Bipartite Tur\'an problem on cographs

A cograph is a graph that contains no induced path $P_4$ on four vertices or equivalently a graph that can be constructed from vertices by sum and product operations. We study the bipartite Tur\'an problem restricted to cographs: for fixed integers $s \leq t$, what is the maximum number of edges in an $n$-vertex cograph that does not contain $K_{s,t}$ as a subgraph? This problem falls within the framework of induced Tur\'an numbers $\text{ex}(n, \{K_{s,t}, P_4\text{-ind}\})$ introduced by Loh, Tait, Timmons, and Zhou. Our main result is a Pumping Theorem: for every $s\le t$ there exists a period $R$ and core cographs such that for all sufficiently large $n$ an extremal cograph is obtained by repeatedly pumping one designated pumping component inside the appropriate core (depending on $n\bmod R$). We determine the linear coefficient of $\text{ex}(n, \{K_{s,t}, P_4\text{-ind}\})$ to be $s-1 + \frac{t-1}{2}$. Moreover, the pumping components are $(t-1)$-regular and have $s-1$ common neighbours in the respecitve core graphs, giving the extremal cographs a particularly rigid extremal star-like shape. Motivated by the rarity of complete classification of extremal configurations, we completely classify all $K_{3,3}$-free extremal cographs by proof. We also develop a dynamic programming algorithm for enumerating extremal cographs for small $n$.