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Forbidden configurations and dominating bicliques in undirected 2-quasi best match graphs

Annachiara Korchmaros, Peter F. Stadler

TL;DR

This work analyzes the undirected counterparts of 2-colored quasi-best-match graphs to uncover their structural constraints via forbidden subgraphs and dominating decompositions. It establishes that un2qBMGs are $P_6$-free and $C_6$-free, placing them in the chordal bipartite class, though this containment is strict. The paper then demonstrates a universal $K extoplus S$ decomposition where the biclique $K$ dominates the graph, and investigates how orientations interact with dominating bicliques under a mild condition, linking tree-based definitions to edge-decomposition concepts. Together, these results advance understanding of the structural and algorithmic implications for un2qBMGs and outline a path toward a full forbidden-subgraph characterization, while highlighting open problems like the role of Sunlet$_4$ in complete characterization.

Abstract

2-quasi best match graphs (2-qBMGs) are directed graphs that capture a notion of close relatedness in phylogenetics. Here, we investigate the undirected underlying graph of a 2-qBMG (un-2qBMG) and show that they contain neither a path $P_l$ nor a cycle $C_l$ of length $l\geq 6$ as an induced subgraph. This property guarantees the existence of specific vertex decompositions with dominating bicliques that provide further insights into their structure.

Forbidden configurations and dominating bicliques in undirected 2-quasi best match graphs

TL;DR

This work analyzes the undirected counterparts of 2-colored quasi-best-match graphs to uncover their structural constraints via forbidden subgraphs and dominating decompositions. It establishes that un2qBMGs are -free and -free, placing them in the chordal bipartite class, though this containment is strict. The paper then demonstrates a universal decomposition where the biclique dominates the graph, and investigates how orientations interact with dominating bicliques under a mild condition, linking tree-based definitions to edge-decomposition concepts. Together, these results advance understanding of the structural and algorithmic implications for un2qBMGs and outline a path toward a full forbidden-subgraph characterization, while highlighting open problems like the role of Sunlet in complete characterization.

Abstract

2-quasi best match graphs (2-qBMGs) are directed graphs that capture a notion of close relatedness in phylogenetics. Here, we investigate the undirected underlying graph of a 2-qBMG (un-2qBMG) and show that they contain neither a path nor a cycle of length as an induced subgraph. This property guarantees the existence of specific vertex decompositions with dominating bicliques that provide further insights into their structure.

Paper Structure

This paper contains 4 sections, 14 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Forbidden induced paths of underlying undirected 2qBMGs
  4. Dominating sets in 2qBMGs

Key Result

Proposition 1.1

schaller2021corrigendumkorchmaros2023quasi A directed graph $(\overrightarrow{G},\sigma)$ with a proper two-coloring $\sigma$ of its vertex set is a 2qBMG if and only if the in-neighborhoods $N^-(v)$, $N^-(u)$ and out-neighborhoods $N^+(v)$, $N^+(u)$ of every two distinct vertices $u$ and $v$ of $\o

Figures (3)

  • Figure 1: A $(P_6,C_6)$-free graph that is not an un2qBMG.
  • Figure 2: (a) 2qBMGs with 5-path-graph un2qBMG: $\overrightarrow{P}_5^{(a)}$ graph (left), $\overrightarrow{P}_5^{(b)}$ graph (middle), and $\overrightarrow{P}_5^{(c)}$ graph (right). The vertex set of $\overrightarrow{P}_5^{(ab)},\overrightarrow{P}_5^{(ac)},\overrightarrow{P}_5^{(abc)}$ coincides with $V(\overrightarrow{P}_5^{(a)})$. The edge sets are $E(\overrightarrow{P}_5^{(ab)}):=E(\overrightarrow{P}_5^{(a)})\cup E(\overrightarrow{P}_5^{(b)}), E(\overrightarrow{P}_5^{(ac)}):=E(\overrightarrow{P}_5^{(a)})\cup E(\overrightarrow{P}_5^{(c)}),$ and $E(\overrightarrow{P}_5^{(abc)}):=E(\overrightarrow{P}_5^{(a)})\cup E(\overrightarrow{P}_5^{(b)})\cup E(\overrightarrow{P}_5^{(c)})$. (b) Case (i) of the proof of Theorem \ref{['thm:P6free']}: an orientation of $\overrightarrow{G}$.
  • Figure 3: 2qBMGs for 4-path-graph and 4-cycle-graph un2qBMGs. Black edges are required. Dashed edges are optional and taken in (a) one at a time (middle-right) or in pairs: north-east, north-west, or east-west (left).

Theorems & Definitions (21)

  • Proposition 1.1
  • Definition 2.1
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • ...and 11 more