Minimally tough chordal graphs with toughness at most 1/2

09/01/2022

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by   Gyula Y Katona, et al.

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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. The toughness of a graph is the largest t for which the graph is t-tough. 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. A graph is chordal if it does not contain an induced cycle of length at least 4. We characterize the minimally t-tough, chordal graphs for all t≤ 1/2. As a corollary, a characterization of minimally t-tough, interval graphs is obtained for t≤ 1/2.