How to show that in every graph, the minimum size of a vertex cover is equal to number of vertices minus the maximum size of an independent set.

According to Vertex cover two problem are not equivalent, but there are should be kind of connection between them.



A subset $C \subseteq V$ is a vertex cover iff

$$\forall (u,v) \in E. u \in C \vee v \in C$$

A subset $I \subseteq V$ is an independent set iff

$$\forall (u,v) \in E. u \notin I \vee v \notin I$$

So as you can see a set is a vertex cover iff its complement is an independent set, and the converse also holds.

Therefore a minimal vertex cover $C$ corresponds to a maximal independent set $V-C$.


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