The phylogeny graphs of double 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 | N D (x) ∩ N D (y) 6= ∅} ∪ {xy | (x, y) ∈ A(D)}, where N D (x) := {v ∈ V (D) | (x, v) ∈ 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 G̃ such that G̃ contains the graph G as an induced subgraph and that G̃ is the phylogeny graph of a doubly partial order.