This paper introduces new ejection chain strategies designed to generate neighborhoods of compound moves with attractive properties for traveling salesman problems. These procedures derive from the principle of creating a reference structure that coordinates the options for the moves generated. We focus on ejection chain processes related to alternating paths, and introduce three reference structures, of progressively greater scope, that produce both classical and non-standard alternating path trajectories. Theorems and examples show that various rules for exploiting these structures can generate moves not available to customary neighborhood search approaches. We also provide a reference structure that makes it possible to generate a collection of alternating paths that precisely expresses the symmetric difference between two tours.

In addition to providing new results related to generalized alternating paths, in directed and undirected graphs, we lay a foundation for achieving a combinatorial leverage effect, where an investment of polynomial effort yields solutions dominating exponential numbers of alternatives. These consequences are explored in a sequel.