On the Probe Problem for $$(r,\ell )$$-Well-Coveredness

Faria, Luérbio; Souza, Uéverton S.

doi:10.1007/978-3-030-89543-3_32

Cited by 2 publications

(1 citation statement)

References 29 publications

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“…In 2020, Alves, Couto, Faria, Gravier, Klein, and Souza [1] studied the complexity of the G S P for the property of being a well-covered graph whose vertex set can be partitioned into k independent sets and into cliques for xed integers k and , i.e., well-covered graphs that are also (k, )-graphs. In 2021, Faria and Souza [15] studied the complexity of P P for the property of being a (k, )-graph that is well-covered. Also, a polynomial-time algorithm for recognizing some sparse-dense graphs that are well-covered can be found in [34].…”

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confidence: 99%

Recognizing well-dominated graphs is coNP-complete

Agrawal¹,

Fernau²,

Kindermann³

et al. 2022

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A graph G is well-covered if every minimal vertex cover of G is minimum, and a graph G is well-dominated if every minimal dominating set of G is minimum. Studies on well-covered graphs were initiated in [Plummer, JCT 1970], and well-dominated graphs were rst introduced in [Finbow, Hartnell and Nowakow, AC 1988]. Well-dominated graphs are well-covered, and both classes have been widely studied in the literature. The recognition of well-covered graphs was proved coNP-complete by [Chvátal and Slater, AODM 1993] and by [Sankaranarayana and Stewart, Networks 1992], but the complexity of recognizing well-dominated graphs has been left open since their introduction. We close this complexity gap by proving that recognizing well-dominated graphs is coNP-complete. This solves a well-known open question (c.f. [Levit and Tankus, DM 2017] and [Gözüpek, Hujdurovic and Milanič, DMTCS 2017]), which was rst asked in [Caro, Sebő and Tarsi, JAlg 1996]. Surprisingly, our proof is quite simple, although it was a long-standing open problem. Finally, we show that recognizing well-totally-dominated graphs is coNP-complete, answering a question of [Bahadır, Ekim, and Gözüpek, AMC 2021].

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confidence: 99%