The phylogeny graphs of doubly partial orders

TL;DR

This paper proves that the phylogeny graph of a doubly partial order is an interval graph and demonstrates that any interval graph can be embedded as an induced subgraph in a phylogeny graph of such an order.

Contribution

It establishes the interval graph property of phylogeny graphs of doubly partial orders and shows the universality of these graphs for all interval graphs.

Findings

Phylogeny graph of a doubly partial order is an interval graph.

Any interval graph can be embedded as an induced subgraph in a phylogeny graph of a doubly partial order.

Abstract

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.