A Framework for Structural Input$/$Output and ControlConfiguration Selection in Large$-$Scale Systems. Sir this is the published paper.

Could you please help me to figure out formulate these conditions into integer linear programming$($ILP$).$ is it possible to give me integer linear progarmming$($ILP$)$with digraph and mathematical Explanation. i dispiritedly need formulate these conditions into intrger lunear programming with graph and mathematical formulation becuase down this theorem $2$ i will write a paper for published.

Theorem $2([30])$: The structural system $(\frac{?}{\text{b}\text{a}\text{r}}(A)\frac{,}{b}ar(B)\frac{,}{b}ar(C))$ associated with $(5)$ has no structurally fixed modes w$.$r$.$t$.$ an information pattern $\frac{?}{\text{b}\text{a}\text{r}}(K),$ if and only if both of the following conditions hold:

a$)$ in $E_{y,U}),$ each state vertex xinx is contained in an SCC which includes an edge of $E_{y,x}$;

b$)$ there exists a finite disjoint union of cycles $C_k=(V_k,E_k)$ in $D(\frac{?}{\text{b}\text{a}\text{r}}(A)\frac{,}{b}ar(B)\frac{,}{b}ar(K)\frac{,}{b}ar(C))$ with kinN such that

xsub$u_k{V}_k.$