A tau-critical graph has no wide pure blockade with cograph pattern

Given: A real $\tau >0$, a $\tau$-critical graph $G$, and an integer $t>0$.

A $\tau$-critical graph satisfies $\kappa (G)<\mathrm{\mid }V(G){\mathrm{\mid }}^{\tau }$, while every proper induced subgraph $H$ satisfies $\kappa (H)\ge \mathrm{\mid }V(H){\mathrm{\mid }}^{\tau }$ (A tau-critical graph).

A pure blockade with cograph pattern has additive $\kappa$ on its support (A pure blockade with a cograph pattern has additive kappa).

If $X\subseteq V(G)$, then every clique or stable set in $G[X]$ is also one in $G$, so $\kappa (G)\ge \kappa (G[X])$ (Subgraphs, induced subgraphs and spanning subgraphs, The parameter kappa(G)=alpha(G)omega(G), Cliques, stable sets, the clique number $\omega (G)$ and stability number $\alpha (G)$).

For positive reals, $(ab{)}^r=a^r{b}^r$ and $(a^r{)}^s=a^{rs}$ (The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents).

Because $\tau >0$, the function $x\mapsto x^{\tau }$ is increasing on $(0,\mathrm{\infty })$: its derivative is $\tau x^{\tau -1}$, the factor $x^{\tau -1}$ is positive for $x>0$ because real powers are defined through $\exp$ and $\exp$ is positive, and the derivative-sign theorem then gives monotonicity (Continuity and derivatives of positive-base real powers, Real powers for positive bases, with the zero-base positive-exponent convention, The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$, On an interval $I$, for $f$ continuous on $I$ and differentiable at every interior point: $f^{\mathrm{\prime }}\ge 0$ throughout gives $f$ nondecreasing, $f^{\mathrm{\prime }}>0$ gives $f$ increasing, $f^{\mathrm{\prime }}\le 0$ and $f^{\mathrm{\prime }}<0$ give the two decreasing forms; conversely a nondecreasing $f$ has $f^{\mathrm{\prime }}\ge 0$ and a nonincreasing $f$ has $f^{\mathrm{\prime }}\le 0$ wherever it is differentiable, and no strict converse is claimed).

If a blockade has width $w$, then each of its blocks has cardinality at least $w$ (Blockades, their length, their width, and their support).