Density of linearity index in the interval of matching numbers

Given integers 2 ≤ p ≤ c ≤ q, we construct a finite simple graph G with ν₁(G) = p and ν(G) = q for which the squarefree power I(G)[k] of the edge ideal $I(G)$ of G has linear quotients for each c ≤ k ≤ q and is not linearly related for each 1 ≤ k < c, where ν₁(G) is the induced matching number of G and ν(G) is the matching number of G.