Representing distance-hereditary graphs with multi-rooted trees

TL;DR

The work establishes that distance-hereditary graphs are exactly the graphs that can be explained by labelled arboreal networks, extending the known correspondence between cographs and labelled rooted trees to a broader arboreal setting with multiple roots. It provides a constructive algorithm to convert any distance-hereditary graph into a labelled arboreal network, analyzes the relationship between the explained graph $G=\mathcal C(N,t)$ and its Ptolemaic completion $G^*=\mathcal A(N)$, and characterizes when such networks can have two roots via GaTEx graphs and 0-basic galled-trees. The results connect network-based representations to classical graph classes (distance-hereditary, Ptolemaic, GaTEx), yielding insights for evolutionary modelling and enabling potential polynomial-time solutions for certain problems on distance-hereditary graphs. Open questions include finding the minimum number of roots required for an optimal arboreal-explanation of a given $G$ and understanding how minimal chordal completions relate to arboreal representations.

Abstract

Arboreal networks are a generalization of rooted trees, defined by keeping the tree-like structure, but dropping the requirement for a single root. Just as the class of cographs is precisely the class of undirected graphs that can be explained by a labelled rooted tree (T,t), we show that the class of distance-hereditary graphs is precisely the class of undirected graphs that can be explained by a labelled arboreal network (N,t).

Figures (9)

• Figure 1: (i) The house. (ii) The gem. (iii) The domino.
• Figure 2: (i) A phylogenetic tree with leaf set $X=\{1,2,3,4,5\}$. (ii) An arboreal network with two roots and leaf set $X$. A network with three roots and leaf set $X$ that is not arboreal. Here and in all subsequent Figures depicting networks, all arcs are assumed to be directed downwards. Note that only the network in (i) is binary.
• Figure 3: (i), (ii), (iii) The shared ancestry graph of the network depicted in Figure \ref{['fig-ntw']} (i), (ii) and (iii), respectively.
• Figure 4: (i) An undirected graph $G$ with vertex set $\{1,2,3,4,5\}$. (ii) to (iv) Three distinct binary labelled arboreal networks explaining $G$, where vertices with label $1$ are indicated with $\bullet$, and vertices with label $0$ are indicated with $\circ$.
• Figure 5: (i) An undirected graph $G$ with vertex set $\{1,2,3,4,5,6\}$. (ii) A supergraph $G^*$ of $G$ on the same vertex set. Although $G$ is distance-hereditary, and therefore, arboreal-explainable, and $G^*$ is connected and Ptolemaic, there exists no labelled-arboreal network $(N,t)$ such that $\mathcal{C}(N,t)=G$ and $\mathcal{A}(N)=G^*$.

Theorems & Definitions (39)

• lemma 1
• lemma 2: HMS24, Lemma 3.1 and Proposition 3.2
• lemma 3
• proof
• lemma 4: HMS24, Proposition 7.1
• lemma 5
• proof
• lemma 6
• proof
• theorem 1: HMS24, Theorem 6.4