Egg-free DAG

Let's call a directed graph *egg-free* if it is acyclic and for any three pairwise distinct vertices $x$, $y$ and $z$, if the graph contains edges $x \rightarrow y$ and $x \rightarrow z$, then it also contains an edge $y \rightarrow z$ and/or an edge $z \rightarrow y$. You are given an undirected graph with $N$ vertices (numbered $1$ through $N$) and $M$ edges (numbered $1$ through $M$). Find a