A bisimulation relates the roots of two rooted directed graphs and matches every successor move in either graph with a successor move in the other leading to related vertices. Multiple successors may be related to the same vertex. This local matching is weaker than graph isomorphism.
The Spoiler takes one successor step from the current vertex of either rooted directed graph; the Duplicator must take a matching successor step in the other. The Duplicator wins by never failing at a finite stage, including when the Spoiler cannot move. A winning strategy gives a bisimulation by collecting endpoint pairs of all finite plays consistent with it.
Two rooted directed graphs are bisimilar if some bisimulation relates their roots. A root with one terminal successor is bisimilar to a root with two terminal successors, although the graphs have different sizes.
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Bisimulation is a concept in the field of concurrency theory and formal methods, particularly in the study of transition systems and processes. It is a relationship between state-transition systems that allows us to determine if two systems behave similarly in a formal sense. The idea is to compare two systems based on their ability to mimic each other's behavior, particularly in terms of their possible state transitions.