Parallel algorithms for finding a/all maximal clique(s) in chordal graphs
What is the best algorithm for finding maximal cliques in a chordal graph. I am aware of Klein's algorithm that works in CRCW[$log^2 n$]. I also come across various versions of algorithms by Ho-Lee...
View ArticleComplexity of computing the simplicial width of a graph
Let $G=(V,E)$ be a finite undirected graph. A tree decomposition $(T,\lambda)$ of $G$ is a tree $T$ with labeling function $\lambda : T \to 2^{V}$ such that:For every edge $\{v_1,v_2\} \in E$, there...
View ArticleEnumerating $k$-Cliques with better than $O(n^k)$ preprocessing
Let us define the problem $k$-Clique enumeration as follows:Input: A Graph $G$ with $n$ nodes.Output: A data structure $D$ which can enumerate all $k$-Cliques in $G$ with $k$-delay.The notion of having...
View Articlewhy is there the need for a reduction between clique and multicoloured clique?
I do not understand why we need to make a reduction from k-clique to k-multicoloured clique.Any k-clique can only be coloured in k different colours since each pair of vertices is adjacent. So it...
View ArticleTree decompositions with unique witness for each edge
In this question I am concerned with tree decompositions of undirected graphs. Recall that a tree decomposition of a graph $G = (V, E)$ is a tree $T$ whose nodes are subsets of $V$ (called bags)...
View ArticleAny value in a formula that calculates (not look up) the 'order' of a...
Any value or interest in a formula that calculates (not look up) the 'integer order' of a given 'Independent Edge Set' OR given an 'Independent Set' calculates the 'integer order' on Complete Graphs?...
View Articleclique problem in graphs with bounded degree
Is the problem of finding a clique of size $d$ in a graph of maximum degree $d$ NP-complete ($d$ part of the input)?
View ArticleFinding cliques in a big graph
I would like to find (all) cliques in a given graph with 8,568 vertices and 12,726,708 edges. The vertex with the lowes degree has 2000, the vertext with the highest degree has 4007.The cliques should...
View ArticleIs Finding an *Unbalanced* Biclique in Bipartite Graphs Hard?
In the balanced biclique in bipartite graphs (MBB) problem we are given a bipartite graph $G = (L,R,E), |L| = |R| = n$ and the goal is to find an induced subgraph of $G$, $G' = (L',R',E')$, with as...
View ArticleThe number of cliques in a graph: the Moon and Moser 1965 result
I'm looking for the full text of the Moon and Moser 1965 clique result On Cliques in Graphs (there exist graphs with a number of maximal cliques exponential in $n$). My university's paywall doesn't...
View ArticleApproximation algorithm for balanced bipartite independent set?
The Problem: Given a bipartite graph $G = (L,R,E)$ with $|L|=|R|=n$, the balanced bipartite independent set problem asks us to output the largest vertex subsets $A\subseteq L, B\subseteq R$ of equal...
View ArticleAlgorithms for online clique detection
Are there any algorithms which let you detect cliques when adding/deleting edges based on previously detected cliques? What would be the time/memory complexity of this approach?
View ArticleHardest problems to approximate
Under some assumptions, it is hard to approximate MAX-CLIQUE within a factor $n^{1-\epsilon}$ for any $\epsilon >0$. Are there any other problems that are known to be equally hard to approximate?...
View ArticleIs finding a very large clique NP-hard?
We know that, unless P=NP, for any $\epsilon$ we can't distinguish in polynomial time between the two cases:There exists a clique of size at least $n^{1-\epsilon}$All cliques have size at most...
View ArticleSeparating 2-SAT from Clique
Since the P vs. NP problem is still an open problem, 2-SAT and Clique might both be in P if P = NP. Is there any known complexity measure whatsoever that is already mathematically proven to distinguish...
View ArticleLinear-time algorithm to test if clique number equals degeneracy bound?
Given a connected simple graph $G=(V,E)$, let $d$ denote its degeneracy and let $\omega$ denote the size of a maximum clique.A well-known bound on the clique number is $\omega\le d+1$, which is helpful...
View ArticleLarge CLIQUE approximation
I am interested in algorithms to identify large cliques in graphs where the largest clique is a large fraction (definitely greater than half, perhaps as great as 4/5) of the total number of...
View ArticleMaximum cliques of the transitive closure of a chordal DAG
Let $G=(V,A)$ be a directed acyclic graph, for which the underlyingundirected graph is chordal (so that every induced cycle in theunderlying undirected graph is a triangle).It is known that in a...
View ArticleAlgorithms for finding all cliques of a given degree in a graph
Consider a graph with $n$ vertices and maximum degree $Δ$. I would like to obtain all $s$ cliques, where $s≤Δ$ and both of them are small compared to $n$. Bron-Kerbosch algorithm gives all maximal...
View ArticleMinimal clique edge cover vs minimalist (assignment-minimum) ones
Given a graph $G=(V,E)$, a clique edge cover is a collection $C$ of subsets of $V$ such that each element $c$ of $C$ is a clique ($c \times c \subseteq E$) and $G$ is the union of these cliques ($E =...
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