Given a graph G $=($V$,$ E$),$ a set of vertices D $\backslash$subseteq V is an independent set if there do not exist i$,$ j $\backslash$in D with ij $\backslash$in E $.$ Given a graph G $=($V$,$ E$)$ with a function w: V $\backslash$rightarrow $\backslash$mathbb$\{$R$\}$ that assigns a weight to each vertex, the maximum weight independent set problem consists of finding an independent set with the maximum total weight. The decision version of this problem is an NP$-$complete problem.

a$)$ Propose a greedy algorithm for this problem. What is the computational complexity of the proposed algorithm?

b$)$ Propose a local search algorithm for this problem.

c$)$ Is the operator on which the local search algorithm proposed in the previous point is based connected?