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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 elements are equivalent. If I have a graph $G$ and…
Thomas Klimpel
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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, we need at least two edges. For example,…
yashirq
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Is there a reasonable algorithm to generate a certain "independent clique graph" with minimal vertices?

In the process of trying to find a solution to the rat and poison puzzle with two rats, I've found myself needing the solve the following problem, in polynomial time: Given any $k_0, k_1, k_2,..., k_{n-1} \in [2,\infty)$, we want to produce a simple…
Fricative Melon
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Number of maximal cliques in a ($2C_4$, $C_5$, $P_5$)-free graph

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 number of maximal cliques in a $(2C_4, C_5,P_5)$-free…
padawan
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Reduce $\sqrt{n}$-CLIQUE to CLIQUE

Recall that $G$ has a clique of size $k$ if it has a complete sub graph consisting of $k$ vertices. Define CLIQUE as the decision problem $$\{ \langle G, c \rangle \mid G \text{ has a clique of size } c\}$$ Define the problem $\sqrt{n}$-CLIQUE as…
MT_
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Why is Clique NP-complete while k-Clique is in P for all k?

I just stumbled upon this question here Why is the clique problem NP-complete? and I am confused by the given answer. The question about about whether $k$-clique is NP-hard for a fixed $k$ and the answer is no. However, we know that clique in…
Cryptonaut
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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 SEPARATE cliques $K_1$ and $K_2$ of size…
Saaber
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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 clique in these graphs, and it has been shown that the…
Austin Buchanan
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Showing a problem is NP complete? Reducing CLIQUE to KITE.

I've got an exam next week all about this sort of thing. Ie: Find polynomial certifier for a problem, give a polynomial reduction, prove problem X reduces to Y and etc. The problem is, there doesn't seem to be anyone available in my department to…
Jay
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Maximum-weight set of cliques of size 3 with no common vertices in undirected graph

I'm looking for an algorithm/insight into a problem that's an extension of the Maximum Weight Matching problem. The maximum weight matching problem looks for the max-weight set of edges that contain 0 common vertices. The problem can be thought of…
TLF
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Prove that "Finishing the degree in three years" problem is NP-Complete

I was asked in an interview the following question: We'll define the "Finishing the degree in three years" problem in the following manner: Given a list of courses $C=\{c_1, c_2,\ldots, c_n\}$, where every course $c_i\in C$ is given a list…
gusfring
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What is the time complexity of the classic Bron-Kerbosch algorithm for finding cliques?

Bron-Kerbosch is an algorithm to find maximal cliques in a undirected graph. In pseudocode it's the following (taken from wikipedia): BronKerbosch1(R, P, X): if P and X are both empty: report R as a maximal clique for each vertex v in…
Juan
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Expected number of independent sets of size $k$ in random graph $G(n,p)$

I am looking for a formula for determining the expected number of independent sets of size $k$ (for arbitrary $k$) in a random graph $G(n,p)$. Here $n$ is the number of vertices and each edge is included with independent probability $p$. I would…
fiftyeight
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Hardness of a special GAP-CLIQUE problem

In the GAP-CLIQUE$(k,\ell)$ problem, we are given a graph $G$ over $n$ vertices and have to decide whether $G$ contains a clique of size $k$ or no clique of size $\ell$. Using a PCP system, it can be shown that GAP-CLIQUE$(k,\varepsilon k)$ is…
Dennis Kraft
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Testing the property of being a union of three disjoint cliques

Design an $\epsilon$-test for the following property in the dense graph model: $G(V,E)$ is a union of three disjoint cliques. I've been sitting for a few hours and I don't have any idea of how to solve this one ... anyone has any ideas / leads? I…
Ploni Almoni
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