  Given a graph G, of arbitrary size and unbounded vertex degree, denote by |G|
the one-complex associated with $G$. The topological space |G| is n-arc
connected (n-ac) if every set of no more than n points of |G| are contained in
an arc (a homeomorphic copy of the closed unit interval).
  For any graph G, we show the following are equivalent: (i) |G| in 7-ac, (ii)
|G| is n-ac for all n, and (iii) G is a subdivision of one of nine graphs. A
graph G has |G| 6-ac if and only if either G is one of the nine 7-ac graphs,
or, after suppressing all degree-2-vertices, the graph G is 3-regular,
3-connected, and removing any 6 edges does not disconnect G into 4 or more
components.
  Similar combinatorial characterizations of graphs G such that |G| is n-ac for
n=3, 4 and 5 are given. Together these results yield a complete classification
of n-ac graphs, for all n.
