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. Subproof, The dipath R contains a subdipath R with initial vertex s in P 2 and terminal vertex in P 1 Let s + be the outneighbour of s in P 2 ? c d . Then a b ? P 1 ? P 2 [b , s] ? R ? ss + is an F 4 -subdivision. ? Henceforth, we assume that a b is not an arc in D. If d + D?{c ,d } (a ) = 0, then there is no (a , b )-dipath in D ? c ? d , and thus no (a , b , c d )-forced F 4 -subdivision. Hence we return 'no'. Inria If d + D?{c ,d } (a ) = 1, then denote by a the unique outneighbour of a in D ? {c , d }. By our assumption, a = b . Let D * be the digraph obtained from D by first removing all arcs entering a and then identifying a and a into a single vertex a *

. Subproof, Now the digraph S * obtained from S by replacing a and a and the four arcs ua , va , a a , a w by the vertex a * and the three arcs ua * , va * , a * w is an (a * , b , c d )-forced F 4 -subdivision in D * , because a = b . Conversely, if S * is an (a * , b , c d )-forced F 4 -subdivision in D * , then the digraph S obtained from S * by replacing a * and its three incident arcs ua * , va * , a * w by vertices a and a and the four arcs ua , va , a a , a w is clearly an (a , b , c d )-forced F 4 -subdivision in D. ? Henceforth, we may assume that d + D?{c ,d } (a ) ? 2. Using a Menger algorithm, we check whether there are two independent (b , {a , c })-dipaths in D ? d , and using a search we check whether there exists an (a , b )-dipath in D ? {c, )-forced F 4 -subdivision in D, and we return 'no'. If four such dipaths exist, then we return 'yes' by virtue of the following claim

. Subproof, Without loss of generality, we may assume that t(P 1 ) = a and t(P 2 ) = c . Let v be the last vertex along Q ? b that is in P 1 ? P 2

?. @bullet-assume-z, vv + is an F 4 subdivision If v + ? V (Q ), then v + = d and so P 2 [v + , c ] is not an empty dipath. Denote by C the directed cycle P 1 [z, a ] ? a z. The dipath Q[v + , b ] ? P 1 [b , z] contains a, Let s + be the outneighbour of s(Q ) in P 2 ?c d . Now C ?Q[a , v + ]?P 2 [v + , s(Q )]? Q ? s(Q )s + is an F 4 -subdivision

?. @bullet-assume-z, Then one can replace the (a , b )-dipath Q by a zQ, This dipath is internally disjoint from P 1 and P 2 , and we get the result by Case 1