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I'm looking for algorithm that, for given undirected graph $G=(V,E)$, find graph $G'=(V,E')$ with minimal amount of edges that have same vertex cover as G. I mean, vertices $U$ are vertex cover of $G$ iff they are vertex cover of $G'$.
I understood that a polynomial algorithm exists, it's mean without finding the vertex cover of $G$, but I found only a non-polynomial one.

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  • $\begingroup$ I thought about edge cover, maybe this is the solution? $\endgroup$ Feb 25, 2021 at 7:49
  • $\begingroup$ No, I checked it and it looks like not related to edge cover $\endgroup$ Feb 25, 2021 at 9:04

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The algorithm to compute $G'$ is really easy: just return $G$.

Indeed, given any $G=(V,E)$ the only graph $G'=(V, E')$ such that $U \subseteq V$ is a vertex cover of $G$ iff $U$ is a vertex cover of $G'$ is $G$ itself.

To prove this you can show that we must simultaneously have $E \subseteq E'$ and $E' \subseteq E$.

Proof that $E \subseteq E'$: Suppose towards a contradiction that $E \nsubseteq E'$. Then $\exists (u,v) \in E \setminus E'$. The set $U = V \setminus \{u,v\}$ is a vertex cover for $G'$ but not for $G$.

Proof that $E' \subseteq E$: Suppose towards a contradiction that $E' \nsubseteq E$. Then $\exists (u,v) \in E' \setminus E$. The set $U = V \setminus \{u,v\}$ is a vertex cover for $G$ but not for $G'$.

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  • $\begingroup$ I'm not sure that you're right, a basic example, cube with one additional vertex. G={V,E}, V={1,2,3,4,5}, E={{1,2},{2,3},{3,4}.{4,5}, {5,2}}, have only one vertex cover U={2,4} same as E'={{1,2},{2,3},{3,4},{4,5}}. $\endgroup$ Feb 25, 2021 at 23:54
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    $\begingroup$ The are many vertex covers of your example graph. A trivial one different from $\{2,4\}$ is $V$ itself. $\endgroup$
    – Steven
    Feb 25, 2021 at 23:56
  • $\begingroup$ Thanks, I'll check how the original question was defined but only tomorrow. Are solution right but too trivial. $\endgroup$ Feb 26, 2021 at 0:02
  • $\begingroup$ Thank you @Steven, but I think you wrong. $G=\{V,E\}, V=\{1,2,3,4\}, E=\{ \{1,2\},\{2,3\},\{3,4\},\{4,1\},\{1,3\},\{2,4\}\}$ have same vertex cover as $E'=\{\{1,2\},\{2,3\},\{3,4\},\{4,1\}\}$ or I missed something $\endgroup$ Feb 26, 2021 at 10:46
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    $\begingroup$ @ChaosPredictor. It's never stated in your question that $G'$ must be a subgraph of $G$, so I didn't assume it. $\endgroup$
    – Steven
    Feb 27, 2021 at 21:15

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