
COMMENTS

Two edges {a,b}, {c,d} are weakly nesting if a <= c < d <= b or c <= a < b <= d. A graph has its weakly nesting edges connected if the graph whose vertices are the edges and whose edges are weakly nesting pairs of edges is connected.


MATHEMATICA

wknXQ[eds_]:=MatchQ[eds, {___, {x_, y_}, ___, {z_, t_}, ___}/; (x<=z&&y>=t)(x>=z&&y<=t)];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Length[csm[Union[List/@#, Select[Subsets[#, {2}], wknXQ]]]]<=1&]], {n, 0, 5}]
