A rearrangement operation makes a small graph-theoretical change to a phylogenetic network to transform it into another one. For unrooted phylogenetic trees and networks, popular rearrangement operations are tree bisection and reconnection (TBR) and prune and regraft (PR) (called subtree prune and regraft (SPR) on trees). Each of these operations induces a metric on the sets of phylogenetic trees and networks. The TBR-distance between two unrooted phylogenetic trees $T$ and $T'$ can be characterised by a maximum agreement forest, that is, a forest with a minimum number of components that covers both $T$ and $T'$ in a certain way. This characterisation has facilitated the development of fixed-parameter tractable algorithms and approximation algorithms. Here, we introduce maximum agreement graphs as a generalisations of maximum agreement forests for phylogenetic networks. While the agreement distance -- the metric induced by maximum agreement graphs -- does not characterise the TBR-distance of two networks, we show that it still provides constant-factor bounds on the TBR-distance. We find similar results for PR in terms of maximum endpoint agreement graphs.

中文翻译：

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无根系统发育网络的一致性距离

重排操作对系统发育网络进行图论上的小改动，以将其转换为另一个网络。对于无根系统发育树和网络，流行的重排操作是树二分和重新连接 (TBR) 和修剪和重新嫁接 (PR)（在树上称为子树修剪和重新嫁接（SPR））。这些操作中的每一个都会在系统发育树和网络集上引入一个度量。两棵无根系统发育树 $T$ 和 $T'$ 之间的 TBR-距离可以用最大一致性森林来表征，即在一个森林中覆盖 $T$ 和 $T'$ 的组件数量最少的森林。某种方式。这种表征促进了固定参数易处理算法和近似算法的开发。这里，我们引入最大一致性图作为系统发育网络最大一致性森林的概括。虽然一致性距离——由最大一致性图引起的度量——不能表征两个网络的 TBR 距离，但我们表明它仍然提供了 TBR 距离的常数因子界限。我们在最大端点一致性图方面发现 PR 的类似结果。