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Minimal toughness in subclasses of weakly chordal graphs

J. Pascal Gollin, Martin Milanič, Laura Ogrin

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, Fernández, Katona, Milanič, and Varga.

Minimal toughness in subclasses of weakly chordal graphs

Abstract

The toughness of a graph is defined as the largest real number such that for any set such that is disconnected, has at least times more elements than has components (unless 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 . 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 , net-free co-chordal graphs, complements of forests, -free graphs, and complete multipartite graphs. Our approach leads to simple proofs of two results on minimally tough graphs due to Dallard, Fernández, Katona, Milanič, and Varga.
Paper Structure (15 sections, 38 theorems, 21 equations, 1 figure, 1 table)

This paper contains 15 sections, 38 theorems, 21 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our approach
  3. Structure of the paper
  4. Preliminaries
  5. Bipartite matchings and local connectivity
  6. Preliminary results on toughness and minimal toughness
  7. Minimal toughness of graphs with dominating edges
  8. Toughness of complete multipartite graphs
  9. Joins of graphs
  10. Graphs with a universal vertex
  11. P_4-free graphs
  12. Co-chordal graphs
  13. Net-free co-chordal graphs
  14. Complements of forests
  15. Open problems

Key Result

Theorem 1.1

A $P_4$-free graph is non-trivially minimally tough if and only if it is isomorphic to one of $K_{2,3}$, $K_{1,\ell}$, $T_{ 2\ell, \ell }$, or $T_{ 2\ell-1, \ell }$, for some ${\ell \geq 2}$.

Figures (1)

  • Figure 7.1: The structure of a co-chordal graph with co-diameter at least $3$. The figure shows all the edges incident with $u$ or $w$, and $U$ and $W$ are independent sets. In addition, there might be some edges between $U$ and $W$ (yellow), between $U\cup W$ and $X$ (purple), and inside $X$ (green).

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: store=cochordal
  • Theorem 1.4: store=netfreecochordal
  • Theorem 1.5: store=coforests
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Definition 3.1
  • ...and 58 more