Minimal toughness in special graph classes

DOI10.46298/DMTCS.10180arXiv1802.00055MaRDI QIDQ6507454

Author name not available (Why is that?)

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

2 K_{2}

-free graphs.

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