I cannot seem to find an answer to this question with Google, so I am  going to ask here: is it required for a good neighbourhood function that it in principle (i. e. by recursively considering all neighbours of a certain solution - which is not practical) can reach all possible solutions?My question is whether there are references in literature that explicitely state it's a requirement - I can see that it is a good property of a neighbourhood.
Neighbourhood in local search metaheuristic
optimization;heuristics
I think the property of being able to reach all the successor states from a current state is crucial and needed in order for the problem to be well-defined. If this was not the case, you might miss out on good solutions. Moreover, you might never reach a goal state and hence run forever.The classic AI book by Russell & Norvig, 3rd edition, page 67 gives a formal definition for a problem. One of the components is a successor function which returns all the successor states for a current state. Together the formal components create a search space, that is the set of all states reachable from the initial state. If one uses a different successor function, namely such that it leaves out some successors, the resulting search space is a different one: goal state(s) might be missing and optimal solutions might very well differ. Perhaps the reason you are unable find it from the literature is that the requirement is obvious.