Showing a problem is in coNP We have the problem $C = \{&lt;G,S&gt;| \text{ S is a minimal cover of G }\}$ and we want to show that $C\in coNP$.

I can easily show that there's a ND TM that decides $coC$ using a guess to check if the drawn vertices are a minimal cover and if they are, return false. But it seems like it's wrong since this can be done with anything. 

So is there another way to show that?
Since coNP is confusing, let us construct an NP machine for $\overline{C}$. The input to the machine is a pair $\langle G,S \rangle$, where $S$ is a cover of $G$. The machine guesses another cover $T$, and verifies that $T$ is a cover of $G$ which is smaller than $S$. The cover $T$ is thus a witness to the fact that $S$ is not a cover of minimum size.

If $S$ is not a minimal cover, then there exists some smaller cover $T$, and so the machine accepts the instance $\langle G,S \rangle$ if it guesses $T$. Conversely, if $S$ is a minimal cover, then no smaller cover exists, and so the machine rejects the instance $\langle G,S \rangle$ whatever the guess is. In total, the machine accepts the language $\overline{C}$.

Here is a another way to look at things. We construct a machine that on input $\langle \langle G,S \rangle,T \rangle$, accepts if $S,T$ are covers of $G$, and $T$ is smaller than $S$. Then


  There exists $T$ such that the machine accepts $\langle \langle G,S \rangle,T \rangle$ if and only if $\langle G,S \rangle \in \overline{C}$.


This machine is polytime, and it accepts $\overline{C}$, showing that $\overline{C} \in \mathsf{NP}$, and so $C \in \mathsf{coNP}$.