Minimally tough chordal graphs with toughness at most 1/2

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. 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.