# On Minimum Dominating Sets in cubic and (claw,H)-free graphs

1 CEDRIC - OC - CEDRIC. Optimisation Combinatoire
CEDRIC - Centre d'études et de recherche en informatique et communications
2 CEDRIC - ROC - CEDRIC. Réseaux et Objets Connectés
CEDRIC - Centre d'études et de recherche en informatique et communications
Abstract : Given a graph $G=(V,E)$, $S\subseteq V$ is a dominating set if every $v\in V\setminus S$ is adjacent to an element of $S$. The Minimum Dominating Set problem asks for a dominating set with minimum cardinality. It is well known that its decision version is $NP$-complete even when $G$ is a claw-free graph. We give a complexity dichotomy for the Minimum Dominating Set problem for the class of $(claw, H)$-free graphs when $H$ has at most six vertices. In an intermediate step we show that the Minimum Dominating Set problem is $NP$-complete for cubic graphs.
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Cited literature [27 references]

https://hal.archives-ouvertes.fr/hal-02493931
Contributor : Valentin Bouquet <>
Submitted on : Wednesday, March 18, 2020 - 6:07:46 PM
Last modification on : Tuesday, March 23, 2021 - 9:28:02 AM
Long-term archiving on: : Friday, June 19, 2020 - 2:42:32 PM

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2002.12232.pdf
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• HAL Id : hal-02493931, version 1
• ARXIV : 2002.12232

### Citation

Valentin Bouquet, François Delbot, Christophe Picouleau, Stephane Rovedakis. On Minimum Dominating Sets in cubic and (claw,H)-free graphs. 2020. ⟨hal-02493931⟩

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