The phylogeny graphs of doubly partial orders

The competition graph of a doubly partial order is known to be an interval graph. The CCE graph and the niche graph of a doubly partial order are also known to be interval graphs if the graphs do not contain a cycle of length four and three as an induced subgraph, respectively. Phylogeny graphs are variant of competition graphs. The phylogeny graph $P(D)$ of a digraph $D$ is the (simple undirected) graph defined by $V(P(D)):=V(D)$ and $E(P(D)):=\{xy \mid N^+_D(x) \cap N^+_D(y) \neq \emptyset \} \cup \{xy \mid (x,y) \in A(D) \}$, where $N^+_D(x):=\{v \in V(D) \mid (x,v) \in A(D)\}$. In this note, we show that the phylogeny graph of a doubly partial order is an interval graph. We also show that, for any interval graph $G$, there exists an interval graph $\tilde{G}$ such that $\tilde{G}$ contains the graph $G$ as an induced subgraph and that $\tilde{G}$ is the phylogeny graph of a doubly partial order.