Geodesic Graphoidal Covering Number of a Graph

A geodesic graphical cover of a graph G is a collection ψ of shortest paths in G such that every path in ψ has at least two vertices, every vertex of G is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. The minimum cardinality of a geodesic graphical cover of G is called the geodesic graphical covering number of G and is denoted by η_g. In this paper, we determine η_g for several classes of graphs. We also prove that η_g≥[q/d(G)] where d(G) is the diameter of G and characterize some classes o f graphs which attain this bound.