Let G = (V,E) be a graph with p vertices and q edges. An acyclicgraphoidal cover of G is a collection of paths in G which are internallydisjointand covering each edge of the graph exactly once. Let f : V !{1, 2, . . . , p} be a bijective labeling of the vertices of G. Let " Gf bethe directed graph obtained by orienting the edges uv of G from u tov provided f(u) < f(v). If the set f of all maximal directed paths in"Gf , with directions ignored, is an acyclic graphoidal cover of G, then fis called a graphoidal labeling of G and G is called a label graphoidal graphand l = min{| f | : f is a graphoidal labeling of G} is called the labelgraphoidal covering number of G. In this paper we characterize graphsfor which (i) l = q − m, where m is the number of vertices of degree 2and (ii) l = q. Also, we determine the value of label graphoidal coveringnumber for unicyclic graphs.