Elsevier

Theoretical Computer Science

Volume 796, 3 December 2019, Pages 70-89
Theoretical Computer Science

Combinatorial properties of Farey graphs

https://doi.org/10.1016/j.tcs.2019.08.022Get rights and content
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Abstract

Combinatorial problems are a fundamental research subject of theoretical computer science, and for a general graph many combinatorial problems are NP-hard and even #P-complete. Thus, it is interesting to seek or design special graphs for which these difficult combinatorial problems can be exactly solved. In this paper, we study some combinatorial problems for the Farey graphs, which are translated from Farey sequences and have received considerable attention from the scientific community. We determine exactly the domination number, the independence number, and the matching number. Moreover, we derive exact or recursive solutions to the number of minimum dominating sets, the number of dominating sets, the number of maximum independent sets, the number of independent sets, the number of maximum matchings, as well as the number of matchings. Finally, we obtain explicit expressions for the number of acyclic orientations and the number of root-connected acyclic orientations. Since the considered combinatorial problems have found wide applications in diverse fields, such as network science and graph data miming, this work is helpful for deepening our understanding of the applications for these combinatorial problems.

Keywords

Farey graph
Combinatorial problem
Minimum dominating set
Maximum independence set
Maximum matching
Domination number
Independence number
Matching number

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