Guides and shortcuts in graphs

The geodesic structure of a graph appears to be a very rich structure. There are many ways to describe this structure, each of which captures only some aspects. Ternary algebras are for this purpose very useful and have a long tradition. We study two instances: signpost systems, and a special case of which, step systems. Signpost systems were already used to characterize graph classes. Here we use these for the study of the geodesic structure of a connected spanning subgraph F with respect to its host graph G. Such a signpost system is called a guide to (F, G). Our main results are: the characterization of the step system of a cycle, the characterization of guides for spanning trees and hamiltonian cycles. (C) 2012 Elsevier B.V. All rights reserved.