Evolutionary events like hybridization and horizontal gene transfer cannot be represented by a tree but require a phylogenetic network instead. Nonetheless, it has been argued that evolution is fundamentally tree-like , in the sense that it can be modeled by a tree with additional arcs. Phylogenetic networks with this property are called tree-based, and this concept has recently attracted significant attention in evolutionary research. But classifying networks into tree-based and non-tree-based ones is usually not sufficient for biological purposes, because sometimes tree-basedness is missed merely due to incomplete sampling. Therefore, measures are needed to quantify how close a non-tree-based network is to being tree-based . Such measuresnaturally also lead to characterizations of tree-based networks, and some measures for this purpose have recently been introduced both for binary and non-binary rooted phylogenetic networks . In the present manuscript, we generalize all these measures to unrooted networks, for which tree-basedness is mathematically more involved, and also present some new measures. This leads to new characterizations of unrooted tree-based networks . We analyze the relationships of the new proximity measures to one another to demonstrate their common properties as well as their respective differences .
Tree-based networks are a class of phylogenetic networks that attempt to formally capture what is meant by `` tree-like '' evolution. A given non-tree-based phylogenetic network, however, might appear to be very close to being tree-based , or very far. In this paper, we formalise the notion of proximity to tree-based for unrooted phylogenetic networks, with a range of proximity measures. These measures also provide characterisations of tree-based networks. One measure in particular, related to the nearest neighbour interchange operation, allows us to define the notion of ``tree-based rank''. This provides a subclassification within the tree-based networks themselves, identifying those networks that are ``very'' tree-based. Finally, we prove results relating tree-based networks in the settings of rooted and unrooted phylogenetic networks, showing effectively that an unrooted network is tree-based if and only if it can be made a rooted tree-based network by rooting it and orienting the edges appropriately . This leads to a clarification of the contrasting decision problems for tree-based networks , which are polynomial in the rooted case but NP complete in the unrooted .
R
Evolutionary events like hybridization and horizontal gene transfer cannot be represented by a tree but require a phylogenetic network instead. Nonetheless, it has been argued that evolution is fundamentally
Tree-based networks are a class of phylogenetic networks that attempt to formally capture what is meant by ``
R
, in the sense that it can be modeled by a tree with additional arcs. Phylogenetic networks with this property are called tree-based, and this concept has recently attracted significant attention in evolutionary research. But classifying networks into tree-based and
'' evolution. A given
R
ones is usually not sufficient for biological purposes, because sometimes tree-basedness is missed merely due to incomplete sampling. Therefore, measures are needed to quantify how close a non-tree-based network is
phylogenetic network, however, might appear to be very close
R
. Such measuresnaturally also lead to characterizations of
, or very far. In this paper, we formalise the notion of proximity to
R
networks, and some measures for this purpose have recently been introduced both for binary and non-binary rooted phylogenetic networks
for unrooted phylogenetic networks, with a range of proximity measures. These measures also provide characterisations of tree-based networks. One measure in particular, related to the nearest neighbour interchange operation, allows us to define the notion of ``tree-based rank''. This provides a subclassification within the tree-based networks themselves, identifying those networks that are ``very'' tree-based. Finally, we prove results relating tree-based networks in the settings of rooted and unrooted phylogenetic networks, showing effectively that an unrooted network is tree-based if and only if it can be made a rooted tree-based network by rooting it and orienting the edges appropriately
D
In the present manuscript, we generalize all these measures to unrooted networks, for which tree-basedness is mathematically more involved, and also present some new measures.
R
new characterizations of unrooted
a clarification of the contrasting decision problems for
R
. We analyze the relationships of the new proximity measures to one another to demonstrate their common properties as well as their respective differences
, which are polynomial in the rooted case but NP complete in the unrooted