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I want to use Vertex Cover as a known $NP$-complete Problem for the reduction. The claim is that if a have a vertex cover in graph $G$ with size $\le k$, I will have a clique cover in $G^\prime$ with total number of cliques $\le k$

I want to reduce vertex cover problem to clique cover problem in the following way:

enter image description here

  1. For each edge in $G$, I will create a corresponding vertex in $G^\prime$. In the picture, each e$_i$ in G corresponds to v$_i$ in $G^\prime$ for $1\le i\le5$

  2. Two vertices in $G^\prime$ will be connected by an edge if their corresponding two edges in G have one vertex in common. For example, there is an edge between v$_1$ to v$_2$ as their corresponding edges e$_1$ and e$_2$ have one vertex in common which is a.

  3. Each clique in $G^\prime$ corresponds to a vertex in G and that vertex belongs to the vertex cover. For example, for the graph in picture I have vertex cover {a,c}. With vertex a, I get a clique {v$_1$, v$_2$, v$_5$} and with vertex c, I get clique {v$_3$, v$_4$} in $G\prime$.

Now I don't know how to prove No instance in opposite direction: If $G^\prime$ doesn't have a clique cover with $\le k$ cliques (i.e it needs at least $k+1$ cliques), $G$ doesn't have a vertex cover size which is $\le k$.

Can anybody shed some light on this issue?

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  • $\begingroup$ Keep trying. Also, try to refute your claim in parallel. $\endgroup$ Feb 24, 2017 at 3:17

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Your reduction is incorrect.

The problem is with your claim "Each clique in $G'$ corresponds to a vertex in $G$". This is false, because there could be a clique associated to two different vertices.

For example, consider a 3-clique $G = (\{v_1, v_2, v_3\}, \{v_1v_2, v_2v_3, v_1v_3\})$. The corresponding graph $G'$ is also a 3-clique, but the 3 edges don't have a unique common vertex. In this case, $G'$ has a clique cover of size $1$, but $G$ doesn't.

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