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Forbidden paths and cycles in the undirected underlying graph of a 2-quasi best match graph

Annachiara Korchmaros

TL;DR

This work analyzes the undirected underlying graphs of 2-quasi best match graphs (2-qBMGs), proving they are both $P_6$-free and $C_6$-free, which allows leveraging the rich theory and polynomial algorithms for these graph classes. It connects these structural properties to dominating bicliques and $K\oplus S$ decompositions, establishing a framework to partition connected 2-qBMGs into type $(A)$ components and enabling topological ordering in certain orientations. The results illuminate how undirected graph theory can inform the study of 2-qBMGs, with potential implications for efficient recognition and decomposition algorithms, and point to future work on larger forbidden subgraphs and related optimization problems. Overall, the paper bridges directed 2-qBMG structure with undirected graph theory to derive new decompositional and algorithmic avenues.

Abstract

The undirected underlying graph of a 2-quasi best match graph (2-qBMG) is proven not to contain any induced graph isomorphic to $P_6$ or $C_6$. This new feature allows for the investigation of 2-BMGs further by exploiting the numerous known results on $P_6$ and $C_6$ free graphs together with the available polynomial algorithms developed for their studies. In this direction, there are also some new contributions about dominating bicliques and certain vertex decompositions of the undirected underlying graph of a 2-qBMG.

Forbidden paths and cycles in the undirected underlying graph of a 2-quasi best match graph

TL;DR

This work analyzes the undirected underlying graphs of 2-quasi best match graphs (2-qBMGs), proving they are both -free and -free, which allows leveraging the rich theory and polynomial algorithms for these graph classes. It connects these structural properties to dominating bicliques and decompositions, establishing a framework to partition connected 2-qBMGs into type components and enabling topological ordering in certain orientations. The results illuminate how undirected graph theory can inform the study of 2-qBMGs, with potential implications for efficient recognition and decomposition algorithms, and point to future work on larger forbidden subgraphs and related optimization problems. Overall, the paper bridges directed 2-qBMG structure with undirected graph theory to derive new decompositional and algorithmic avenues.

Abstract

The undirected underlying graph of a 2-quasi best match graph (2-qBMG) is proven not to contain any induced graph isomorphic to or . This new feature allows for the investigation of 2-BMGs further by exploiting the numerous known results on and free graphs together with the available polynomial algorithms developed for their studies. In this direction, there are also some new contributions about dominating bicliques and certain vertex decompositions of the undirected underlying graph of a 2-qBMG.
Paper Structure (5 sections, 11 theorems, 2 equations, 3 figures)

This paper contains 5 sections, 11 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Forbidden induced path graphs of underlying undirected 2-qBMGs
  4. Dominating sets in 2-qBMGs
  5. Future directions

Key Result

Theorem 3.1

The underlying undirected graph of a 2-qBMG is $P_6$-free.

Figures (3)

  • Figure 1: Case (i): all edges of $\overrightarrow{G}$ up to possible symmetric edges.
  • Figure 2: Tree topology explaining all digraphs in Theorem \ref{['thm:P5']}. Leaves $v_1,v_3,v_5$ have the same color opposite to the color of leaves $v_2,v_4$.
  • Figure 3: (a) Tree topology explaining a 2-qBMG with an induced $P_4$. (b) Tree topology explaining a 2-qBMG with an induced $P_3$.

Theorems & Definitions (21)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Example 3.4
  • Theorem 3.5
  • Corollary 3.6
  • Proposition 3.7
  • proof
  • ...and 11 more