Proof .
By Lemma 2.1, Theorem 4.2 is clearly true in the case when D is bipartite, and so we assume in what follows that β(D)≥1 and D is nonbipartite. Let T be a minimum direct cut of D and let F be a minimum arc-set of D such that D-F=(X,Y;E) is a bipartite graph with bipartition X and Y. Then β(D)=|F| and, λ(D)=|T|≥1 by Lemma 2.1.

Claim 1.
λ
(
K
2
→
×
D
)
≤
min
{
2
λ
(
D
)
,
β
(
D
)
,
min
{
j
+
β
j
:
λ
(
D
)
≤
j
≤
δ
}
.

Let A(K2→)={(a,b)}. By Theorem 1.1, K2→×(D-F) consists of two components. The vertex sets of these two components are {(a,x):x∈X}∪{(b,y):y∈Y} and {(b,x):x∈X}∪{(a,y):y∈Y}. It is not difficult to see that {((a,x),(b,y)):(x,y)∈F∩A(D[X])}∪{((b,x),(a,y)):(x,y)∈F∩A(D[Y])} is a directed cut of K2→× D, since its removal makes {(b,x):x∈X}∪{(a,y):y∈Y} not reachable from {(a,x):x∈X}∪{(b,y):y∈Y}. And so, λ(K2←× D)≤|F|=β(D).

By Lemma 4.1, D-T contains a strongly connected component D1 that has no outer neighbors. By the minimality of T, we have (D1,D1¯)=T. Obviously, {((a,x),(b,y)):(x,y)∈T}∪{((a,y),(b,x)):(x,y)∈T} is a directed cut of K2→× D. Hence λ(K2→× D)≤2|T|=2λ(D).

Let Tj be a directed cut of D that has size j, Cj be a strongly connected component of D-Tj with (Cj,D-Tj-Cj)=∅ (by Lemma 4.1 such components exist), Fj be an arc set of Cj such that Cj-Fj=(Xj,Yj;Ej) is a bipartite subgraph with bipartition (Xj,Yj). Then K2→× (Cj-Fj) consists of two components, whose vertex sets are {(a,x):x∈Xj}∪{(b,y):y∈Yj} and {(b,x):x∈Xj}∪{(a,y):y∈Yj}. It’s not difficult to see that the union of {((a,x),(b,y)):(x,y)∈Fj∩A(D[Xj])}, {((b,x),(a,y)):(x,y)∈Fj∩A(D[Yj])}, {((a,x),(b,y)):(x,y)∈(Xj,D-Cj)} and {((b,x),(a,y)):(x,y)∈(Yj,D-Cj)} is a directed cut of K2→× D, since its removal makes {(a,y):y∈Yj}∪{(b,x):x∈Xj} not reachable from {(a,x):x∈Xj}∪{(b,y):y∈Yj}. Noticing that |{((a,x),(b,y)):(x,y)∈Fj∩A(D[Xj])}∪{((b,x),(a,y)):(x,y)∈Fj∩A(D[Yj])}|=|Fj|, |{((a,x),(b,y)):(x,y)∈(Xj,D-Cj)}∪{((b,x),(a,y)):(x,y)∈(Yj,D-Cj)}|=|Tj| and |Tj∪Fj|=|Tj|+|Fj|=j+βj, we deduce that λ(K2→× D)≤min {j+βj:λ(D)≤j≤δ}. And so, Claim 1 follows.

Claim 2.
λ
(
K
2
→
×
D
)
≥
min
{
2
λ
(
D
)
,
β
(
D
)
,
min
{
j
+
β
j
:
λ
(
D
)
≤
j
≤
δ
}
.

Let S be a minimum direct cut of K2→× D, V1={x∈V(D): vertices (a,x) and (b,x) lie in common strongly connected component of K2→× D-S}, V2={x∈V(D): vertices (a,x) and (b,x) lie in different components of K2→× (D-S)}. Then (V1,V2) is a partition of V(D). From the minimality of S it follows that K2→× D-S consists of two strongly connected components, say C1 and C2. Noticing that (C1,C2)⊆S or (C2,C1)⊆S, we assume without loss of generality that (C1,C2)⊆S. Now three different cases occur: V1≠∅=V2; V1=∅≠V2; V1≠∅≠V2.

Consider at first the case when V1≠∅=V2. By Lemma 2.1, we deduce that {x∈V(D):(a,x)∈V(C1)} induces a strongly connected component of D as well as {y∈V(D):(a,y)∈V(C2)} in this case. Furthermore, ({x∈V(D):(a,x)∈V(C1)},{y∈V(D):(a,y)∈V(C2)}) is a directed cut of D and ({(a,x):(a,x)∈V(C1)},{(b,y):(b,y)∈V(C2)})∪({(b,x):(b,x)∈V(C1)},{(a,y):(a,y)∈V(C2)})⊆S. It follows from these observations that
(4.1)|S|≥|{(a,x)∈V(C1)},{(b,y)∈V(C2)}| +|{(b,x)∈V(C1)},{(a,y)∈V(C2)}|≥2λ(D).

Consider secondly the case when V1=∅≠V2. Let M={x∈V(D):(a,x)∈V(C1)} and N={y∈V(D):(b,y)∈V(C1)}. Then (M,N) is a partition of V(D) and (C1,C2)={((a,x),(b,y)):(x,y)∈A(D[M])}∪{((b,x),(a,y)):(x,y)∈A(D[N])}. Since D-A(D[M])-A(D[N]) is a bipartite subgraph of D, it follows that |(C1,C2)|=|A(D[M])|+|A(D[N])|≥β(D). Recalling that (C1,C2)⊆S, we have |S|≥|(C1,C2)|≥β(D).

Consider finally the case when V1≠∅≠V2. Let
(4.2)H={x∈V(D):(a,x)∈V(C1), x∈V1},Q={x∈V(D):(a,x)∈V(C2), x∈V1},W={x∈V(D):(a,x)∈V(C1), x∈V2},Z={x∈V(D):(a,x)∈V(C2), x∈V2}.

Then (H,Q,W,Z) is a partition of V(D), refer to (1) of Figure 2. Since (C1,C2)⊆S, the arcs in this set is removed in Figure 2. If H≠∅≠Q, then the set of arcs from {(a,x):x∈H}∪{(b,x):x∈H} to C2 is a directed cut of K2→× D, but it has less arcs than S. This contradiction shows that either H=∅ or Q=∅. Assume without loss of generality that Q=∅. Then K2→× D-S can be depicted as (2) of Figure 2.

On the one hand, for every arc (x,y)∈(H,D-H), if y∈W then ((a,x),(b,y))∈S; if y∈Z then ((b,x),(a,y))∈S. On the other hand, for every arc (x,y)∈A(D[W]) the arc ((a,x),(b,y))∈S and for every arc (x,y)∈A(D[Z]) the arc ((b,x),(a,y))∈S. It follows from these observations that
(4.3)|S|≥|[H,D-H]|+(|A(D[W])|+|A(D[Z])|)≥min{j+βj:λ(D)≤j≤δ(D)},
where [H,D-H] represents the set of arcs with ends in H and D-H, respectively. And so, the theorem follows from Claims 1 and 2.