Setting
Definition(1). $\mathcal{M} \models T$ is an existentially closed (e.c.) model of $T$ if whenever $\mathcal{N} \models T$, $\mathcal{N} \supseteq \mathcal{M}$, and $\mathcal{N}\models \exists \bar{v} \phi(\bar{v},\bar{a})$, where $\bar{a} \in M$ and $\phi$ is a quantifier free formula, then $\mathcal{M} \models \exists \bar{v} \phi(\bar{v},\bar{a})$.
Definition(2). Let $T$ be a theory. A theory $T^*$ is called the model companion of $T$ if $$\mathcal{M}\models T^* \text{ iff } \mathcal{M} \text{ is an e.c. model of } T$$
So $T^*= \bigcap_{\mathcal{M} \text{ is e.c.} } Th(\mathcal{M})$, where $Th(\mathcal{M})$ is the full theory of $\mathcal{M}$.
Case of study
Let $T$ be the theory of bowtie-free graphs (i.e. $T$ is the theory of graphs plus the axiom that expresses every five points does not form a bowtie), and $mod(T)$ be the class of all models of $T$.
Question(1). Is $mod(T)$ closed under the union of chain? (equivalently we may ask, does $T$ have $\forall\exists$-axiomatization?)
If the answer of the first question is positive, we can ask the following questions.
Question(2). What are the $e.c.$ models of $T$?
Question(3). Does $T$ have model-companion?