The rigid hybrid number for two phylogenetic trees

Area of Science:

• Evolutionary biology
• Computational phylogenetics
• Bioinformatics

Background:

• Phylogenetic networks model complex evolutionary histories beyond tree structures.
• Traditional methods of displaying trees in networks can be overly restrictive for phenomena like incomplete lineage sorting.
• Minimum hybrid number seeks the simplest network displaying given trees, but this can be too strict.

Purpose of the Study:

• Introduce and analyze the concept of "rigidly displaying" phylogenetic trees within networks.
• Characterize the conditions under which two trees can be rigidly displayed by a temporal tree-child network.
• Investigate the relationship between the rigid hybrid number and other hybrid number definitions.

Main Methods:

• Definition and study of "rigidly displaying" for phylogenetic trees in networks.
• Characterization using novel "fork-picking sequences" derived from tree subconfigurations.
• Analysis of minimum weight fork-picking sequences for determining the rigid hybrid number.
• Comparison of rigid hybrid number with weak, beaded, and temporal hybrid numbers.

Main Results:

• Characterization of when two trees can be rigidly displayed by temporal tree-child networks via fork-picking sequences.
• Demonstration that the rigid hybrid number is given by a minimum weight fork-picking sequence.
• Identification that weak, beaded, and temporal hybrid numbers can differ even for the same pair of trees.
• Proof that the difference between rigid and temporal hybrid numbers can grow linearly with the number of leaves.

Conclusions:

• Rigidly displaying offers a more flexible framework for phylogenetic network construction than strict tree display.
• Fork-picking sequences provide a computational tool for analyzing rigid display and hybrid numbers.
• The distinctions between various hybrid number definitions highlight the complexity of modeling reticulate evolution.