Characterizations of undirected 2-quasi best match graphs

TL;DR

This work characterizes undirected 2-quasi-best-match graphs (un2qBMGs) as the $(P_6,C_6,Sunlet_4)$-free bipartite graphs, equivalently the $(P_6,C_6)$-free bi-cographs, and shows they admit a tree-based explanation via least-resolved trees. It introduces the HEART-TREE algorithm, achieving cubic-time recognition and constructive explanations, while also enabling linear-time recognition through existing $(P_6,C_6)$-free bi-cograph decompositions. The results reveal a precise structural framework centered on heart-vertices (vertices adjacent to all opposite-color vertices) and establish heredity, which underpins both the algorithmic approach and the subgraph analyses. This advances the understanding of how phylogenetic-type relations project onto undirected graph classes and paves the way for efficient, decomposition-based algorithms in this domain.

Abstract

Bipartite best match graphs (BMG) and their generalizations arise in mathematical phylogenetics as combinatorial models describing evolutionary relationships among related genes in a pair of species. In this work, we characterize the class of \emph{undirected 2-quasi-BMGs} (un2qBMGs), which form a proper subclass of the $P_6$-free chordal bipartite graphs. We show that un2qBMGs are exactly the class of bipartite graphs free of $P_6$, $C_6$, and the eight-vertex Sunlet$_4$ graph. Equivalently, a bipartite graph $G$ is un2qBMG if and only if every connected induced subgraph contains a ``heart-vertex'' which is adjacent to all the vertices of the opposite color. We further provide a $O(|V(G)|^3)$ algorithm for the recognition of un2qBMGs that, in the affirmative case, constructs a labeled rooted tree that ``explains'' $G$. Finally, since un2qBMGs coincide with the $(P_6,C_6)$-free bi-cographs, they can also be recognized in linear time.

Figures (6)

• Figure 1: $\text{Sunlet}_4$ with vertex bipartion induced by the vertex-coloring.
• Figure 2: Example of a 2qBMG and 2BMG explained by a rooted phylogenetic tree. Black edges represent the edge-set of the 2qBMG. Black and dashed edges form the edge-set of the 2BMG. The truncation map $u$ of the 2qBMG is $u(x_2)=x_2$ and $u(x)=\rho$ otherwise.
• Figure 3: Example of non-monochromatic least-resolved tree. $(T,\sigma,u)$ and $(T,\sigma,u')$ explain $G$, where $u$ and $u'$ map every leaf to the root of $T$ and $T'$, respectively. Since $T'$ can be obtained from $T$ by contracting the arc $vw$, $T$ is not least-resolved, although it is not monochromatic.
• Figure 4: All connected non-biclique un2qBMGs with at most five vertices, together with the trees explaining them. Bipartitions $\sigma$ are shown as vertex-colorings. The truncation maps $u$ are: (a) $u(x_1)=x_1$ and $u(x_i)=\rho_T$ for $i \in \{2,3,4\}$; (b) $u(x_i)=\rho_T$ for $i \in \{1, \ldots, 5\}$; (c) $u(x_1)=x_1$, $u(x_5)=x_5$ and $u(x_i)=\rho_T$ for $i \in \{2,3,4\}$; (d) $u(x_5)=x_5$ and $u(x_i)=\rho_T$ for $i \in \{1, \ldots, 4\}$.
• Figure 5: Three intermediate steps of the proof of Proposition \ref{['pr:heartless']} - see text for details. Note that in all stages, the depicted graph is an induced subgraph of $G$.

Theorems & Definitions (32)

• Definition 3.1
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Corollary 3.4
• Proposition 3.5
• proof