How to survive while visiting a graph

A visit of a graph is a permutation of its vertices which establishes the order in which these are considered. In this paper we deal with the problem of determining visits that perform well in terms of accumulated edges, where at a given step of the visit an edge is accumulated iff both its endpoints have already been considered at that step. Several visits of this kind are here defined, the notion of strong and weak visitability presented, and the problem of deciding whether a graph can be correctly visited or not is considered. Such a problem is computationally hard in various versions. We then analyze necessary and sufficient conditions for graph visitability, characterize some a.e. visitable classes of graphs, describe polynomial-time algorithms to solve particular problem instances, and give heuristics and approximation algorithms for some intractable cases.