On the challenge of reconstructing level-1 phylogenetic networks from triplets and clusters

Phylogenetic networks have gained prominence over the years due to their ability to represent complex non-treelike evolutionary events such as recombination or hybridization. Popular combinatorial objects used to construct them are triplet systems and cluster systems, the motivation being that any network N induces a triplet system R(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal R(N)$$\end{document} and a softwired cluster system S(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal S(N)$$\end{document}. Since in real-world studies it cannot be guaranteed that all triplets/softwired clusters induced by a network are available, it is of particular interest to understand whether subsets of R(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal R(N)$$\end{document} or S(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal S(N)$$\end{document} allow one to uniquely reconstruct the underlying network N. Here we show that even within the highly restricted yet biologically interesting space of level-1 phylogenetic networks it is not always possible to uniquely reconstruct a level-1 network N, even when all triplets in R(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal R(N)$$\end{document} or all clusters in S(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal S(N)$$\end{document} are available. On the positive side, we introduce a reasonably large subclass of level-1 networks the members of which are uniquely determined by their induced triplet/softwired cluster systems. Along the way, we also establish various enumerative results, both positive and negative, including results which show that certain special subclasses of level-1 networks N can be uniquely reconstructed from proper subsets of R(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal R(N)$$\end{document} and S(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal S(N)$$\end{document}. We anticipate these results to be of use in the design of algorithms for phylogenetic network inference.