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If it says no, then $v$ is in every clique of size at least $k$. Output $v$ and decrease $k$ by one and lather, rinse, repeat. …

The graphs with clique size at least $k$ and VC at most $k$ have a particular structure: They can be partitioned into three sets, $C$, $I$ and a singleton $\{s\}$; $G[C]$ is a clique, $G[I]$ is an independent … Proof: If a graph $G$ contains a clique of size $k$, then that clique uses up (at least) $k-1$ of your vertex cover budget. Let $G,k$ be a yes instance and $C$ any $k$-clique. …

I am assuming you mean the number of maximal cliques, as the number of cliques of a complete graph is trivially $2^n$ (any subset of the vertices forms a clique). …

Thus I'm still curious if there are other more efficient approaches to solving the problem. If you take the complement graph $\overline{G}$, then your problem corresponds to a coloring problem. …

Yes, and the algorithm is straight forward branching. Suppose that you are given a graph $G$ and an integer $k$. If $k = 0$, we simply return whether $G$ is an independent set. If $G$ is an indepen …

The problem is called Defective $k$-Clique [Yu et al., Bioinformatics (2006)]. The optimization problem is: Problem: Defective $k$-Clique Input: A graph $G$ and $k \in \mathbb{N}$. … Output: The minimum number, $\ell$, of edges to add to $G$ such that $G$ has a $k$-clique. Clearly, $G$ has a $k$-clique if and only if $\ell = 0$, hence the problem is NP-complete. …

Is it correct to say that it doesn't exist because clique is NP-Complete? No, that would not be correct. … Since we currently do not know whether P=NP, it could be that Clique is polynomial time solvable. However, if we assume that P != NP, there is no polynomial time algorithm for Clique. …

Consider a graph which is the disjoint union of $K_i$ and $K_1$, the isolated vertex. Let the order of selection of vertices be such that the isolated vertex is last. Then your heuristic outputs sol …