$n$-Arc connected graphs
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 $|G|$ is contained in an arc (a homeomorphic copy of the closed unit interval). For any graph $G$, we show the following are equivalent: (i) $|G|$ is $7$-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, is $3$-regular, $3$-connected, and removing any $6$ edges does not disconnect $G$ into four 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 $n$.