Injective split systems

Hellmuth, Marc, Huber, Katharina T., Moulton, Vincent ORCID:, Scholz, Guillaume and Stadler, Peter F. (2023) Injective split systems. Graphs and Combinatorics. ISSN 0911-0119 (In Press)

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A split system $\mathcal S$ on a finite set $X$, $|X|\ge3$, is a set of bipartitions or splits of $X$ which contains all splits of the form $\{x,X-\{x\}\}$, $x \in X$. To any such split system $\mathcal S$ we can associate the Buneman graph $\mathcal B(\mathcal S)$ which is essentially a median graph with leaf-set $X$ that displays the splits in $\mathcal S$. In this paper, we consider properties of injective split systems, that is, split systems $\mathcal S$ with the property that $\med_{\mathcal B(\mathcal S)}(Y) \neq \med_{\mathcal B(\mathcal S)}(Y')$ for any 3-subsets $Y,Y'$ in $X$, where $\med_{\mathcal B(\mathcal S)}(Y)$ denotes the median in $\mathcal B(\mathcal S)$ of the three elements in $Y$ considered as leaves in $\mathcal B(\mathcal S)$. In particular, we show that for any set $X$ there always exists an injective split system on $X$, and we also give a characterization for when a split system is injective. We also consider how complex the Buneman graph $\mathcal B(\mathcal S)$ needs to become in order for a split system $\mathcal S$ on $X$ to be injective. We do this by introducing a quantity for $|X|$ which we call the injective dimension for $|X|$, as well as two related quantities, called the injective 2-split and the rooted-injective dimension. We derive some upper and lower bounds for all three of these dimensions and also prove that some of these bounds are tight. An underlying motivation for studying injective split systems is that they can be used to obtain a natural generalization of symbolic tree maps. An important consequence of our results is that any three-way symbolic map on $X$ can be represented using Buneman graphs.

Item Type: Article
Uncontrolled Keywords: median graph,split system,buneman graph
Faculty \ School: Faculty of Science > School of Computing Sciences
Depositing User: LivePure Connector
Date Deposited: 02 May 2023 17:30
Last Modified: 25 May 2023 08:31

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