Intersection graphs of non-crossing paths

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). Consequently, we obtain a linear time algorithm to detect 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 study dominating sets and spanning subgraphs. For example, minimum connected dominating sets are characterized (leading to a linear time algorithm to compute one), and we observe that there is always an independent dominating set which is a minimum dominating set (again leading to linear time algorithms for these). Regarding spanning subgraphs, each such graph G is shown to have a Hamiltonian cycle if it is 2-connected, and when G is not 2-connected, a minimum-leaf spanning tree of G has ℓ leaves if G's block-cutpoint tree has exactly ℓ leaves (e.g., implying that the block-cutpoint tree is a path if and only if the graph has a Hamiltonian path).