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We introduce the concept of the primitivity of independent set invertex-transitive graphs, and investigate the relationship between theprimitivity and the structure of maximum independent sets in direct products ofvertex-transitive graphs. As a consequence of our main results, we positivelysolve an open problem related to the structure of independent sets in powers ofvertex-transitive graphs.

Primitivity and Independent Sets in Direct Products of Vertex-Transitive Graphs