Pith. sign in

REVIEW 4 minor 49 references

The $\ominus$-metric to compare phylogenetic networks

T0 review · 0 major / 4 minor · reviewed 2026-07-10 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Two new distances compare phylogenetic networks by counting vertex removals that leave isomorphic ancestry structure.

desk verdict Clean operational metrics that actually work as metrics, extend RF, and come with a sharp complexity split; solid combinatorial paper ready for referees. read the letter →

arxiv 2607.08259 v1 pith:6YKTTO2O submitted 2026-07-09 cs.DM

Marc Hellmuth , Manuel Lafond , Guillaume E. Scholz This is my paper · ORCID
classification cs.DM MSC 05C8568Q2592B1005C90
keywords phylogeneticnetworksnetworkcomparisonoperationaldistanceRobinson-Fouldshardwiredclustersshortcut-freedistinct-clustervertexcover
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Phylogenetic networks describe evolutionary histories with reticulation, but comparing them is hard: feature-based scores often fail to be true metrics, and most operational distances are restricted or NP-hard. This paper defines two distances from the ⊖-operator, which deletes a vertex and reconnects its parents to its children while preserving ancestor relations among the survivors. The distance d_⊖ is the fewest total internal vertices that must be removed from both networks until the results are leaf-fixing isomorphic; the relaxed distance d_⊖^- ignores shortcut arcs and therefore compares only the induced ancestry. Both are genuine metrics (up to the natural equivalences), both recover the classical Robinson–Foulds distance on trees, and both are bounded below by hardwired cluster distances. On broad practical classes—tree-child, normal, level-1, regular—d_⊖^- equals the multiset cluster distance and can be computed in polynomial time; d_⊖ remains hard even on those classes. For distinct-cluster networks the computation of d_⊖^- reduces exactly to Vertex Cover, giving FPT algorithms and a 2-approximation. Optimal deletion sets also serve as interpretable witnesses that localize which vertices and ancestry relations differ.

What carries the argument

The ⊖-operator (remove a vertex and join every parent to every child) together with the bad-ancestry graph of two DC-similar networks, whose minimum vertex covers are exactly the minimum bypassing sets that realize d_⊖^-.

What would settle it

Exhibit two networks inside one of the listed classes that have identical cluster multisets yet non-isomorphic shortcut-free versions, or two DC-similar networks whose minimum bypassing set is strictly smaller than twice the size of a minimum vertex cover of their bad-ancestry graph.

Watch

Extended reading notes

Core claim

The ⊖-operator yields two operational distances that are true metrics on rooted phylogenetic networks (up to leaf-fixing isomorphism for d_⊖ and shortcut-free isomorphism for d_⊖^-), both extend Robinson–Foulds, and d_⊖^- is polynomial-time computable on the main network classes used in practice while reducing to Vertex Cover on distinct-cluster networks.

Load-bearing premise

The claim that d_⊖^- equals the multiset cluster distance on tree-child, normal, level-1 and regular networks rests on those classes being both closed under shortcut-free ⊖-reduction and uniquely determined by their cluster multisets.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper introduces two operational distances on rooted phylogenetic networks based on the ⊖-operator (vertex deletion that reconnects parents to children while preserving ancestry). d_⊖ is the minimum total number of internal vertices that must be removed from two networks so that the resulting networks are leaf-fixing isomorphic; d_⊖^- requires only that the shortcut-free reductions are isomorphic. Theorems 4.3 and 4.4 establish that these are metrics up to the respective equivalence relations. Both distances extend the Robinson–Foulds distance on trees and are bounded below by the hardwired (multi)cluster distances (Lemma 5.1). For networks satisfying PCC (hence tree-child, normal, level-1, semi-regular and regular networks) d_⊖^- equals the multiset cluster distance and is therefore polynomial-time computable (Theorem 5.6). Computing d_⊖ is NP-hard, W[2]-hard and inapproximable even on DC-similar tree-child networks (Theorem 6.1). Computing d_⊖^- is NP-hard in general (Theorem 8.3) but, on distinct-cluster networks, reduces to Vertex Cover on the bad-ancestry graph, yielding FPT and 2-approximation algorithms (Theorems 7.8–7.11).

Significance. The work supplies genuine metrics (rather than pseudometrics) that apply to arbitrary rooted phylogenetic networks, recover the classical Robinson–Foulds distance on trees, and admit a transparent operational interpretation via common ancestry cores. The algorithmic dichotomy is clean: d_⊖^- is tractable on the most commonly studied network classes and reduces to a well-studied combinatorial problem on distinct-cluster networks, while d_⊖ is hard even on those classes. All central claims are accompanied by complete, self-contained proofs; the Vertex-Cover formulation immediately supplies practical exact, FPT and approximation methods. The contribution is therefore both theoretically solid and of direct utility for network comparison in computational phylogenetics.

minor comments (4)
  1. [global] Throughout: several typographical slips remain (e.g. “retriculation”, “shotcut-removal”, “bypassing sets” used before Definition 7.1 is fully motivated). A careful copy-edit would improve readability.
  2. [Figure 1] Figure 1 caption and surrounding text: the concrete numerical example (d_⊖=3, d_⊖^-=1) is helpful; adding the corresponding cluster multisets would make the lower-bound relation of Lemma 5.1 immediately visible.
  3. [§9] Section 9: the open questions on c-distinct-cluster networks and on the realizability of arbitrary graphs as bad-ancestry graphs are well-posed; a one-sentence pointer to the corresponding complexity status of coloured isomorphism would help the reader.
  4. [References] References: a few arXiv preprints are cited without final publication data (e.g. [24]); update where possible.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity; metric and complexity claims are proved from definitions and standard reductions, with only minor self-citation of prior uniqueness characterizations.

  1. uniqueness imported from authors [Proposition 5.5 (semi-regular case)]
    "By [23, Thm 5], a semi-regular network is uniquely determined, up to isomorphism, by its cluster multiset. Consequently, N^- ≃ (N')^- and we can conclude that Γ is multicluster-shortcut-encoded."

    The multicluster-shortcut-encoded property (needed for Prop. 5.4 / Thm. 5.6 equality and poly-time claim) is obtained by importing a uniqueness theorem from prior work whose author list overlaps with the present paper. The uniqueness is treated as an external fact that forces the implication 𝔐_N = 𝔐_N' ⇒ N^- ≃ (N')^-. This is a mild instance of pattern 4; it is not self-definitional inside the present paper and does not affect the metric or hardness results.

full rationale

The distances are defined operationally via the ⊖-operator (Def. 4.1 and the relaxed W^- version) independently of any target property. Metric axioms (Thms. 4.3–4.4) are verified directly by exhibiting admissible deletion sets and using order-independence of ⊖ (Lem. 3.2–3.3). The RF/multiset lower bounds (Lem. 5.1) follow by cluster preservation under ⊖ and shortcut removal. Equality d_⊖^- = |𝔐_N Δ 𝔐_N'| on the listed classes (Prop. 5.4, Thm. 5.6) is obtained by constructing explicit deletion sets that equalize the multisets and invoking that the classes are ⊖-shortcut-closed and multicluster-shortcut-encoded; the latter property for semi-regular networks cites a uniqueness theorem from the authors’ prior work [23, Thm. 5], but this is ordinary reuse of a published characterization rather than a self-definitional loop or fitted prediction. Hardness (Thm. 6.1, 8.3) and the Vertex-Cover reduction (Thm. 7.8) are ordinary many-one reductions with explicit constructions. No quantity is fitted and then re-predicted; no ansatz is smuggled; the central claims do not reduce to their inputs by construction. The single self-citation of uniqueness is non-load-bearing for the metric axioms or the hardness results and does not force the poly-time claims beyond what the paper re-proves from its own lemmas on cluster/ancestor preservation.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The paper works entirely inside standard DAG/network combinatorics. The only external ingredients are previously published characterizations of network classes and the classical NP-completeness of Set Cover and Vertex Cover; no free parameters or ad-hoc physical entities are introduced.

assumptions (3)
  • domain assumption The ⊖-operator is order-independent and preserves the ancestor relation among remaining vertices (Lemmas 3.2–3.3, taken from prior work).
    Used throughout the metric and reduction arguments; cited from [34,43].
  • domain assumption Regular / semi-regular / (PCC) networks are uniquely determined (up to shortcut removal) by their cluster multisets.
    Invoked in Proposition 5.5 and Theorem 5.6 to obtain the poly-time equality with multiset cluster distance.
  • standard math Set Cover is NP-complete and W[2]-hard; Vertex Cover is NP-complete on 3-connected cubic girth>3 graphs and admits a 2-approximation and FPT algorithms.
    Used for the hardness and algorithmic results in Sections 6–8.
invented entities (2)
  • ⊖-distance d_⊖ and ancestry distance d_⊖^-
    purpose: Provide operational metrics that count vertex deletions needed to make two networks isomorphic (or ancestry-isomorphic).
    Core objects of the paper; defined from the existing ⊖-operator.
  • bad ancestry graph B_{N,N'}
    purpose: Encode pairwise ancestry disagreements so that minimum bypassing sets become ordinary vertex covers.
    Enables the FPT/2-approx results for distinct-cluster networks.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The $\ominus$-metric to compare phylogenetic networks." pith.science (2026). https://pith.science/paper/6YKTTO2O

@misc{pith2026260708259,
  author       = {Pith},
  title        = {Pith review of: The $\ominus$-metric to compare phylogenetic networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6YKTTO2O}},
  note         = {Machine review of arXiv:2607.08259}
}
abstract

We introduce two novel distances for comparing rooted phylogenetic networks based on the $\ominus$-operator, which removes a vertex while preserving the ancestor relations among the remaining vertices. The distance $d_{\ominus}$ measures the minimum number of such removals needed to obtain isomorphic networks, whereas $d_{\ominus}^-$ ignores shortcut arcs and therefore compares the induced ancestry structures. We show that $d_{\ominus}$ is a metric up to leaf-fixing isomorphism and that $d_{\ominus}^-$ is a metric up to shortcut-free isomorphism. Moreover, both distances extend the Robinson--Foulds distance on phylogenetic trees and are bounded below by the hardwired cluster distances. For several broad network classes, including tree-child, normal, level-$1$, and regular networks, $d_{\ominus}^-$ can be computed in polynomial time. In contrast, computing $d_{\ominus}$ is NP-hard, W[2]-hard when parameterized by the distance value, and admits no polynomial-time constant-factor approximation unless $\mathrm{P}=\mathrm{NP}$. Although computing $d_{\ominus}^-$ is NP-hard in general, for distinct-cluster networks it reduces to \textsc{Vertex Cover}, yielding a fixed-parameter algorithm and a polynomial-time $2$-approximation.

Figures

Figures reproduced from arXiv: 2607.08259 by the authors.

Figure 1
Figure 1. A network 𝑁, a phylogenetic tree 𝑇, and several ⊖-reductions. The arc 𝜌 → 𝑐 in 𝑁 ⊖ 𝑤 is a shortcut, since there is also the directed path 𝜌 → 𝑢 → 𝑐. Here, (𝑁 ⊖ 𝑤) − ≃ 𝑇, and therefore 𝑑 − ⊖ (𝑁, 𝑇) = 1. In contrast, 𝑁 ⊖ 𝑤 is not isomorphic to 𝑇. To obtain isomorphic networks without removing shortcuts, we must instead “suppress” both 𝑢 and 𝑤 from 𝑁 and 𝑢 from 𝑇. This yields 𝑁 ⊖ {𝑢, 𝑤} ≃ 𝑇 ⊖ 𝑢, where both reduced netw… view at source ↗
Figure 2
Figure 2. A network 𝑁, the ⊖-reductions 𝑁 ⊖ 𝑣, the shortcut-free ⊖-reduction (𝑁 ⊖ 𝑣) − , and the star-tree 𝑁 ⊖ 𝑉 0 (𝑁). The following result shows that, when computing 𝑁 ⊖ 𝑊, shortcuts can be removed at any stage of the sequence of ⊖-operations. Lemma 3.3. Let 𝑁 be a network and let 𝑊 ⊆ 𝑉(𝑁). Then (𝑁 ⊖ 𝑊) − = (𝑁 − ⊖ 𝑊) − . Proof. Both (𝑁 ⊖ 𝑊) − and (𝑁 − ⊖ 𝑊) − have vertex set 𝑉(𝑁) \ 𝑊. Moreover, removing shortcuts does not ch… view at source ↗
Figure 3
Figure 3. A network 𝑁 and a phylogenetic tree 𝑇. The cluster of non-leaf vertices are indicated next to the respective vertex. Both 𝑁 and 𝑇 are distinct-cluster networks. In addition, the networks (𝑁 ⊖ 𝑤) − ≃ 𝑇 (where shortcuts of 𝑁⊖𝑤 are highlighted by gray-dashed arcs) and the star-tree 𝑁⊖𝑉 0 (𝑁) ≃ 𝑇⊖𝑉 0 (𝑇) are shown. Here, |ℭ𝑁 Δℭ𝑁′ | = |𝔐𝑁 Δ𝔐𝑁′ | = |{{𝑏, 𝑐}}| = 1 = 𝑑 − ⊖ (𝑇, 𝑁) = |{𝑤}| + |∅| < 𝑑⊖ (𝑇, 𝑁) = |𝑉 0 (𝑁)| + |𝑉 0… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: A phylogenetic tree 𝑇 and 𝑇 ′ together with the phylogenetic tree 𝑇 ⊖ 𝑤 ≃ 𝑇 ⊖ 𝑤 ′ . Both 𝑇 and 𝑇 ′ are regular. The cluster of non-leaf vertices are indicated next to the respective vertex. Following the construction used in Proposition 5.4, the vertices corresponding …
Figure 5
Figure 5. Figure 5: Shown are two networks 𝑁 and 𝑁b that serve as generic instances for which the problem of deciding as whether 𝑑⊖ (𝑁, 𝑁′ ) ≤ 𝛿 for some integer 𝛿 ≥ 0 is NP-hard, see Theorem 6.1. We claim that the Set Cover instance has a solution of size at most 𝑘 if and only if 𝑑⊖ (𝑁, …
Figure 6
Figure 6. Figure 6: A phylogenetic tree 𝑇 and a network 𝑁. Here, 𝑇 and 𝑁 are DC-similar and have canonically identified vertex sets. In this example, (𝑤, 𝑤′ ) is the unique bad ancestry pair of 𝑇 and 𝑁. The bad ancestry graph 𝐵𝑇,𝑁 consists of the vertices 𝑤 and 𝑤 ′ together with the undir…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

  1. [2]

    Algorithmica 71(1):120–138

    Arvind V, Das B, K ¨obler J, Toda S (2015) Colored hypergraph isomorphism is fixed parameter tractable. Algorithmica 71(1):120–138

    work page 2015

  2. [1]

    Theoretical Computer Science 609:211–225, DOI 10.1016/j.tcs.2015.09.023

    Akiba T, Iwata Y (2016) Branch-and-reduce exponential/FPT algorithms in practice: A case study of vertex cover. Theoretical Computer Science 609:211–225, DOI 10.1016/j.tcs.2015.09.023

    work page doi:10.1016/j.tcs.2015.09.023 2016

  3. [3]

    Annals of Combina- torics 8:391–408, DOI 10.1007/s00026-004-0228-0

    Baroni M, Semple C, Steel M (2004) A framework for representing reticulate evolution. Annals of Combina- torics 8:391–408, DOI 10.1007/s00026-004-0228-0

    work page doi:10.1007/s00026-004-0228-0 2004

  4. [4]

    (1976) Graph theory with applications, vol 290

    Bondy JA, Murty USR, et al. (1976) Graph theory with applications, vol 290. Macmillan London

    work page 1976

  5. [5]

    Journal of theoretical biology 423:1–12

    Bordewich M, Linz S, Semple C (2017) Lost in space? generalising subtree prune and regraft to spaces of phylogenetic networks. Journal of theoretical biology 423:1–12

    work page 2017

  6. [6]

    IEEE/ACM Transactions on Computational Biology and Bioinformatics 6:552–569, DOI 10.1109/TCBB.2007.70270

    Cardona G, Rossell ´o F, Valiente G (2007) Comparison of tree-child phylogenetic networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics 6:552–569, DOI 10.1109/TCBB.2007.70270

    work page doi:10.1109/tcbb.2007.70270 2007

  7. [7]

    IEEE/ACM Transactions on Computational Biology and Bioinformatics 6(1):46– 61

    Cardona G, Llabr ´es M, Rossell´o F, Valiente G (2008) Metrics for phylogenetic networks i: Generalizations of the robinson-foulds metric. IEEE/ACM Transactions on Computational Biology and Bioinformatics 6(1):46– 61

    work page 2008

  8. [8]

    IEEE/ACM Transactions on Computational Biology and Bioinformatics 6(4):552–569

    Cardona G, Rossell ´o F, Valiente G (2008) Comparison of tree-child phylogenetic networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics 6(4):552–569

    work page 2008

Show all 49 references
  1. [9]

    Mathematical Biosciences 211(2):356–370

    Cardona G, Rossell ´o F, Valiente G (2008) Tripartitions do not always discriminate phylogenetic networks. Mathematical Biosciences 211(2):356–370

    2008

  2. [10]

    The Scientific World Journal 2014(1):254279

    Cardona G, Llabr ´es M, Rossell´o F, Valiente G (2014) The comparison of tree-sibling time consistent phylo- genetic networks is graph isomorphism-complete. The Scientific World Journal 2014(1):254279

    2014

  3. [11]

    IEEE/ACM Transactions on Computational Biology and Bioinformatics 21(3):501–507

    Cardona G, Pons JC, Ribas G, Coronado TM (2024) Comparison of orchard networks using their extended 𝜇-representation. IEEE/ACM Transactions on Computational Biology and Bioinformatics 21(3):501–507

    2024

  4. [12]

    Theoretical Computer Science 411(40–42):3736–3756, DOI 10.1016/j.tcs.2010.06.026

    Chen J, Kanj IA, Xia G (2010) Improved upper bounds for vertex cover. Theoretical Computer Science 411(40–42):3736–3756, DOI 10.1016/j.tcs.2010.06.026

    2010 doi

  5. [13]

    MIT Press

    Cormen TH, Leiserson CE, Rivest RL, Stein C (2009) Introduction to Algorithms, 3rd edn. MIT Press

    2009

  6. [14]

    In: Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pp 624–633

    Dinur I, Steurer D (2014) Analytical approach to parallel repetition. In: Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pp 624–633

    2014

  7. [15]

    Springer

    Downey RG, Fellows MR (2013) Fundamentals of Parameterized Complexity. Springer

    2013

  8. [16]

    Discrete Applied Mathematics 294:205–213

    Erd ˝os PL, Francis A, Mezei TR (2021) Rooted NNI moves and distance-1 tail moves on tree-based phylogenetic networks. Discrete Applied Mathematics 294:205–213

    2021

  9. [17]

    Journal of mathe- matical biology 65(1):157–180

    Gambette P, Huber KT (2012) On encodings of phylogenetic networks of bounded level. Journal of mathe- matical biology 65(1):157–180

    2012

  10. [18]

    Journal of mathematical biology 74(7):1729–1751

    Gambette P, Huber KT, Kelk S (2017) On the challenge of reconstructing level-1 phylogenetic networks from triplets and clusters. Journal of mathematical biology 74(7):1729–1751

    2017

  11. [19]

    PLoS computational biology 13(8):e1005611

    Gambette P, Van Iersel L, Jones M, Lafond M, Pardi F, Scornavacca C (2017) Rearrangement moves on rooted phylogenetic networks. PLoS computational biology 13(8):e1005611

    2017

  12. [20]

    Garey MR, Johnson DS (1979) Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco 25

    1979

  13. [21]

    Harris DG, Narayanaswamy NS (2024) A faster algorithm for vertex cover parameterized by solution size. In: Proceedings of the 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024), Schloss Dagstuhl – Leibniz-Zentrum f¨ ur Informatik, Leibniz Inte...

    2024 doi

  14. [22]

    Discrete Applied Mathematics 378:584–593, DOI 10.1016/j.dam.2025.08.037

    Hellmuth M, Lindeberg A (2026) Characterizing and transforming DAGs within the J-lca framework. Discrete Applied Mathematics 378:584–593, DOI 10.1016/j.dam.2025.08.037

    2026 doi

  15. [23]

    Theory in Biosciences 142(4):301–358, DOI 10.1007/s12064-023-00398-w

    Hellmuth M, Schaller D, Stadler PF (2023) Clustering systems of phylogenetic networks. Theory in Biosciences 142(4):301–358, DOI 10.1007/s12064-023-00398-w

    2023 doi

  16. [24]

    URLhttps://arxiv.org/abs/2605.21725,2605.21725

    Hellmuth M, Lindeberg A, Moulton V (2026) Regularizing and normalizing DAGs and phylogenetic networks. URLhttps://arxiv.org/abs/2605.21725,2605.21725

    2026 arXiv

  17. [25]

    In: Proceedings of the 21st Workshop on Algorithm Engineering and Experiments (ALENEX 2020), SIAM, pp 1–13, DOI 10.1137/1.9781611976007.1

    Hespe D, Lamm S, Schulz C, Strash D (2020) WeGotYouCovered: The winning solver from the PACE 2019 implementation challenge, vertex cover track. In: Proceedings of the 21st Workshop on Algorithm Engineering and Experiments (ALENEX 2020), SIAM, pp 1–13, DOI 10.1137/1.9781611976007.1

    2020 doi

  18. [26]

    In: Hochbaum DS (ed) Approximation Algorithms for NP-Hard Problems, PWS Publishing Company, Boston, pp 94–143

    Hochbaum DS (1997) Approximating covering and packing problems: Set cover, vertex cover, independent set and related problems. In: Hochbaum DS (ed) Approximation Algorithms for NP-Hard Problems, PWS Publishing Company, Boston, pp 94–143

    1997

  19. [27]

    Cambridge University Press, Cambridge, UK

    Huson DH, Rupp R, Scornavacca C (2010) Phylogenetic Networks: Concepts, Algorithms and Applications, 1st edn. Cambridge University Press, Cambridge, UK

    2010

  20. [28]

    Bulletin of Mathematical Biology 80:2177–2208, DOI 10.1007/s11538-018-0452-0

    Janssen R, Jones M, Erd ˝os PL, van Iersel L, Scornavacca C (2018) Exploring the tiers of rooted phylogenetic network space using tail moves. Bulletin of Mathematical Biology 80:2177–2208, DOI 10.1007/s11538-018-0452-0

    2018 doi

  21. [29]

    Algorithmica 83(6):1786–1828

    Jansson J, Mampentzidis K, Rajaby R, Sung WK (2021) Computing the rooted triplet distance between phylogenetic networks. Algorithmica 83(6):1786–1828

    2021

  22. [30]

    ACM Trans Algorithms 5(4), DOI 10.1145/1597036.1597045

    Karakostas G (2009) A better approximation ratio for the vertex cover problem. ACM Trans Algorithms 5(4), DOI 10.1145/1597036.1597045

    2009 doi

  23. [31]

    The Electronic Journal of Combinatorics pp P2–3

    Klawitter J, Linz S (2019) On the subnet prune and regraft distance. The Electronic Journal of Combinatorics pp P2–3

    2019

  24. [32]

    Systematic Biology 74(1):53–69

    Kong S, Swofford DL, Kubatko LS (2025) Inference of phylogenetic networks from sequence data using composite likelihood. Systematic Biology 74(1):53–69

    2025

  25. [33]

    IEEE/ACM Transactions on Computational Biology and Bioinformatics

    Landry K, Teodocio A, Lafond M, Tremblay-Savard O (2022) Defining phylogenetic network distances using cherry operations. IEEE/ACM Transactions on Computational Biology and Bioinformatics

    2022

  26. [34]

    Bulletin of Mathematical Biology 87(3):44, DOI 10.1007/s11538-025-01419-z

    Lindeberg A, Hellmuth M (2025) Simplifying and characterizing DAGs and phylogenetic networks via least common ancestor constraints. Bulletin of Mathematical Biology 87(3):44, DOI 10.1007/s11538-025-01419-z

    2025 doi

  27. [35]

    BMC genomics 18(Suppl 2):111

    Lu B, Zhang L, Leong HW (2017) A program to compute the soft robinson–foulds distance between phyloge- netic networks. BMC genomics 18(Suppl 2):111

    2017

  28. [36]

    Bioinformatics 38(15):3725–3733

    Lutteropp S, Scornavacca C, Kozlov AM, Morel B, Stamatakis A (2022) Netrax: accurate and fast maximum likelihood phylogenetic network inference. Bioinformatics 38(15):3725–3733

    2022

  29. [37]

    Algorithms for Molecular Biology 20(1):18

    Marchand B, Tahiri N, Fard SG, Tremblay-Savard O, Lafond M (2025) Finding maximum common contractions between phylogenetic networks. Algorithms for Molecular Biology 20(1):18

    2025

  30. [38]

    In: RECOMB International Workshop on Comparative Genomics, Springer, pp 234–264 26

    Marchand B, Tahiri N, Tremblay-Savard O, Lafond M (2026) On the comparison of lgt networks and tree-based networks. In: RECOMB International Workshop on Comparative Genomics, Springer, pp 234–264 26

    2026

  31. [39]

    IEEE Transactions on Computational Biology and Bioinformatics 22(2):684–696

    Maxfield M, Xu J, An ´e C (2025) A dissimilarity measure for semidirected networks. IEEE Transactions on Computational Biology and Bioinformatics 22(2):684–696

    2025

  32. [40]

    Advances in Applied Mathematics 172:102953

    Reichling C, van Iersel L, Murakami Y (2026) Metrics for classes of semi-binary phylogenetic networks using 𝜇-representations. Advances in Applied Mathematics 172:102953

    2026

  33. [41]

    Mathematical Biosciences 53(1):131–147, DOI 10.1016/0025-5564(81)90043-2

    Robinson D, Foulds L (1981) Comparison of phylogenetic trees. Mathematical Biosciences 53(1):131–147, DOI 10.1016/0025-5564(81)90043-2

    1981 doi

  34. [42]

    Oxford University Press, Oxford

    Semple C, Steel M (2003) Phylogenetics. Oxford University Press, Oxford

    2003

  35. [43]

    In: Kalyanasundaram S, Maheshwari A (eds) Algorithms and Discrete Applied Mathematics, Springer Nature Switzerland, Cham, pp 148–161, DOI 10.1007/978-3-031-52213-0 11

    Shanavas A V, Changat M, Hellmuth M, Stadler PF (2024) Unique least common ancestors and clusters in directed acyclic graphs. In: Kalyanasundaram S, Maheshwari A (eds) Algorithms and Discrete Applied Mathematics, Springer Nature Switzerland, Cham, pp 148–161, DOI 10.1007/978-3...

    2024 doi

  36. [44]

    PLoS Genetics 12(3):e1005896

    Sol ´ıs-Lemus C, An´e C (2016) Inferring phylogenetic networks with maximum pseudolikelihood under incom- plete lineage sorting. PLoS Genetics 12(3):e1005896

    2016

  37. [45]

    Uehara R (1996) NP-complete problems on a 3-connected cubic planar graph and their applications. Tech. Rep. TWCU-M-0004, Tokyo Woman’s Christian University, URLhttps://www.jaist.ac.jp/˜uehara/ pdf/triangle.pdf

    1996

  38. [46]

    Briefings in Bioinformatics 20(6):1972–1983, DOI 10.1093/bib/bby044

    Wang J (2019) A review of metrics measuring dissimilarity for rooted phylogenetic networks. Briefings in Bioinformatics 20(6):1972–1983, DOI 10.1093/bib/bby044

    2019 doi

  39. [47]

    Systematic biology 67(4):735–740

    Wen D, Yu Y, Zhu J, Nakhleh L (2018) Inferring phylogenetic networks using phylonet. Systematic biology 67(4):735–740

    2018

  40. [48]

    Bulletin of Mathematical Biology 72(2):340– 358, DOI 10.1007/s11538-009-9449-z

    Willson SJ (2010) Properties of normal phylogenetic networks. Bulletin of Mathematical Biology 72(2):340– 358, DOI 10.1007/s11538-009-9449-z

    2010 doi

  41. [49]

    Annals of Combinatorics 20(4):917–938, DOI 10.1007/s00026-016-0324-y 27

    Willson SJ (2016) Comparing and simplifying distinct-cluster phylogenetic networks. Annals of Combinatorics 20(4):917–938, DOI 10.1007/s00026-016-0324-y 27

    2016 doi

Pith tools

Reviewed July 10, 2026 · model on record in the stance chip above.