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Automorphisms and quotients of 2-colored quasi best match graphs

Annachiara Korchmaros

TL;DR

This work analyzes the automorphism structure of two-color quasi-best match graphs ($2$-qBMGs), a phylogeny-inspired class of directed bipartite graphs defined by axioms $(N1)$–$(N3)$ and known to be $P_6$-free in their undirected underlying graphs. It establishes that $2$-qBMGs are hereditary under color-preserving quotients, with the classical quotient $partial \,igl\overrightarrow{G}$ encoding the action of the automorphism group as a product of symmetric groups on $ ode{dot}{ ext{sim}}$-classes, and that normal quotients induce subgroup quotients of automorphism groups. The paper constructs large automorphism groups via blow-up techniques and shows that thin $2$-qBMGs exhibit strong orbit-structure constraints, enabling explicit realizations of substantial $ ext{Sym}_m$-type automorphism subgroups. It also proves that orientations of $2$-qBMGs preserve the defining properties, with the UW-orientation preserving color-preserving automorphisms, thus allowing automorphism analysis on oriented graphs without symmetric edges. Collectively, these results provide concrete methods to analyze, bound, and engineer automorphism groups in $2$-qBMGs and illuminate their quotient and orientation behavior.

Abstract

2-colored quasi best match graphs (2-qBMGs) are directed graphs that arose in phylogenetics. Investigations of 2-qBMGs have mostly focused on computational issues. However, 2-qBMGs also have relevant properties for structural graph theory; in particular, their undirected underlying graph is free from induced paths and cycles of size at least 6. In this paper, results on the structure of the automorphism groups of 2-qBMGs are obtained, which shows how to construct 2-qBMGs with large automorphism groups.

Automorphisms and quotients of 2-colored quasi best match graphs

TL;DR

This work analyzes the automorphism structure of two-color quasi-best match graphs (-qBMGs), a phylogeny-inspired class of directed bipartite graphs defined by axioms – and known to be -free in their undirected underlying graphs. It establishes that -qBMGs are hereditary under color-preserving quotients, with the classical quotient encoding the action of the automorphism group as a product of symmetric groups on -classes, and that normal quotients induce subgroup quotients of automorphism groups. The paper constructs large automorphism groups via blow-up techniques and shows that thin -qBMGs exhibit strong orbit-structure constraints, enabling explicit realizations of substantial -type automorphism subgroups. It also proves that orientations of -qBMGs preserve the defining properties, with the UW-orientation preserving color-preserving automorphisms, thus allowing automorphism analysis on oriented graphs without symmetric edges. Collectively, these results provide concrete methods to analyze, bound, and engineer automorphism groups in -qBMGs and illuminate their quotient and orientation behavior.

Abstract

2-colored quasi best match graphs (2-qBMGs) are directed graphs that arose in phylogenetics. Investigations of 2-qBMGs have mostly focused on computational issues. However, 2-qBMGs also have relevant properties for structural graph theory; in particular, their undirected underlying graph is free from induced paths and cycles of size at least 6. In this paper, results on the structure of the automorphism groups of 2-qBMGs are obtained, which shows how to construct 2-qBMGs with large automorphism groups.

Paper Structure

This paper contains 11 sections, 13 theorems, 21 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Graph theory
  4. Graphs and Groups
  5. Two-quasi-best match graphs
  6. -quotients of 2-qBMGs
  7. Automorphism groups of digraphs containing equivalent vertices
  8. Automorphism groups of thin 2-qBMGs
  9. Orbits of on vertices
  10. Examples for Theorem \ref{['pro20102024B']}
  11. Orientations and automorphisms of 2-qBMGs

Key Result

Theorem 3.1

If $\overrightarrow{G}$ is a 2-qBMG and $\Gamma$ is a subgroup of $\rm{Aut}_I(\overrightarrow{G})$ then the $\Gamma$-quotient $\overrightarrow{G}/\Gamma$ is also a 2-qBMG.

Figures (8)

  • Figure 1: Non-bi-transitive quotient of 2-qBMG. The graph $\overrightarrow{G}$ is a 2-qBMG with color classes $U=\{1,2,3\}$ and $W=\{4,5,6\}$, and covering $U_1=\{1\},U_2=\{2\},U_3=\{3\}$, $W_1=\{4\}, W_2=\{5,6\}$. The quotient bipartite digraph $\overrightarrow{G}_0$ has color classes $\mathcal{U}=\{U_1,U_2,U_3\}$ and $\mathcal{W}=\{W_1,W_2\}$ and edge set $\mathcal{E}=\{W_1U_1,W_1U_2,U_2W_2,W_2U_3\}$. $\overrightarrow{G}_0$ does not satisfy (N2) as $W_1U_2,U_2W_2,W_2U_3 \in \mathcal{E}$ but $W_1U_3\not\in \mathcal{E}$.
  • Figure 2: Non-bi-transitive digraph of 2-qBMG $\Gamma$-quotient. $\{1,2,3,4\}$ and $\{5,6,7,8\}$ are the color classes of $\overrightarrow{G}$. $[2,6]\notin E$, (N2) does not hold in $\overrightarrow{G}$. $\rm{Aut}_I(\overrightarrow{G})$ is a group of order $3$ and it has four orbits: $o_1=\{1\},o_2=\{8\},o_3=\{2,3,4\},o_4=\{5,6,7\}$. For $\Gamma=\rm{Aut}_I(\overrightarrow{G})$, the $\Gamma$-covering is the 2-qBMG with color classes $\{o_1,o_3\},\,\{o_2,o_4\}$ and edge-set $\{[o_1,o_2],[o_2,o_3],[o_3,o_4],[o_1,o_4]\}$.
  • Figure 3: Non-trivial automorphisms for $\Gamma$-quotient. $\{1,2,3,4\}$ and $\{5,6,7,8\}$ are the color classes of $\overrightarrow{G}$. $|\rm{Aut}_I(\overrightarrow{G})|=36$ and $\rm{Aut}_I(\overrightarrow{G})$ has four orbits: $o_1=\{1\}$, $o_2=\{8\}$, $o_3=\{2,3,4\}$ and $o_4=\{5,6,7\}$. The quotient $\overrightarrow{G}/\rm{Aut}_I(\overrightarrow{G})$ is a 2-qBMG on $\{o_1,o_2,o_3,o_4\}$ with color classes $\{o_1,o_3\}$ and $\{o_2,o_4\}$, and edge-set $\{o_1o_2,o_1o_4,o_3o_2,o_4o_3\}$. $\rm{Aut}_I(\overrightarrow{G}/\rm{Aut}_I(\overrightarrow{G}))$ contains the involutory permutations $(o_1o_3),(o_2,o_4)$.
  • Figure 4: First blow-up. $\overrightarrow{G}$ is the 2-qBMG with color classes $\{1,3,5\},\,\{2,4\}$, and edge-set $E:=\{[1,2],[2,1],[3,2],[3,4],[4,5]\}$ (left). $\overrightarrow{G}[1]$ is the 2-qBMG with color classes $\{1,3,5,6\}$, $\{2,4\}$, and edge-set $\{[1,2],[2,1],[3,2],[3,4],[4,5],[6,2],[2,6]\}$, obtained by adding vertex $7$ to $\overrightarrow{G}[1]$. $\overrightarrow{G}[1]$ is the blow-up at vertex $1$ applied to $\overrightarrow{G}$ (right).
  • Figure 5: Second blow-up. $\overrightarrow{G}[1][2]$ is the 2-qBMG with color classes $\{1,3,5,6\}$, $\{2,4,7\}$ and edge-set $\{[1,2],[2,1],[3,2],[3,4],[4,5],[6,2],[2,6]\}$, obtained by adding vertex $7$ to $\overrightarrow{G}[1]$. $\overrightarrow{G}[1][2]$ is the blow-up at vertex $2$ applied to $\overrightarrow{G}[1]$ (left). $\overrightarrow{G}[1,2]$ is the digraph, obtained by adding $6$ and $7$ simultaneously, with color classes $\{1,3,5,6\}$, $\{2,4,7\}$, and edge-set $E\cup \{[6,2],[2,6],[7,1],[1,7],[3,7]\}$ is not a 2-qBMG (right).
  • ...and 3 more figures

Theorems & Definitions (23)

  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Theorem 4.3
  • Proposition 4.4
  • proof
  • Remark 4.5
  • ...and 13 more