size 2*: Let $G = (Q \cup S, E)$ with $Q = \{u, u'\}$, $S = \{v_1,\ldots,v_r,v'_1,\ldots,v'_t\}$ and $E = \{(u, u')\} \cup \{(u,v_i) : i \in [r]\} \cup \{(u',v'_i) : i \in [t]\}$. If $r \neq t$, we are in the previous case. If $r = t$, $\eta(G) = 2$ which is the value given by Theorem \[COTASUPSPLIT\].

-   *Complete splits*: Let $G = (Q' \cup S', E)$ with $|Q'| \geq 1$, $|S'| \geq 2$, $Q'$ is a clique of $G$ and there are edges $(u, v)$ for all $u \in Q'$ and $v \in S'$. $G$ is known as *complete split*. Since $G$ has $|Q'|$ true twins, $\eta(G) \geq |Q'|$. On the other hand, let $v \in S'$ and $Q = Q' \cup \{v\}$. Here, $Q$ is a maximal clique of $G$. Consider $T = \{u, v\}$ where $u \in Q'$. In virtue of Theorem \[COTASUPSPLIT\], $\eta(G) = |Q|-1 = |Q'|$.

Now, we will see families of split graphs where the bound given by Theorem \[COTASUPSPLIT\] is not tight. We study two of them here and another one in the next section (called complete suns).

A *thin headless spider* of orden $q \geq 2$ is a split graph 