## The Phylogeny Graphs of Doubly Partial Orders

 Boram Park National Institute for Mathematical Sciences Daejeon 305-811, Korea Yoshio Sano Division of Information Engineering Faculty of Engineering, Information and Systems University of TsukubaIbaraki 305-8573, Japan

## 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 | N+D(x) ∩N+D(y) ≠ ∅} ∪{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 G˜ such that G˜ contains the graph G as an induced subgraph and that G˜ is the phylogeny graph of a doubly partial order.

Keywords: competition graph, phylogeny graph, doubly partial order, interval graph

2010 Mathematics Subject Classification: 05C20, 05C75.

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Received 2 November 2011
Revised 26 July 2012
Accepted 30 July 2012