex-set of the given graph can be covered with at most "k" bicliques (complete bipartite subgraphs). It has been shown that "Biclique Vertex-...
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### Distributed MST Construction in O(log log n) Rounds in a Clique

I'm reading the paper MST Construction in O(log log n) Communication Rounds in a Clique and trying to understand the correctness analysis, in page 5. It shows by induction on k (phase number), that ...
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### Question about the lower bound for $k \times k$-clique (under ETH) shown in “Slightly Superexponential Parameterized Problems”

I am reading the paper Slightly Superexponential Parameterized Problems at the moment and have two questions about it: First question: The paper gives a proof of the following statement Theorem ...
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### Finding maximum weighted n disjoint cliques

Maximum weight clique problem has some attention but i could not find any efficient approaches to this problem yet. I acknowledge that it is np-hard, but are there any known approximations? Given a ...
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### Maximal cliques in a multipartite graph - efficient?

I am looking at a combinatorial optimisation problem where I have N classes and k objects of each class. Now I am looking for the optimal subset such that each of the N classes is represented exactly ...
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### Does the k-clique problem became easier on sparse graphs?

Some definitions, just to not create confusion: A sparse graph is a graph that contains a number of edges less or equal than the number of vertices. In $k$-clique problem we are given a graph and an ...
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### Can I find a clique with more than 2 nodes in a bipartite graph?

As in the title, is it possible to find a clique with more than 2 nodes in a bipartite graph?
### Is $\frac{n}{3}$-CLIQUE NP-complete?
Consider the problem $\frac{n}{3}$-CLIQUE: determining whether a graph contains a clique with at least $n/3$ vertices. I want to prove it is NP-complete using a polynomial transformation from CLIQUE. ...