Fixed Points and 2-Cycles of Synchronous Dynamic Coloring Processes on Trees

TL;DR

This paper provides a complete graph-theoretic characterization of fixed points and 2-cycles in synchronous threshold models on trees, enabling efficient enumeration and bounds for these states.

Contribution

It introduces the first complete characterization of fixed points and 2-cycles for trees in the threshold model, facilitating efficient algorithms and bounds.

Findings

Characterization of fixed points and 2-cycles for trees

Efficient output-sensitive algorithms for generating states

Bounds on the number of fixed points and 2-cycles

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.