The Feasibility Problem -- the family ${\cal F}$(G)$of all induced$G$-free graphs

Abstract: An infinite family of graphs ${\cal F}$ is called feasible if for any pair of integers $(n,m)$, $n \geq 1$, $0 \leq m \leq \binom{n}{2}$, there is a member $G \in {\cal F}$ such that $G$ has $n$ vertices and $m$ edges. We prove that given a graph $G$, the family ${\cal F}$(G)$of all induced$G$-free graphs is feasible if and only if$G$is not$K_k$,$K_k\backslash K_2$,$\overline{K_k}$,$\overline{K_k\backslash K_2}$, for$k \geq 2$.