Computer Science > Discrete Mathematics
[Submitted on 18 Nov 2019 (v1), last revised 30 Jan 2020 (this version, v3)]
Title:Generalized Fitch Graphs II: Sets of Binary Relations that are explained by Edge-labeled Trees
View PDFAbstract:Fitch graphs $G=(X,E)$ are digraphs that are explained by $\{\emptyset, 1\}$-edge-labeled rooted trees $T$ with leaf set $X$: there is an arc $(x,y) \in E$ if and only if the unique path in $T$ that connects the last common ancestor $\mathrm{lca}(x,y)$ of $x$ and $y$ with $y$ contains at least one edge with label "1". In practice, Fitch graphs represent xenology relations, i.e., pairs of genes $x$ and $y$ for which a horizontal gene transfer happened along the path from $\mathrm{lca}(x,y)$ to $y$.
In this contribution, we generalize the concept of Fitch graphs and consider trees $T$ that are equipped with edge-labeling $\lambda: E\to \mathcal{P}(M)$ that assigns to each edge a subset $M'\subseteq M$ of colors. Given such a tree, we can derive a map $\varepsilon_{(T,\lambda)}$ (or equivalently a set of not necessarily disjoint binary relations), such that $i\in \varepsilon_{(T,\lambda)}(x,y)$ (or equivalently $(x,y)\in R_i$) with $x,y\in X$, if and only if there is at least one edge with color $i$ from $\mathrm{lca}(x,y)$ to $y$.
The central question considered here: Is a given map $\varepsilon$ a Fitch map, i.e., is there there an edge-labeled tree $(T,\lambda)$ with $\varepsilon_{(T,\lambda)} = \varepsilon$, and thus explains $\varepsilon$? Here, we provide a characterization of Fitch maps in terms of certain neighborhoods and forbidden submaps. Further restrictions of Fitch maps are considered. Moreover, we show that the least-resolved tree explaining a Fitch map is unique (up to isomorphism). In addition, we provide a polynomial-time algorithm to decide whether $\varepsilon$ is a Fitch map and, in the affirmative case, to construct the (up to isomorphism) unique least-resolved tree $(T^*,\lambda^*)$ that explains $\varepsilon$.
Submission history
From: Marc Hellmuth [view email][v1] Mon, 18 Nov 2019 07:41:39 UTC (66 KB)
[v2] Wed, 15 Jan 2020 13:12:59 UTC (66 KB)
[v3] Thu, 30 Jan 2020 16:20:27 UTC (94 KB)
Current browse context:
cs.DM
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
Connected Papers (What is Connected Papers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.