Intersection Graphs of Non-crossing Paths

Abstract: 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).