Path-Additions of Graphs
  Path-addition is an operation that takes a graph and adds an internally vertex-disjoint path between two vertices together with a set of supplementary edges. Path-additions are just the opposite of taking minors.   We show that some classes of graphs are closed under path-addition, including non-planar, right angle crossing, fan-crossing free, quasi-planar, (aligned) bar 1-visibility, and interval graphs, whereas others are not closed, including all subclasses of planar graphs, bounded treewidth, k-planar, fan-planar, outer-fan planar, outer-fan-crossing free, and bar (1,j)-visibility graphs. 
Franz J. Brandenburg, Alexander Esch, Daniel Neuwirth
cs.DM