$(6$ points$)$ A stable set of a graph $G=(V,E)$ is defined as a set of nodes SsubeV with the property that

the nodes in $S$ are pairwise non$-$adjacent $($two nodes are adjacent if there is an edge between them$).$

$($a$)(2$ points$)$ Formulate the problem of finding a maximum cardinality stable set as an MILP.

$($b$)(4$ points$)$ An odd hole is a cycle with an odd number of nodes and no edges between non$-$adjacent

nodes of the cycle. Let $x_j$ be a binary variable that takes a value $1$ if a node jinS. If $H$ denotes

the set of nodes in an odd hole, show that the following inequality is valid: ${\displaystyle \underset{\text{j}\text{i}\text{n}\text{H}}{\overset{?}{}}}{x}_j\frac{|H|-1}{2}.$