## Abstract

We introduce a general class of algorithms and analyse their application to regular graphs of large girth. In particular, we can transfer several results proved for random regular graphs into (deterministic) results about all regular graphs with sufficiently large girth. This reverses the usual direction, which is from the deterministic setting to the random one. In particular, this approach enables, for the first time, the achievement of results equivalent to those obtained on random regular graphs by a powerful class of algorithms which contain prioritised actions. As a result, we obtain new upper or lower bounds on the size of maximum independent sets, minimum dominating sets, maximum *k*-independent sets, minimum *k*-dominating sets and maximum *k*-separated matchings in *r*-regular graphs with large girth.

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## Additional information

Supported by FAPERGS (Proc. 2233-2551/14-0), CNPq (Proc. 448754/2014-2 and 308539/2015-0) and FAPESP (Proc. 2013/03447-6).

Research supported by the Canada Research Chairs Program, NSERC and the Australian Laureate Fellowship program of the ARC.

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Hoppen, C., Wormald, N. Local Algorithms, Regular Graphs of Large Girth, and Random Regular Graphs.
*Combinatorica* **38, **619–664 (2018). https://doi.org/10.1007/s00493-016-3236-x

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### Mathematics Subject Classification (2000)

- 05C35
- 05C80
- 05C69
- 05C85