Shared ancestry graphs and symbolic arboreal maps

Huber, Katharina T., Moulton, Vincent ORCID: https://orcid.org/0000-0001-9371-6435 and Scholz, Guillaume E. (2024) Shared ancestry graphs and symbolic arboreal maps. SIAM Journal on Discrete Mathematics, 38 (4). pp. 2553-2577. ISSN 0895-4801

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Abstract

A network N on a finite set X, | X| geq 2, is a connected directed acyclic graph with leaf set X in which every root in N has outdegree at least 2 and no vertex in N has indegree and outdegree equal to 1; N is arboreal if the underlying unrooted, undirected graph of N is a tree. Networks are of interest in evolutionary biology since they are used, for example, to represent the evolutionary history of a set X of species whose ancestors have exchanged genes in the past. For M some arbitrary set of symbols, d : ( X 2 ) → M ∪ {⊙ } is a symbolic arboreal map if there exists some arboreal network N whose vertices with outdegree 2 or more are labeled by elements in M and so that d({ x, y} ), { x, y} ∈ ( X 2 ) , is equal to the label of the least common ancestor of x and y in N if this exists, and ⊙ otherwise. Important examples of symbolic arboreal maps include the symbolic ultrametrics, which arise in areas such as game theory, phylogenetics, and cograph theory. In this paper we show that a map d : ( X 2 ) → M ∪ { ⊙} is a symbolic arboreal map if and only if d satisfies certain 3- and 4-point conditions and the graph with vertex set X and edge set consisting of those pairs { x, y} ∈ ( X 2 ) with d({ x, y} ) not =odot is Ptolemaic (i.e., its shortest path distance satisfies Ptolemy's inequality). To do this, we introduce and prove a key theorem concerning the shared ancestry graph for a network N on X, where this is the graph with vertex set X and edge set consisting of those { x, y} ∈ ( X 2 ) such that x and y share a common ancestor in N. In particular, we show that for any connected graph G with vertex set X and edge clique cover K in which there are no two distinct sets in K with one a subset of the other, there is some network with | K| roots and leaf set X whose shared ancestry graph is G.

Item Type:	Article
Uncontrolled Keywords:	ptolemaic graphs,arboreal networks,cographs,symbolic maps,ultrametrics,mathematics(all) ,/dk/atira/pure/subjectarea/asjc/2600
Faculty \ School:	Faculty of Science > School of Computing Sciences
UEA Research Groups:	Faculty of Science > Research Groups > Computational Biology Faculty of Science > Research Groups > Norwich Epidemiology Centre Faculty of Medicine and Health Sciences > Research Groups > Norwich Epidemiology Centre
Related URLs:	http://www.scopus.com/inward/record.url?...
Depositing User:	LivePure Connector
Date Deposited:	28 Jun 2024 12:31
Last Modified:	16 Dec 2024 01:42
URI:	https://ueaeprints.uea.ac.uk/id/eprint/95704
DOI:	10.1137/23M1573318

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