  We study graph classes modeled by families of non-crossing (NC) connected
sets. Two classic graph classes in this context are disk graphs and proper
interval graphs. We focus on the cases when the sets are paths and the host is
a tree (generalizing proper interval graphs). Forbidden induced subgraph
characterizations and linear time certifying recognition algorithms are given
for intersection graphs of NC paths of a tree (and related subclasses). A
direct consequence of our certifying algorithms is a linear time algorithm
certifying the presence/absence of an induced claw $(K_{1,3})$ in a chordal
graph.
  For the intersection graphs of NC paths of a tree, we characterize the
minimum connected dominating sets (leading to a linear time algorithm to
compute one). We further observe that there is always an independent dominating
set which is a minimum dominating set, leading to the dominating set problem
being solvable in linear time. Finally, each such graph $G$ is shown to have a
Hamiltonian cycle if and only if it is 2-connected, and when $G$ is not
2-connected, a minimum-leaf spanning tree of $G$ has $\ell$ leaves if and only
if $G$'s block-cutpoint tree has exactly $\ell$ leaves (e.g., implying that the
block-cutpoint tree is a path if and only if the graph has a Hamiltonian path).
