A local subgraph with respect to a graph is a subgraph isomorphic to that contains a given vertex. Harary and Palmer (1973, p. 263)
pose the problem of enumerating graphs of order in which every vertex lies in such a local subgraph.
A local subgraph is distinct from a local graph, which
is induced by the neighbors of a specified vertex.
Harary, F. and Palmer, E. M. "A Survey of Graphical Enumeration Problems." In A Survey of Combinatorial Theory (Ed. J. N. Srivastava).
Amsterdam, Netherlands: North-Holland, pp. 259-275, 1973.
is a subgraph isomorphic to 
that contains a given vertex. Harary and Palmer (1973, p. 263)
pose the problem of enumerating graphs of order 
in which every vertex lies in such a local subgraph.
