A pure blockade with a cograph pattern has additive kappa

We argue by induction on $t$. If $t=1$, then $V(\mathcal{B})=B_1$, so $\kappa (G[V(\mathcal{B})])=\kappa (G[B_1]),$ and the claim is immediate.

Assume now that $t>1$ and that the theorem is known for shorter pure blockades with cograph pattern. By [L2], the cograph $P$ is disconnected or its complement is disconnected. Choose either a connected component $I$ of $P$ in the first case, or an anticomponent $I$ of $P$ in the second case, and let $J:=[t]\setminus I$. Then $I$ and $J$ are nonempty. Put $U:={\bigcup }_{i\in I}{B}_i,W:={\bigcup }_{j\in J}{B}_j,V:=U\cup W=V(\mathcal{B})\mathrm{.}$

The induced pattern subgraphs $P[I]$ and $P[J]$ are cographs by [L1]. If $I$ is a component, then [L3] and [L4] make $U$ anticomplete to $W$; if $I$ is an anticomponent, then [L3] and [L4] make $U$ complete to $W$.

Applying the induction hypothesis to the subblockades indexed by $I$ and $J$ gives $\kappa (G[U])\ge {\sum }_{i\in I}\kappa (G[B_i]),\kappa (G[W])\ge {\sum }_{j\in J}\kappa (G[B_j])\mathrm{.}$

If $U$ is anticomplete to $W$, then a stable set in $G[U]$ together with a stable set in $G[W]$ is stable in $G[V]$, while every clique in $G[V]$ lies in one side. Thus $\alpha (G[V])\ge \alpha (G[U])+\alpha (G[W]),\omega (G[V])\ge \max \{\omega (G[U]),\omega (G[W])\},$ and [L5] yields $\kappa (G[V])\ge \kappa (G[U])+\kappa (G[W])$. If $U$ is complete to $W$, the same reasoning with cliques and stable sets exchanged again gives $\kappa (G[V])\ge \kappa (G[U])+\kappa (G[W])$.

Combining steps 3.1 and 3.2 gives $\kappa (G[V(\mathcal{B})])=\kappa (G[V])\ge {\sum }_{i=1}^t\kappa (G[B_i])\mathrm{.}$ Together with step 1.1, this closes the induction.