Well-Partitioned Chordal Graphs: Obstruction Set and Disjoint Paths

Ahn, Jung-Ho; Jaffke, Lars; Kwon, O-joung; Lima, Paloma T.

doi:10.1007/978-3-030-60440-0_12

Cited by 5 publications

(8 citation statements)

References 22 publications

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“…Additionally, {4-FAN, A}-free chordal graphs contain also the well-partitioned chordal graphs coined by Ahn et.al. [3] very recently.…”

Section: Hendry's Conjecture Hamiltonian Chordal Graphs Are Cycle Ext...mentioning

confidence: 99%

Cycle Extendability of Hamiltonian Strongly Chordal Graphs

Rong¹,

Li²,

Wang³

et al. 2020

Preprint

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In 1990, Hendry conjectured that all Hamiltonian chordal graphs are cycle extendable. After a series of papers confirming the conjecture for a number of graph classes, the conjecture is yet refuted by Lafond and Seamone in 2015. Given that their counterexamples are not strongly chordal graphs and they are all only 2-connected, Lafond and Seamone asked the following two questions:(1) Are Hamiltonian strongly chordal graphs cycle extendable? (2) Is there an integer k such that all k-connected Hamiltonian chordal graphs are cycle extendable? Later, a conjecture stronger than Hendry's is proposed. In this paper, we resolve all these questions in the negative. On the positive side, we add to the list of cycle extendable graphs two more graph classes, namely, Hamiltonian 4-leaf powers and Hamiltonian {4-FAN, A}-free chordal graphs.

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“…Additionally, {4-FAN, A}-free chordal graphs contain also the well-partitioned chordal graphs coined by Ahn et.al. [3] very recently.…”

Section: Hendry's Conjecture Hamiltonian Chordal Graphs Are Cycle Ext...mentioning

confidence: 99%

Cycle Extendability of Hamiltonian Strongly Chordal Graphs

Rong¹,

Li²,

Wang³

et al. 2020

Preprint

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“…In this work, we deepen the structural and algorithmic understanding of the recently introduced class of well-partitioned chordal graphs [1]. This subclass of chordal graphs generalizes split graphs in two ways.…”

Section: Introductionmentioning

confidence: 99%

“…There are several examples of algorithmic problems in the literature that are efficiently solvable in split graphs but hard on chordal graphs, see [1] and the references therein. In such cases it is worthwhile to narrow down the complexity gap between split and chordal graphs, especially due to the structural difference between the two classes.…”

Section: Introductionmentioning

confidence: 99%

“…In such cases it is worthwhile to narrow down the complexity gap between split and chordal graphs, especially due to the structural difference between the two classes. For several variants of vertex-coloring problems that are NP-hard on chordal graphs and polynomial-time solvable on split graphs, it was observed [1] that they remain NP-hard on well-partitioned chordal graphs. However, there was no example of such a problem that becomes polynomial-time solvable on well-partitioned chordal graphs.…”

Section: Introductionmentioning

confidence: 99%

“…As alluded to above, it has a positive answer on split graphs [25], and this result has been generalized to starlike graphs [10]. Both split graphs and starlike graphs are subclasses of well-partitioned chordal graphs [1]. It remains a challenging open problem to determine whether all chordal graphs admit a longest path transversal of size one.…”

Section: Introductionmentioning

confidence: 99%

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Three Problems on Well-Partitioned Chordal Graphs

Ahn

Jaffke

Kwon

et al. 2021

Lecture Notes in Computer Science

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In this work, we solve three problems on well-partitioned chordal graphs. First, we show that every connected (resp., 2-connected) well-partitioned chordal graph has a vertex that intersects all longest paths (resp., longest cycles). It is an open problem [Balister et al., Comb. Probab. Comput. 2004] whether the same holds for chordal graphs. Similarly, we show that every connected well-partitioned chordal graph admits a (polynomial-time constructible) tree 3-spanner, while the complexity status of the Tree 3-Spanner problem remains open on chordal graphs [Brandstädt et al., Theor. Comput. Sci. 2004]. Finally, we show that the problem of finding a minimum-size geodetic set is polynomial-time solvable on well-partitioned chordal graphs. This is the first example of a problem that is NP-hard on chordal graphs and polynomial-time solvable on well-partitioned chordal graphs. Altogether, these results reinforce the significance of this recently defined graph class as a tool to tackle problems that are hard or unsolved on chordal graphs.

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