Geodesic
The Phylogeny Graphs of Doubly Partial Orders
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 4, pp. 657-664.

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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) ¹ ⊘ ⋃ 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 G̃, there exists an interval graph G such that G̃ contains the graph G as an induced subgraph and that G̃ is the phylogeny graph of a doubly partial order.
Keywords: competition graph, phylogeny graph, doubly partial order, interval graph 
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Park, Boram; Sano, Yoshio. The Phylogeny Graphs of Doubly Partial Orders. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 4, pp. 657-664. http://geodesic.mathdoc.fr/item/DMGT_2013_33_4_a2/

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