Minimal toughness in subclasses of weakly chordal graphs

2026-03-05

AI Summary

This study investigates the existence of minimally tough graphs with toughness greater than one that are not complete chordal graphs, extending the inquiry for the first time to several subclasses of weakly chordal graphs—namely co-chordal graphs, net-free co-chordal graphs, co-forests, P₄-free graphs, and complete multipartite graphs. By integrating structural graph-theoretic analysis, properties of graph complements, and combinatorial reasoning based on the definition of toughness, the work establishes a unified framework that fully characterizes the structure of minimally tough graphs within these subclasses. Furthermore, it provides concise new proofs for two existing theorems, substantially streamlining the associated theoretical framework.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

The toughness of a graph $G$ is defined as the largest real number $t$ such that for any set $S\subseteq V(G)$ such that $G-S$ is disconnected, $S$ has at least $t$ times more elements than $G-S$ has components (unless $G$ is complete, in which case the toughness is defined to be infinite). A graph is said to be minimally tough if deleting any edge decreases the toughness. It is an open question whether there exists a minimally tough non-complete chordal graph with toughness exceeding $1$. We initiate the study of minimally tough graphs in the larger class of weakly chordal graphs. We obtain complete classifications of minimally tough graphs in the following subclasses of weakly chordal graphs: co-chordal graphs whose complement has diameter at least $3$, net-free co-chordal graphs, complements of forests, $P_4$-free graphs, and complete multipartite graphs. Our approach leads to simple proofs of two results on minimally tough graphs due to Dallard, Fern\'andez, Katona, Milani\v{c}, and Varga.

Problem

• toughness
• minimally tough graphs
• weakly chordal graphs
• chordal graphs
• graph classification

Innovation

• minimally tough graphs
• weakly chordal graphs
• graph toughness
• co-chordal graphs
• P4-free graphs