Minimal toughness in special graph classes

Abstract

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs.

Figures (7)

• Figure 1: The set $S = S(e)$ consisting of the vertices of $Q \setminus \{ u, v \}$ and the common neighbors of $u$ and $v$.
• Figure 2: The set $R = R(a)$ consisting of those vertices of $Q$ that are not adjacent to $a$.
• Figure 3: The class of minimally $1/b$-tough, split graphs.
• Figure 4: A minimally 3/2-tough, claw-free graph and its edge $e$ for which there exist no vertex sets $S=S(e)$ guaranteed by Proposition \ref{['prop:minttoughlemma']} with $|S| \le 3$.
• Figure 5: Creating a minimally $1/2$-tough, claw-free graph from a tree according to Theorem \ref{['thm:characterization_of_min_1/2_tough_clawfree']}.

Theorems & Definitions (44)

• Definition 1.1
• Definition 1.2
• Conjecture 1.3: Kriesell, article:kriesell_conjecture
• Conjecture 1.4: Generalized Kriesell conjecture
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Definition 3.1
• Definition 3.2