MHB NP Problems: Weighted Vertex Cover vs Weighted Independent Set

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The discussion revolves around a theorem connecting independent sets and vertex covers in graphs, specifically addressing the scenario where vertices have positive integer weights. It posits that if S is an independent set with maximum weight, then the complement V - S should be a vertex cover with minimum weight. The argument presented demonstrates that if S is indeed a maximum weight independent set, then any alternative vertex cover T with a lower weight would lead to a contradiction regarding the maximality of S. This reasoning is applied in both directions, confirming that the properties of maximum weight independent sets and minimum weight vertex covers are intrinsically linked under the conditions specified.
JohnDoe2013
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There is a Theorem that states that if G = (V, E) is a graph, then S is an independent set $\Leftarrow\Rightarrow$ V - S is a vertex cover.

Suppose the vertices have positive integer weights. Does it follow from the theorem that:

S is an independent set with maximum weight $\Leftarrow\Rightarrow$ V - S is a vertex cover with minimum weight?
 
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I think so. If $S\subseteq V$, let $w(S)$ denote the sum of weights of vertices is $S$, and let $W=w(V)$. Suppose that $S$ is an independent set with maximum weight, but $V-S$ is not minimal, i.e., there exists a $T\subseteq V$ such that $T$ is a vertex cover and $w(T)<w(V-S)=W-w(S)$. Then $w(S)<W-w(T)=w(V-T)$ and $V-T$ is an independent set, a contradiction with maximality of $S$. The other direction is shown similarly.
 
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