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Phylogeny numbers for graphs with two triangles
Journal article   Open access   Peer reviewed

Phylogeny numbers for graphs with two triangles

Fred S. Roberts and Li Sheng
Discrete Applied Mathematics, v 103(1), pp 191-207
2000
url
https://doi.org/10.1016/s0166-218x(99)00209-7View
Published, Version of Record (VoR)Open Access (Publisher-Specific) Open

Abstract

Competition graphs Competition numbers Phylogenetic tree reconstruction Phylogeny digraphs Phylogeny graphs Phylogeny numbers
Motivated by problems of phylogenetic tree reconstruction, Roberts and Sheng introduced notions of phylogeny graph and phylogeny number. These notions are analogous to and can be considered as natural generalizations of notions of competition graph and competition number that arise from problems of ecology. Given an acyclic digraph D=( V, A), define its phylogeny graph G= P( D) by taking the same vertex set as D and, for x≠ y, letting xy∈ E( G) if and only if ( x, y)∈ A or ( y, x)∈ A or ( x, a),( y, a)∈ A for some vertex a∈ V. Given a graph G=( V, E), we shall call the acyclic digraph D a phylogeny digraph for G if G is an induced subgraph of P( D) and D has no arcs from vertices outside of G to vertices in G. The phylogeny number p( G) is defined to be the smallest r such that G has a phylogeny digraph D with | V( D)|−| V( G)|= r. In this paper, we obtain results about phylogeny number for graphs with exactly two triangles analogous to those for competition number.

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Mathematics, Applied
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