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The clique‐based structure of chordal graphs allows the development of efficient solutions for many algorithmic problems. In this context, the minimal vertex separators play a decisive role. In this paper, we study properties of these structures, presenting a linear time determination of the toughness of strictly chordal graphs. We prove that every 1‐tough strictly chordal graph is Hamiltonian. This result leads to the characterization of a new class, the Hamiltonian strictly chordal graphs. We also prove that graphs belonging to this class are cycle extendable.

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