Toughness and Hamiltonicity of strictly chordal graphs

Abstract

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.

Linear time determination of the scattering number for strictly chordal graphs

“…Few classes of graphs present polynomial results when dealing with the determination of the toughness: interval graphs and trapezoid graphs [24], cocomparability graph with τ (G) ≥ 1 [14] and claw-free graphs, split graphs and 2K 2 -free graphs [10]. Markenzon and Waga [32] presented a linear time determination of the toughness of strictly chordal graphs. The scattering number of interval graphs [11], grid graphs and cartesian product of two complete graphs [35] and gear graphs [2] can be solved in linear time; however, trapezoid graphs maintain the polynomial result to the scattering number [24].…”

“…Theorem 2 [32] Let G = (V, E) be a chordal graph and S be the set of minimal vertex separators of G. G is a strictly chordal graph if and only if for any distinct S, S ′ ∈ S, S ∩ S ′ = ∅.…”

“…Theorem 4 [32] Let G be a non-complete strictly chordal graph and S be the set of minimal vertex separators of G. Then τ (G) = min S∈S |S| µ(S)+1 .…”

Toughness and Hamiltonicity in Random Apollonian Networks

“…He has also conjectured that there exists a t such that every t-tough graph is Hamiltonian. Some papers prove Chvátal's conjecture for different graph classes: τ (G) ≥ 3/2 for a split graph [13], τ (G) > 1 for planar chordal graphs [4], τ (G) ≥ 3/2 for spider graphs [11] and τ (G) ≥ 1 for strictly chordal graphs [18]. In particular for k-trees, Broersma et al [5] presented important results, showing that if G is a k-tree, k ≥ 2, with toughness at least (k + 1)/3, then G is Hamiltonian.…”