proposition: Lemma connected_cp_closed (x y : G) (V : {set G}) : connected V -> [set x; y] \subset V -> cp x y \subset V.
proof: Proof. Proof using. move=> V_conn. rewrite subUset !sub1set. case/andP=> Hx Hy. case: (altP (x =P y)) => [<-|xNy]; first by rewrite cpxx sub1set. have[p _ /subsetP pV] := connect_irredRP xNy (V_conn x y Hx Hy). by apply/subsetP=> u /cpP/(_ p)/pV. Qed.
imports: mathcomp(all_ssreflect) GraphTheory(edone preliminaries digraph) GraphTheory(sgraph connectivity)