On normal networks

Bickner, Devin Robert

doi:10.31274/etd-180810-1598

Cited by 5 publications

(8 citation statements)

References 22 publications

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“…The two parts of the next lemma are established in [4] and [1], respectively. However, we include its short proof for completeness.…”

Section: Parameters Of Networkmentioning

confidence: 99%

“…Note that the bounds in Lemma 2.2 are best possible [4,1]. Lemma 2.2 and (5) characterise the possible parameters of a treechild and normal network; that is, given integers r ≥ 0 and ≥ 1 with r + ≥ 2, there is a tree-child (respectively, normal) network with these parameters provided r ≤ − 1 (respectively, r ≤ − 2).…”

Section: Parameters Of Networkmentioning

confidence: 99%

“…Mathematically, phylogenetic networks provide a much more significant challenge and, indeed, relatively little is known about these objects. For example, the number of leaf-labelled binary phylogenetic trees with leaves has been known since Schröder's work in 1870, and this also gives the number of such trees on n labelled vertices-see (1) and (3) below. In contrast, the number of binary phylogenetic networks on n labelled vertices is unknown, and similarly for subclasses like tree-child networks.…”

Section: Introductionmentioning

confidence: 99%

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Counting Phylogenetic Networks

2015

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Abstract. We give approximate counting formulae for the numbers of labelled general, tree-child, and normal (binary) phylogenetic networks on n vertices. These formulae are of the form 2 γn log n+O(n) , where the constant γ is 3 2 for general networks, and 5 4 for tree-child and normal networks. We also show that the number of leaf-labelled tree-child and normal networks with leaves are both 2 2 log +O( ) . Further we determine the typical numbers of leaves, tree vertices, and reticulation vertices for each of these classes of networks.

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“…The two parts of the next lemma are established in [4] and [1], respectively. However, we include its short proof for completeness.…”

Section: Parameters Of Networkmentioning

confidence: 99%

Section: Parameters Of Networkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Counting Phylogenetic Networks

2015

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“…A single postorder transversal of D is sufficient to determine C ρ . Since N has at most O(|X|) vertices and, therefore, at most O(|X|) edges in total [2] (also see [15]), it takes O(|X|) time to find the visibility set of u, and so it takes O(|X|…”

Section: Proof (Proof Of Theorem 2(i))mentioning

confidence: 99%

Caterpillars on three and four leaves are sufficient to reconstruct normal networks

Linz,

Semple

2020

Preprint

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While every rooted binary phylogenetic tree is determined by its set of displayed rooted triples, such a result does not hold for an arbitrary rooted binary phylogenetic network. In particular, there exist two non-isomorphic rooted binary temporal normal networks that display the same set of rooted triples. Moreover, without any structural constraint on the rooted phylogenetic networks under consideration, similarly negative results have also been established for binets and trinets which are rooted subnetworks on two and three leaves, respectively. Hence, in general, piecing together a rooted phylogenetic network from such a set of small building blocks appears insurmountable. In contrast to these results, in this paper, we show that a rooted binary normal network is determined by its sets of displayed caterpillars (particular type of subtrees) on three and four leaves. The proof is constructive and realises a polynomial-time algorithm that takes the sets of caterpillars on three and four leaves displayed by a rooted binary normal network and, up to isomorphism, reconstructs this network.

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“…When we recurse, the distance matrix D and distance vector d inputted to the recursive call is the minimum distance matrix and maximum distance outgroup vector of a normal network with either one less leaf or one less reticulation than (N , w). As normal networks have at most |X| − 2 reticulations, and therefore O(|X|) vertices in total [2], the total number of iterations is at most O(|X|). Hence Reticulation-Pair Normal completes in O(|X| 3 ) time, thereby completing the proof of Theorem 2.4.…”

Section: We Call a Leaf A Candidate Leaf Ifmentioning

confidence: 99%

Recovering normal networks from shortest inter-taxa distance information

et al. 2018

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Phylogenetic networks are a type of leaf-labelled, acyclic, directed graph used by biologists to represent the evolutionary history of species whose past includes reticulation events. A phylogenetic network is tree-child if each non-leaf vertex is the parent of a tree vertex or a leaf. Up to a certain equivalence, it has been recently shown that, under two different types of weightings, edge-weighted tree-child networks are determined by their collection of distances between each pair of taxa. However, the size of these collections can be exponential in the size of the taxa set. In this paper, we show that, if we have no "shortcuts", that is, the networks are normal, the same results are obtained with only a quadratic number of inter-taxa distances by using the shortest distance between each pair of taxa. The proofs are constructive and give cubic-time algorithms in the size of the taxa sets for building such weighted networks.

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On normal networks

Cited by 5 publications

References 22 publications

Counting Phylogenetic Networks

Counting Phylogenetic Networks

Caterpillars on three and four leaves are sufficient to reconstruct normal networks

Recovering normal networks from shortest inter-taxa distance information

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