Fixed Points and 2-Cycles of Synchronous Dynamic Coloring Processes on Trees
Abstract
This paper considers synchronous discrete-time dynamical systems on graphs based on the threshold model. It is well known that after a finite number of rounds these systems either reach a fixed point or enter a 2-cycle. The problem of finding the fixed points for this type of dynamical system is in general both NP-hard and #P-complete. In this paper we give a surprisingly simple graph-theoretic characterization of fixed points and 2-cycles for the class of finite trees. Thus, the class of trees is the first nontrivial graph class for which a complete characterization of fixed points exists. This characterization enables us to provide bounds for the total number of fixed points and pure 2-cycles. It also leads to an output-sensitive algorithm to efficiently generate these states.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2022
- DOI:
- 10.48550/arXiv.2202.01580
- arXiv:
- arXiv:2202.01580
- Bibcode:
- 2022arXiv220201580T
- Keywords:
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- Computer Science - Discrete Mathematics;
- Computer Science - Distributed;
- Parallel;
- and Cluster Computing
- E-Print:
- 15 pages, 7 figures