1 Introduction

The generalized k-connectivity \(\kappa _k(G)\) of a graph \(G=(V,E)\) was introduced by Hager [8] in 1985 (\(2\le k\le |V|\)). For a graph \(G=(V,E)\) and a set \(S\subseteq V\) of at least two vertices, an S-Steiner tree or, simply, an S-tree is a subgraph T of G which is a tree with \(S\subseteq V(T)\). Two S-trees \(T_1\) and \(T_2\) are said to be internally disjoint if \(E(T_1)\cap E(T_2)=\emptyset\) and \(V(T_1)\cap V(T_2)=S\). The generalized local connectivity \(\kappa _S(G)\) is the maximum number of internally disjoint S-trees in G. For an integer k with \(2\le k\le n\), the generalized k-connectivity is defined as

$$\begin{aligned} \kappa _k(G)=\min \{\kappa _S(G)\mid S\subseteq V(G), |S|=k\}. \end{aligned}$$

Observe that \(\kappa _2(G)=\kappa (G)\). If G is disconnected and vertices of S are placed in different connectivity components, we have \(\kappa _S(G)=0\). Thus, \(\kappa _k(G)=0\) for a disconnected graph G. Generalized connectivity of graphs has become an established area in graph theory, see a recent monograph [9] by Li and Mao on generalized connectivity of undirected graphs.

To extend generalized k-connectivity to directed graphs, Sun et al. [13] observed that in the definition of \(\kappa _S(G)\), one can replace “an S-tree” by “a connected subgraph of G containing S”. Therefore, Sun et al. [13] defined strong subgraph k-connectivity by replacing “connected” with “strongly connected” (or, simply, “strong”) as follows. Let \(D=(V,A)\) be a digraph of order n, S a subset of V of size k and \(2\le k\le n\). A subgraph H of D is called an S-strong subgraph if \(S\subseteq V(H)\). A pair of S-strong subgraphs \(D_1\) and \(D_2\) are said to be arc-disjoint if \(A(D_1)\cap A(D_2)=\emptyset\). A pair of arc-disjoint S-strong subgraphs \(D_1\) and \(D_2\) are said to be internally disjoint if \(V(D_1)\cap V(D_2)=S\). Let \(\kappa _S(D)\) be the maximum number of internally disjoint S-strong subgraphs in D. The strong subgraph k -connectivity [13] is defined as

$$\begin{aligned} \kappa _k(D)=\min \{\kappa _S(D)\mid S\subseteq V(D), |S|=k\}. \end{aligned}$$

By definition, \(\kappa _2(D)=0\) if D is not strong.

As a natural counterpart of the strong subgraph k-connectivity, we now introduce the concept of strong subgraph k-arc-connectivity. Let \(\lambda _S(D)\) be the maximum number of arc-disjoint S-strong digraphs in D. The strong subgraph k-arc-connectivity is defined as

$$\begin{aligned} \lambda _k(D)=\min \{\lambda _S(D)\mid S\subseteq V(D), |S|=k\}. \end{aligned}$$

By definition, \(\lambda _2(D)=0\) if D is not strong.

For a digraph D, its reverse \(D^{\mathrm{rev}}\) is a digraph with same vertex set and such that \(xy\in A(D^{\mathrm{rev}})\) if and only if \(yx\in A(D)\). A digraph D is symmetric if \(D^{\mathrm{rev}}=D\). In other words, a symmetric digraph D can be obtained from its underlying undirected graph G by replacing each edge of G with the corresponding arcs of both directions, that is, \(D=\overleftrightarrow {G}.\)

The strong subgraph k-(arc-)connectivity is not only a natural extension of the concept of generalized k-(edge-)connectivity, but also relates to important problems in graph theory. For \(k=2\), \(\kappa _2(\overleftrightarrow {G})=\kappa (G)\) [13] and \(\lambda _2(\overleftrightarrow {G})=\lambda (G)\) (Theorem 3.6). Hence, \(\kappa _k(D)\) and \(\lambda _k(D)\) could be seen as generalizations of connectivity and edge-connectivity of undirected graphs, respectively. For \(k=n\), \(\kappa _n(D)=\lambda _n(D)\) is the maximum number of arc-disjoint spanning strong subgraphs of D. Moreover, since \(\kappa _S(G)\) and \(\lambda _S(G)\) are the number of internally disjoint and arc-disjoint strong subgraphs containing a given set S, respectively, these parameters are relevant to the problem of finding the maximum number of strong spanning arc-disjoint subgraphs in a digraph studied, e.g., in [3,4,5, 12].

In what follows, n will denote the number of vertices of the digraph under consideration.

A digraph \(D=(V(D), A(D))\) is called minimally strong subgraph \((k,\ell )\) -connected if \(\kappa _k(D)\ge \ell\) but for any arc \(e\in A(D)\), \(\kappa _k(D-e)\le \ell -1\). Similarly, a digraph \(D=(V(D), A(D))\) is called minimally strong subgraph \((k,\ell )\)-arc-connected if \(\lambda _k(D)\ge \ell\) but for any arc \(e\in A(D)\), \(\lambda _k(D-e)\le \ell -1\).

A 2-cycle xyx of a strong digraph D is called a bridge if \(D-\{xy,yx\}\) is disconnected. Thus, a bridge corresponds to a bridge in the underlying undirected graph of D. An orientation of a digraph D is a digraph obtained from D by deleting an arc in each 2-cycle of D. A digraph D is semicomplete if for every distinct \(x,y\in V(D)\) at least one of the arcs xyyx is in D. A digraph D is k-regular if the in- and out-degree of every vertex of D is equal to k. We refer the readers to [2] for graph theoretical notation and terminology not given here.

Let \(k \ge 2\) and \(\ell \ge 2\) be fixed integers. By reduction from the Directed 2-Linkage problem, Sun et al. [13] proved that deciding whether \(\kappa _S(D)\ge \ell\) is NP-complete for a k-subset S of V(D). Thomassen [14] showed that for every positive integer p there are digraphs which are strongly p-connected, but which contain a pair of vertices not belonging to the same cycle. This implies that for every positive integer p there are strongly p-connected digraphs D such that \(\kappa _2(D)=1\) [13].

The above negative results motivate studying strong subgraph k-connectivity for special classes of digraphs. In Sun et al. [13], showed that the problem of deciding whether \(\kappa _k(D)\ge \ell\) for every semicomplete digraphs is polynomial-time solvable for fixed k and \(\ell\). The main tool used in their proof is a recent Directed k -Linkage theorem of Chudnovsky, Scott and Seymour [7]. Sun et al. [13] showed that for any connected graph G, the parameter \(\kappa _2(\overleftrightarrow {G})\) can be computed in polynomial time. This result is best possible in the following sense. Let D be a symmetric digraph and \(k\ge 3\) a fixed integer. Then it is NP-complete to decide whether \(\kappa _S(D)\ge \ell\) for \(S\subseteq V(D)\) with \(|S|=k\) [13]. Let D be a strong digraph with n vertices. Sun et al. [13] proved that \(1\le \kappa _k(D)\le n-1\) for \(2\le k\le n\). The bounds are sharp; Sun et al. [13] also characterized those digraphs D for which \(\kappa _k(D)\) attains the upper bound. The main tool used in their proof is a Hamiltonian cycle decomposition theorem of Tillson [15].

In this paper, we prove that for fixed integers \(k,\ell \ge 2\), the problem of deciding whether \(\lambda _S(D)\ge \ell\) is NP-complete for a digraph D and a set \(S\subseteq V(D)\) of size k. This result is proved in Sect. 3 using the corresponding result for \(\kappa _S(D)\) proved in [13]. In the same section, we also consider classes of digraphs. We characterize when \(\lambda _k(D)\ge 2\), \(2\le k\le n\), for both semicomplete and symmetric digraphs D of order n. The characterizations imply that the problem of deciding whether \(\lambda _k(D)\ge 2\) is polynomial-time solvable for both semicomplete and symmetric digraphs. For fixed \(\ell \ge 3\) and \(k\ge 2\), the complexity of deciding whether \(\lambda _k(D)\ge \ell\) remains an open problem for both semicomplete and symmetric digraphs. It was proved in [13] that for fixed \(k, \ell \ge 2\) the problem of deciding whether \(\kappa _k(D)\ge \ell\) is polynomial-time solvable for both semicomplete and symmetric digraphs, but it appears that the approaches to prove the two results cannot be used for \(\lambda _k(D)\). In fact, we would not be surprised if the \(\lambda _k(D)\ge \ell\) problem turns out to be NP-complete at least for one of the two classes of digraphs.

In Sect. 4, we first give sharp upper bounds for the parameters \(\kappa _k(D)\) and \(\lambda _k(D)\) in terms of classical connectivity. Then we get some lower and upper bounds for the parameter \(\lambda _k(D)\) including a lower bound whose analog for \(\kappa _k(D)\) does not hold as well as Nordhaus-Gaddum type bounds.

In Sect. 5, we characterize minimally strong subgraph \((2,n{-}2)\)-connected digraphs and minimally strong subgraph \((2,n{-}2)\)-arc-connected digraphs. Also, we bound the sizes of minimally strong subgraph \((2,n{-}2)\)-connected digraphs.

We conclude the paper in Sect. 6 by discussing open problems.

2 Preliminaries

Let us start this section from observations that can be easily verified using definitions of \(\lambda _{k}(D)\) and \(\kappa _k(D)\). Note that the first inequality of the following inequalities (2) can be found in [13].

Proposition 2.1

Let D be a digraph of order n, and let \(k\ge 2\) be an integer. Then

$$\begin{aligned} \lambda _{k+1}(D)\le \lambda _{k}(D) \text{ for } \text{ every } k\le n-1 \end{aligned}$$
(1)

For a spanning subgraph \(D'\) of D,  we have

$$\begin{aligned} \kappa _k(D')\le \kappa _k(D), \lambda _k(D')\le \lambda _k(D) \end{aligned}$$
(2)
$$\begin{aligned} \kappa _k(D)\le \lambda _k(D) \le \min \{\delta ^+(D), \delta ^-(D)\} \end{aligned}$$
(3)

The inequality (1) means that the parameter \(\lambda _{k}\) has a monotonically non-increasing with respect to k. However, this property may not hold for \(\kappa _{k}\), that is, \(\kappa _n(D)\le \kappa _{n-1}(D)\le \cdots \le \kappa _3(D)\le \kappa _2(D)=\kappa (D)\) may not be true. Consider the following example: Let D be a digraph obtained from two copies \(D_1\) and \(D_2\) of the complete digraph \(\overleftrightarrow {K}_{t}~(t\ge 4)\) by identifying one vertex in each of them. Clearly, D is a strong digraph with a cut vertex, say u. For \(2\le k\le 2t-2\), let S be a subset of \(V(D)\setminus \{u\}\) with \(|S|=k\) such that \(S\cap V(D_i)\ne \emptyset\) for every \(i\in \{1,2\}.\) Since each S-strong subgraph must contain u, we have \(\kappa _k(D)\le 1\), furthermore, we deduce that \(\kappa _k(D)= 1\) for \(2\le k\le 2t-2\). Let \(G_i\) be the underlying undirected graph of \(D_i\) for \(i\in \{1,2\}.\) Each \(G_i\) contains \(\lfloor \frac{t}{2}\rfloor\) edge-disjoint spanning trees, say \(T_{i,j}~(1\le j\le \lfloor \frac{t}{2}\rfloor )\), since \(G_i\) is a complete graph of order t (see, e.g., (3.1) in [10]). Now in D, let \(H_j\) be a subgraph of D obtained from the tree \(T_j\) which is the union of \(T_{1,j}\) and \(T_{2,j}\) by replacing each edge with two arcs of the opposite directions. Clearly, these subgraphs are strong, spanning and arc-disjoint. Hence, \(\kappa _{2t-1}(D)\ge \lfloor \frac{t}{2}\rfloor >1=\kappa _k(D)\) for \(2\le k\le 2t-2\).

We will use the following decomposition theorem by Tillson.

Theorem 2.2

[15] The arcs of \(\overleftrightarrow {K}_n\) can be decomposed into Hamiltonian cycles if and only if \(n\ne 4,6\).

3 Complexity

Yeo proved that it is an NP-complete problem to decide whether a 2-regular digraph has two arc-disjoint hamiltonian cycles (see, e.g., Theorem 6.6 in [5]). Thus, the problem of deciding whether \(\lambda _n(D)\ge 2\) is NP-complete, where n is the order of D. We will extend this result in Theorem 3.1.

Let D be a digraph and let \(s_1,s_2,\ldots {},s_k,t_1,t_2,\ldots {},t_k\) be a collection of not necessarily distinct vertices of D. A weak k-linkage from \((s_1,s_2,\ldots {},s_k)\) to \((t_1,t_2,\ldots {},t_k)\) is a collection of k arc-disjoint paths \(P_1,\ldots {},P_k\) such that \(P_i\) is an \((s_i,t_i)\)-path for each \(i\in [k]\). A digraph \(D=(V,A)\) is weakly k-linked if it contains a weak k-linkage from \((s_1,s_2,\ldots {},s_k)\) to \((t_1,t_2,\ldots {},t_k)\) for every choice of (not necessarily distinct) vertices \(s_1,\ldots {},s_k,t_1,\ldots {},t_k\). The weak k -linkage problem is the following. Given a digraph \(D=(V,A)\) and distinct vertices \(x_1,x_2,\ldots {},x_k, y_1,y_2,\ldots {},y_k\); decide whether D contains k arc-disjoint paths \(P_1,\ldots {},P_k\) such that \(P_i\) is an \((x_i,y_i)\)-path. The problem is well-known to be NP-complete already for \(k=2\) [2].

Fig. 1
figure 1

The digraph \(D'\)

Theorem 3.1

Let \(k\ge 2\) and \(\ell \ge 2\) be fixed integers. Let D be a digraph and \(S \subseteq V(D)\) with \(|S|=k\). The problem of deciding whether \(\lambda _S(D)\ge \ell\) is NP-complete.

Proof

Clearly, the problem is in NP. We will show that it is NP-hard using a reduction similar to that in Theorem 2.1 of [13]. Let us first deal with the case of \(\ell =2\) and \(k=2\). Consider the digraph \(D'\) used in the proof of Theorem 2.1 of [13] (see Fig. 1), where D is an arbitrary digraph, xy are vertices not in D, and \(t_1x,xs_1, t_2y,ys_2, xs_2,s_2x,yt_1,t_1y\) are additional arcs. To construct a new digraph \(D''\) from \(D'\), replace every vertex u of D by two vertices \(u^-\) and \(u^+\) such that \(u^-u^+\) is an arc in \(D''\) and for every \(uv\in A(D)\) add an arc \(u^+v^-\) to \(D''\). Also, for \(z\in \{x,y\}\), for every arc zu in \(D'\) add an arc \(zu^-\) to \(D''\) and for every arc uz add an arc \(u^+z\) to \(D''\).

Let \(S=\{x,y\}\). It was proved in Theorem 2.1 of [13] that \(\kappa _S(D')\ge 2\) if and only if there are vertex-disjoint paths from \(s_1\) to \(t_1\) and from \(s_2\) to \(t_2\). It follows from this result and definition of \(D''\) that \(\lambda _S(D'')\ge 2\) if and only if there are arc-disjoint paths from \(s_1^-\) to \(t^+_1\) and from \(s_2^-\) to \(t^+_2\). Since the weak 2-linkage problem is NP-complete, we conclude that the problem of deciding whether \(\lambda _S(D'')\ge 2\) is NP-hard.

Now let us consider the case of \(\ell \ge 3\) and \(k=2\). Add to \(D''\) \(\ell -2\) copies of the 2-cycle xyx and subdivide the arcs of every copy to avoid parallel arcs. Let us denote the new digraph by \(D'''\). Similarly to that in Theorem 2.1 of [13], we can show that \(\lambda _S(D''')\ge \ell\) if and only if \(\lambda _S(D'')\ge 2\).

It remains to consider the case of \(\ell \ge 2\) and \(k\ge 3\). Add to \(D'''\) (where \(D'''=D''\) for \(\ell =2\)) \(k-2\) new vertices \(x_1,\dots ,x_{k-2}\) and arcs of \(\ell\) 2-cycles \(xx_ix\) for each \(i\in [k-2]\). Subdivide the new arcs to avoid parallel arcs. Denote the obtained digraph by \(D''''\). Let \(S=\{x,y,x_1,\dots ,x_{k-2}\}\). Similarly to that in Theorem 2.1 of [13], we can show that \(\lambda _S(D'''')\ge \ell\) if and only if \(\lambda _S(D'')\ge 2\).

Bang-Jensen and Yeo [5] conjectured the following:

Conjecture 1

For every \(\lambda \ge 2\) there is a finite set \(\mathcal{S}_{\lambda }\) of digraphs such that a \(\lambda\)-arc-strong semicomplete digraph D contains \(\lambda\) arc-disjoint spanning strong subgraphs unless \(D\in \mathcal{S}_{\lambda }\).

Fig. 2
figure 2

Digraph \(S_4\)

Bang-Jensen and Yeo [5] proved the conjecture for \(\lambda =2\) by showing that \(|\mathcal{S}_2|=1\) and describing the unique digraph \(S_4\) of \(\mathcal{S}_2\) of order 4. Now we have the following characterization:

Theorem 3.2

For a semicomplete digraph D, of order n and an integer k such that \(2\le k\le n\), \(\lambda _k(D)\ge 2\) if and only if D is 2-arc-strong and the following does not hold: \(D\cong S_4\) and \(k=4\).

Proof

We first consider the direction “only if”. Suppose that D is not a 2-arc-strong and \(xy\in A(D)\) such that \(D-xy\) is not strong. Thus, for \(S=\{x,y\}\) we have \(\lambda _S(D)=1.\) Hence \(\lambda _2(D)=1\) and by (1) \(\lambda _k(D)=1\) for each \(k,\ 2\le k\le n.\) Furthermore, by the result of Bang-Jensen and Yeo, the following does not hold: \(D\cong S_4\) and \(k=4\).\(\square\)

We next prove the direction “if”. If D is 2-arc-strong and \(D\not \cong S_4\), then D contains two arc-disjoint spanning strong subgraphs by the result of Bang-Jensen and Yeo, that is, \(\lambda _n(D)\ge 2\). Furthermore, we have \(\lambda _k(D)\ge 2\) for all \(2\le k\le n\) by (1). Now we consider the case that \(D\cong S_4\). Let S be any subset of V(D) with \(|S|=3\); by symmetry of \(S_4\) it suffices to assume that \(S=\{v_1, v_2, v_3\}\) (see Fig. 2). Let \(D_1\) be the cycle \(v_1, v_2, v_3, v_1\) and \(D_2\) be subgraph of D with \(A(D_2)= A(D)\setminus A(D_1)\). It can be easily checked that both \(D_1\) and \(D_2\) are S-strong subgraphs, so \(\lambda _3(D)\ge 2\). Furthermore by (1), we have \(\lambda _2(D)\ge 2\).

Now we turn our attention to symmetric digraphs. We start from characterizing symmetric digraphs D with \(\lambda _k(D)\ge 2\), an analog of Theorem 3.2. To prove it we will use the following result of Boesch and Tindell [6] translated from the language of mixed graphs to that of digraphs.

Theorem 3.3

A strong digraph D has a strong orientation if and only if D has no bridge.

Here is our characterization.

Theorem 3.4

For a strong symmetric digraph D of order n and an integer k such that \(2\le k\le n\), \(\lambda _k(D)\ge 2\) if and only if D has no bridge.

Proof

Let D have no bridge. Then, by Theorem 3.3, D has a strong orientation H. Since D is symmetric, \(H^{\mathrm{rev}}\) is another orientation of D. Clearly, \(H^{\mathrm{rev}}\) is strong and hence \(\lambda _k(D)\ge 2\).\(\square\)

Suppose that D has a bridge xyx. Choose a set S of size k such that \(\{x,y\}\subseteq S\) and observe that any strong subgraph of D containing vertices x and y must include both xy and yx. Thus, \(\lambda _S(D)=1\) and \(\lambda _k(D)=1\).

Theorems 3.2 and 3.4 imply the following complexity result, which we believe to be extendable from \(\ell =2\) to any natural \(\ell\).

Corollary 3.5

The problem of deciding whether \(\lambda _k(D)\ge 2\) is polynomial-time solvable if D is either semicomplete or symmetric digraph of order n and \(2\le k\le n.\)

Now we give a lower bound on \(\lambda _k(D)\) for symmetric digraphs D.

Theorem 3.6

For every graph G, we have

$$\begin{aligned} \lambda _k(\overleftrightarrow {G})\ge \lambda _k(G). \end{aligned}$$

Moreover, this bound is sharp. In particular, we have \(\lambda _2(\overleftrightarrow {G})=\lambda _2(G)\).

Proof

We may assume that G is a connected graph. Let \(S=\{x,y\}\), where xy are distinct vertices of \(\overleftrightarrow {G}\). Observe that \(\lambda _S(G)\ge \lambda _S(\overleftrightarrow {G})\). Indeed, let \(p=\lambda _S(\overleftrightarrow {G})\) and let \(D_1,\dots ,D_p\) be arc-disjoint S-strong subgraphs of \(\overleftrightarrow {G}\). Thus, by choosing a path from x to y in each \(D_i\), we obtain p arc-disjoint paths from x to y, which correspond to p arc-disjoint paths between x and y in G. Thus, \(\lambda (G)=\lambda _2(G)\ge \lambda _2(\overleftrightarrow {G})\).

We now consider the general k. Let \(\lambda _S(\overleftrightarrow {G})=\lambda _k(\overleftrightarrow {G})\) for some \(S\subseteq V(\overleftrightarrow {G})\) with \(|S|=k\). We know that there are at least \(\lambda _k(G)\) edge-disjoint trees containing S in G, say \(T_i~(i\in [\lambda _k(G)])\). For each \(i\in [\lambda _k(G)]\), we can obtain a strong subgraph containing S, say \(D_i\), in \(\overleftrightarrow {G}\) by replacing each edge of \(T_i\) with the corresponding arcs of both directions. Clearly, any two such subgraphs are arc-disjoint, so we have \(\lambda _k(\overleftrightarrow {G})=\lambda _S(\overleftrightarrow {G})\ge \lambda _k(G)\), and we also have \(\lambda _2(\overleftrightarrow {G})=\lambda _2(G)=\lambda (G)\).

For the sharpness of the bound, consider the tree T with order n. Clearly, we have \(\lambda _k(T)=1\). Furthermore, \(1\le \lambda _k(\overleftrightarrow {T})\le \min \{\delta ^+(D), \delta ^-(D)\}=1\) by Inequality (3).\(\square\)

Note that for the case that \(3\le k\le n\), the equality \(\lambda _k(\overleftrightarrow {G})=\lambda _k(G)\) does not always hold. For example, consider the cycle \(C_n\) of order n; it is not hard to check that \(\lambda _k(\overleftrightarrow {C}_n)=2\), but \(\lambda _k(C_n)=1\).

Theorem 3.6 immediately implies the next result, which follows from the fact that \(\lambda (G)\) can be computed in polynomial time.

Corollary 3.7

For a symmetric digraph D, \(\lambda _2(D)\) can be computed in polynomial time.

4 Sharp bounds of \(\kappa _k(D)\) and \(\lambda _k(D)\)

To prove a new bound on \(\kappa _k(D)\) in Theorem 4.2, we will use the following result of Sun et al. [13].

Theorem 4.1

Let \(2\le k\le n\). For a strong digraph D of order n, we have

$$\begin{aligned} 1\le \kappa _k(D)\le n-1. \end{aligned}$$

Moreover, both bounds are sharp, and the upper bound holds if and only if \(D\cong \overleftrightarrow {K}_n\), \(2\le k\le n\) and \(k\not \in \{4,6\}\).

The following result concerns the relation between \(\kappa _k(D)\) (resp. \(\lambda _k(D)\)) and \(\kappa (D)\) (resp. \(\lambda (D)\)).

Theorem 4.2

Let \(k\in \{2,\dots ,n\}\). The following assertions hold:

  1. (i)

     For \(n\ge \kappa (D)+k,\) we have \(\kappa _k(D)\le \kappa (D)\);

  2. (ii)

     \(\lambda _k(D)\le \lambda (D).\) Moreover, both bounds are sharp.

Proof

Part (i). For \(k=2\), assume that \(\kappa (D)=\kappa (x,y)\) for some \(\{x,y\}\subseteq V(D)\). It follows from the strong subgraph connectivity definition that \(\kappa _{\{x,y\}}(D)\le \kappa (x,y)\), so \(\kappa _2(D)\le \kappa _{\{x,y\}}(D)\le \kappa (x,y)= \kappa (D).\)

We now consider the case of \(k\ge 3\). If \(\kappa (D)=n-1\), then we have \(\kappa _k(D)\le n-1=\kappa (D)\) by Theorem 4.1. If \(\kappa (D)=n-2\), then there are two vertices, say u and v, such that \(uv\not \in A(D)\). So we have \(\kappa _k(D)\le n-2=\kappa (D)\) by Theorem 4.1. If \(1\le \kappa (D)\le n-3\), then there exists a \(\kappa (D)\)-vertex cut, say Q, for two vertices uv in D such that there is no \(u-v\) path in \(D-Q\). Let \(S=\{u,v\}\cup S'\) where \(S'\subseteq V(D)\setminus (Q\cup \{u,v\})\) and \(|S'|=k-2\). Since u and v are in different strong components of \(D-Q\), any S-strong subgraph in D must contain a vertex in Q. By the definition of \(\kappa _S(D)\) and \(\kappa _k(D)\), we have \(\kappa _k(D)\le \kappa _S(D)\le |Q|=\kappa (D)\).

For the sharpness of the bound, consider the following digraph D. Let D be a symmetric digraph whose underlying undirected graph is \(K_{k}\bigvee {\overline{K}}_{n-k}\) (\(n\ge 3k\)), i.e. the graph obtained from disjoint graphs \(K_{k}\) and \({\overline{K}}_{n-k}\) by adding all edges between the vertices in \(K_{k}\) and \({\overline{K}}_{n-k}\).

Let \(V(D)=W\cup U\), where \(W=V(K_k)=\{w_i\mid 1\le i\le k\}\) and \(U=V({\overline{K}}_{n-k})=\{u_j\mid 1\le j\le n-k\}\). Let S be any k-subset of vertices of V(D) such that \(|S\cap U|=s\) (\(s\le k\)) and \(|S\cap W|=k-s\). Without loss of generality, let \(w_i\in S\) for \(1\le i\le k-s\) and \(u_j\in S\) for \(1\le j\le s\). For \(1\le i\le k-s\), let \(D_i\) be the symmetric subgraph of D whose underlying undirected graph is the tree \(T_i\) with edge set

$$\begin{aligned} \{w_iu_1, w_iu_2, \ldots , w_iu_s, u_{k+i}w_1, u_{k+i}w_2, \ldots , u_{k+i}w_{k-s}\}. \end{aligned}$$

For \(k-s+1\le j\le k\), let \(D_j\) be the symmetric subgraph of D whose underlying undirected graph is the tree \(T_j\) with edge set

$$\begin{aligned} \{w_ju_1, w_ju_2, \ldots , w_ju_s, w_jw_1, w_jw_2, \ldots , w_jw_{k-s}\}. \end{aligned}$$

Observe that \(\{D_i\mid 1\le i\le k-s\}\cup \{D_j\mid k-s+1\le j\le k\}\) is a set of k internally disjoint S-strong subgraph, so \(\kappa _S(D)\ge k\), and then \(\kappa _k(D)\ge k\). Combining this with the bound that \(\kappa _k(D)\le \kappa (D)\) and the fact that \(\kappa (D)\le \min \{\delta ^+(D), \delta ^-(D)\}=k\), we can get \(\kappa _k(D)= \kappa (D)=k\).

Part (ii) Let A be a \(\lambda (D)\)-arc-cut of D, where \(1\le \lambda (D)\le n-1\). We choose \(S\subseteq V(D)\) such that at least two of these k vertices are in different strong components of \(D-A\). Thus, any S-strong subgraph in D must contain an arc in A. By the definition of \(\lambda _S(D)\) and \(\lambda _k(D)\), we have \(\lambda _k(D)\le \lambda _S(D)\le |A|=\lambda (D)\).

For the sharpness of the bound, consider the the digraph D in part (i). Recall that \(\{D_i\mid 1\le i\le k\}\) is a set of k internally disjoint S-strong subgraph, so \(\lambda _S(D)\ge \kappa _S(D)\ge k\), and then \(\lambda _k(D)\ge k\). Combining this with the bound that \(\lambda _k(D)\le \lambda (D)\) and the fact that \(\lambda (D)\le \min \{\delta ^+(D), \delta ^-(D)\}=k\), we can get \(\lambda _k(D)= \lambda (D)=k\).\(\square\)

Note that the condition “\(n\ge \kappa (D)+k\)” in Theorem 4.2 cannot be removed. Consider the example after Proposition 2.1. We have \(n=2t-1< 2t=\kappa (D)+k\) when \(k=n\), but now \(\kappa _n(D)>\kappa (D)\).

In the proof of Theorem 4.1, they used the following result on \(\kappa _k(\overleftrightarrow {K}_n)\).

Lemma 4.3

[13] For \(2\le k\le n\), we have

$$\begin{aligned} \kappa _k(\overleftrightarrow {K}_n)=\left\{ \begin{array}{ll} {n-1}, &{} \text{ if } k\not \in \{4,6\}\text{; }\\ {n- 2}, &{}\text{ otherwise. } \end{array} \right. \end{aligned}$$

We can now compute the exact values of \(\lambda _k(\overleftrightarrow {K}_n)\).

Lemma 4.4

For \(2\le k\le n\), we have

$$\begin{aligned} \lambda _k(\overleftrightarrow {K}_n)=\left\{ \begin{array}{ll} {n-1}, &{} \text{ if } k\not \in \{4,6\}\text{, } \text{ or, } k\in \{4,6\} \text{ and } k<n\text{; }\\ {n- 2}, &{}\text{ if } k=n\in \{4,6\}\text{. } \end{array} \right. \end{aligned}$$

Proof

For the case that \(2\le k\le n\) and \(k\not \in \{4,6\}\), by (3) and Lemma 4.3, we have \(n-1\le \kappa _k(\overleftrightarrow {K}_n)\le \lambda _k(\overleftrightarrow {K}_n)\le n-1\). Hence, \(\lambda _k(\overleftrightarrow {K}_n)= n-1\) and in the following argument we assume that \(2\le k\le n\) and \(k\in \{4,6\}\).

We first consider the case of \(2\le k=n\). For \(n=4\), since \(K_n\) contains a Hamiltonian cycle, the two orientations of the cycle imply that \(\lambda _n(\overleftrightarrow {K}_n) \ge 2 = n-2\). To see that there are at most two arc-disjoint strong spanning subgraphs of \(\overleftrightarrow {K}_n\), suppose that there are three arc-disjoint such subgraphs. Then each such subgraph must have exactly four arcs (as \(|A(\overleftrightarrow {K}_n)|=12\)), and so all of these three subgraphs are Hamiltonian cycles, which means that the arcs of \(\overleftrightarrow {K}_n\) can be decomposed into Hamiltonian cycles, a contradiction to Theorem 2.2). Hence, \(\lambda _n(\overleftrightarrow {K}_n)= n-2\) for \(n=4\). Similarly, we can prove that \(\lambda _n(\overleftrightarrow {K}_n)= n-2\) for \(n=6\), as \(K_n\) contains two edge-disjoint Hamiltonian cycles, and therefore \(\overleftrightarrow {K}_n\) contains four arc-disjoint Hamiltonian cycles.

We next consider the case of \(2\le k\le n-1\). We assume that \(k=6\) as the case of \(k=4\) can be considered in a similar and simpler way. Let \(S\subseteq V(\overleftrightarrow {K}_n)\) be any vertex subset of size six. Let \(S=\{u_i\mid 1\le i\le 6\}\) and \(V(\overleftrightarrow {K}_n)\setminus S=\{v_j\mid 1\le j\le n-6\}\). Let \(D_1\) be the cycle \(u_1u_2u_3u_4u_5u_6u_1\); let \(D_2=D_1^{\mathrm{rev}}\); let \(D_3\) be the cycle \(u_1u_3u_6u_4u_2u_5u_1\); let \(D_4=D_3^{\mathrm{rev}}\); let \(D_5\) be a subgraph of \(\overleftrightarrow {K}_n\) with vertex set \(S\cup \{v_1\}\) and arc set \(\{u_1v_1, v_1u_2, u_2u_6, u_6v_1, v_1u_5, u_5u_3, u_3v_1, v_1u_4, u_4u_1\}\); let \(D_6=D_5^{\mathrm{rev}}\); for each \(x\in \{v_j\mid 2\le j\le n-6\}\), let \(D_x\) be a subgraph of \(\overleftrightarrow {K}_n\) with vertex set \(S\cup \{x\}\) and arc set \(\{xu_i, u_ix\mid 1\le i\le 6\}\). Hence, we have \(\lambda _S(D)\ge n-1\) for any \(S\subseteq V(\overleftrightarrow {K}_n)\) with \(|S|=6\) and so \(\lambda _k(D)\ge n-1\). We clearly have \(\lambda _k(D)\le n-1\) by (3), then our result holds.\(\square\)

Now we obtain sharp lower and upper bounds for \(\lambda _k(D)\) for \(2\le k\le n\).

Theorem 4.5

Let \(2\le k\le n\). For a strong digraph D of order n, we have

$$\begin{aligned} 1\le \lambda _k(D)\le n-1. \end{aligned}$$

Moreover, both bounds are sharp, and the upper bound holds if and only if \(D\cong \overleftrightarrow {K}_n\), where \(k\not \in \{4,6\}\), or, \(k\in \{4,6\}\) and \(k<n\).

Proof

The lower bound is clearly correct by the definition of \(\lambda _k(D)\), and for the sharpness, a cycle is our desired digraph. The upper bound and its sharpness hold by (2) and Lemma 4.4.

If D is not equal to \(\overleftrightarrow {K}_n\) then \(\delta ^+(D) \le n-2\) and by (3) we observe that \(\lambda _k(D) \le \delta ^+(D) \le n-2\). Therefore, by Lemma 4.4, the upper bound holds if and only if \(D\cong \overleftrightarrow {K}_n\), where \(k\not \in \{4,6\}\), or, \(k\in \{4,6\}\) and \(k<n\).

Shiloach [11] proved the following:

Theorem 4.6

[11] A digraph D is weakly k-linked if and only if D is k-arc-strong.

Using Shiloach’s Theorem, we will prove the following lower bound for \(\lambda _k(D).\) Such a bound does not hold for \(\kappa _k(D)\) since it was shown in [13] using Thomassen’s result in [14] that for every \(\ell\) there are digraphs D with \(\kappa (D)=\ell\) and \(\kappa _2(D)=1\).

Proposition 4.7

Let \(k\le \ell =\lambda (D)\). We have \(\lambda _k(D)\ge \lfloor \ell /k\rfloor\).

Proof

Choose an arbitrary vertex set \(S=\{s_1,\ldots ,s_k\}\) of D and let \(t=\lfloor \ell /k\rfloor\). By Theorem 4.6, there is a weak kt-linkage L from \(x_1,x_2,\ldots {},x_{kt}\) to \(y_1,y_2,\ldots {},y_{kt}\), where \(x_i= s_{i \mod k}\) and \(y_i=s_{i \mod k +1}\) and \(s_{k+1}=s_1\). Note that the paths of L form t arc-disjoint strong subgraphs of D containing S.\(\square\)

For a digraph \(D=(V(D), A(D))\), the complement digraph, denoted by \(D^c\), is a digraph with vertex set \(V(D^c)=V(D)\) such that \(xy\in A(D^c)\) if and only if \(xy\not \in A(D)\).

Given a graph parameter f(G), the Nordhaus-Gaddum Problem is to determine sharp bounds for (a) \(f(G) + f(G^c)\) and (b) \(f(G)f(G^c)\), and characterize the extremal graphs. The Nordhaus-Gaddum type relations have received wide attention; see a recent survey paper [1] by Aouchiche and Hansen. Theorem 4.9 concerns such type of a problem for the parameter \(\lambda _k\). To prove the theorem, we will need the following:

Proposition 4.8

A digraph D with order n is strong if and only if \(\lambda _k(D)\ge 1\), where \(2\le k\le n\).

Proof

If D is strong, then for every vertex set S of size kD has a strong subgraph containing S. If \(\lambda _k(D)\ge 1\), for each vertex set S of size k construct \(D_S,\) a strong subgraph of D containing S. The union of all \(D_S\) is a strong subgraph of D as there are sets \(S_1, S_2, \dots , S_p\) such that the union of \(S_1, S_2, \dots , S_p\) is V(D) and for each \(i\in [p-1],\) \(D_{S_i}\) and \(D_{S_{i+1}}\) share a common vertex.\(\square\)

Theorem 4.9

For a digraph D with order n, the following assertions hold:

  1. (i)

     \(0\le \lambda _k(D)+\lambda _k(D^c)\le n-1\). Moreover, both bounds are sharp. In particular, the lower bound holds if and only if \(\lambda (D)=\lambda (D^c)=0\).

  2. (ii)

     \(0\le \lambda _k(D){\lambda _k(D^c)}\le \left( \frac{n-1}{2}\right) ^2\). Moreover, both bounds are sharp. In particular, the lower bound holds if and only if \(\lambda (D)=0\) or \(\lambda (D^c)=0\).

Proof

We first prove (i). Since \(D\cup D^c=\overleftrightarrow {K}_n\), by definition of \(\lambda _k\), \(\lambda _k(D)+\lambda _k(D^c)\le \lambda _k(\overleftrightarrow {K}_n)\). Thus, by Lemma 4.4, the upper bound for the sum \(\lambda _k(D)+\lambda _k(D^c)\) holds. Let \(H\cong \overleftrightarrow {K}_n\). When \(k\not \in \{4,6\}\), or, \(k\in \{4,6\}\) and \(k<n\), by Lemma 4.4, we have \(\lambda _k(H)=n-1\) and we clearly have \(\lambda _k(H^c)=0\), so the upper bound is sharp.

The lower bound is clear. Clearly, the lower bound holds, if and only if \(\lambda _k(D)=\lambda _k(D^c)=0\), if and only if \(\lambda (D)=\lambda (D^c)=0\) by Proposition 4.8.

We now prove (ii). The lower bound is clear, and it holds, if and only if \(\lambda _k(D)=0\) or \(\lambda _k(D^c)=0\), if and only if \(\lambda (D)=0\) or \(\lambda (D^c)=0\) by Proposition 4.8. For the upper bound, we have

$$\begin{aligned} \lambda _k(D){\lambda _k(D^c)}\le \left( \frac{\lambda _k(D)+\lambda _k(D^c)}{2}\right) ^2\le \left( \frac{n-1}{2}\right) ^2. \end{aligned}$$

Let \(H\cong \overleftrightarrow {K}_{n}\) with \(n=2h+1\ge 7\). By Theorem 2.2, H contains 2h arc-disjoint Hamiltonian cycles: \(H_1, \ldots , H_{2h}\). Let \(D_1\) be the union of the former h cycles, and \(D_2\) be the union of the remaining h cycles. Clearly, \(D_1^c=D_2\) and \(\lambda _n(D_i)\ge h\) and so \(\lambda _k(D_i)\ge h\) for \(1\le i\le 2, 2\le k \le n\) by (1). Furthermore, \(D_i\) is h-regular, so \(\lambda _k(D_i)\le h\) by (3). Hence, \(\lambda _k(D_i)= h\) for \(1\le i\le 2, 2\le k \le n\). Now \(\lambda _k(D_1){\lambda _k(D_1^c)}=\lambda _k(D_1){\lambda _k(D_2)}= h^2=\left( \frac{n-1}{2}\right) ^2\), so the upper bound is sharp.\(\square\)

5 Minimally Strong Subgraph \((k,\ell )\)-(arc-)connected Digraphs

In this section, we will first study the minimally strong subgraph \((k,\ell )\)-connected digraphs. By the definition of a minimally strong subgraph \((k,\ell )\)-connected digraph, we can get the following observation.

Proposition 5.1

A digraph D is minimally strong subgraph \((k,\ell )\)-connected if and only if \(\kappa _k(D)= \ell\) and \(\kappa _k(D-e)= \ell -1\) for any arc \(e\in A(D)\).

Proof

The direction “if” is clear by definition, and we only need to prove the direction “only if”. Let D be a minimally strong subgraph \((k,\ell )\)-connected digraph. By definition, we have \(\kappa _k(D)\ge \ell\) and \(\kappa _k(D-e)\le \ell -1\) for any arc \(e\in A(D)\). Then for any set \(S \subseteq V(D)\) with \(|S|=k\), there is a set \({\mathcal {D}}\) of \(\ell\) internally disjoint S-strong subgraphs. As e must belong to one and only one element of \({\mathcal {D}}\), we are done.\(\square\)

A digraph D is minimally strong if D is strong but \(D-e\) is not for every arc e of D.

Proposition 5.2

The following assertions hold:

  1. (i)

     A digraph D is minimally strong subgraph (k, 1)-connected if and only if D is a minimally strong digraph;

  2. (ii)

     For \(k\ne 4,6\), a digraph D is minimally strong subgraph \((k,n-1)\)-connected if and only if \(D\cong \overleftrightarrow {K}_n\).

Proof

To prove (i), it suffices to show that a digraph D is strong if and only if \(\kappa _k(D)\ge 1.\) If D is strong, then for every vertex set S of size kD has an S-strong subgraph. If \(\kappa _k(D)\ge 1\), for each vertex set S of size k construct \(D_S,\) an S-strong subgraph of D. The union of all \(D_k\) is a strong subgraph of D as there are sets \(S_1, S_2, \dots , S_p\) such that the union of \(S_1, S_2, \dots , S_p\) is V(D) and for each \(i\in [p-1],\) \(D_{S_i}\) and \(D_{S_{i+1}}\) share a common vertex.

Part (ii) follows from Theorem 4.1.\(\square\)

The following result characterizes minimally strong subgraph \((2,n-2)\)-connected digraphs.

Theorem 5.3

A digraph D is minimally strong subgraph \((2,n-2)\)-connected if and only if D is a digraph obtained from the complete digraph \(\overleftrightarrow {K}_n\) by deleting an arc set M such that \(\overleftrightarrow {K}_n[M]\) is a 3-cycle or a union of \(\lfloor n/2\rfloor\) vertex-disjoint 2-cycles. In particular, we have \(f(n,2,n-2)=n(n-1)-2\lfloor n/2\rfloor\), \(F(n,2,n-2)=n(n-1)-3\).

Proof

Let \(D\cong \overleftrightarrow {K}_n-M\) be a digraph obtained from the complete digraph \(\overleftrightarrow {K}_n\) by deleting an arc set M. Let \(V(D)=\{u_i\mid 1\le i\le n\}\).

Firstly, we will consider the case that \(\overleftrightarrow {K}_n[M]\) is a 3-cycle \(u_1u_2u_3u_1\). We now prove that \(\kappa _2(D)=n-2\). By (3), we have \(\kappa _2(D)\le \min \{\delta ^+(D), \delta ^-(D)\}=n-2\). Let \(S=\{u,v\} \subseteq V(D)\); we just consider the case that \(u=u_1,v=u_2\) since the other cases are similar. Let \(D_1\) be a subgraph of D with \(V(D_1)=\{u_1,u_2,u_3\}\) and \(A(D_1)=\{u_1u_3, u_3u_2, u_2u_1\}\); for \(2\le i\le n-2\), let \(D_i\) be a subgraph of D with \(V(D_i)=\{u_1,u_2,u_{i+2}\}\) and \(A(D_i)=\{u_1u_{i+2}, u_2u_{i+2}, u_{i+2}u_1, u_{i+2}u_2\}\). Clearly, \(\{D_i\mid 1\le i\le n-2\}\) is a set of \(n-2\) internally disjoint S-strong subgraphs, so \(\kappa _S(D)\ge n-2\) and \(\kappa _2(D)\ge n-2\). Hence, \(\kappa _2(D)= n-2\).

For any \(e\in A(D)\), without loss of generality, one of the two digraphs in Fig. 3 is a subgraph of \(\overleftrightarrow {K}_n[M\cup \{e\}]\), so if the following claim holds, then we must have \(\kappa _2(D-e)\le \kappa _2(D') \le n-3\) by Proposition 4.3, and so D is minimally strong subgraph \((2,n-2)\)-connected. Now it suffices to prove the following claim. \(\square\)

Fig. 3
figure 3

Two graphs for Claim 1

Claim 1

If \(\overleftrightarrow {K}_n[M']\) is isomorphic to one of two graphs in Fig. 3, then \(\kappa _2(D')\le n-3\), where \(D'=\overleftrightarrow {K}_n-M'\).

Proof of Claim 1

We first show that \(\kappa _2(D')\le n-3\) if \(M'\) is the digraph of Fig. 3a. Let \(S=\{u_2, u_4\}\); we will prove that \(\kappa _S(D')\le n-3\), and then we are done. Suppose that \(\kappa _S(D')\ge n-2\), then there exists a set of \(n-2\) internally disjoint S-strong subgraphs, say \(\{D_i\mid 1\le i\le n-2\}\). If both of the two arcs \(u_2u_4\) and \(u_4u_2\) belong to the same \(D_i\), say \(D_1\), then for \(2\le i\le n-2\), each \(D_i\) contains at least one vertex and at most two vertices of \(\{u_i\mid 1\le i\le n, i\ne 2,4\}\). Furthermore, there is at most one \(D_i\), say \(D_2\), contains (exactly) two vertices of \(\{u_i\mid 1\le i\le n, i\ne 2,4\}\). We just consider the case that \(u_1,u_3\in V(D_2)\) since the other cases are similar. In this case, we must have that each vertex of \(\{u_i\mid 5\le i\le n\}\) belongs to exactly one digraph from \(\{D_i\mid 3\le i\le n-2\}\) and vice versa. However, this is impossible since the vertex set \(\{u_2, u_4, u_5\}\) cannot induce an S-strong subgraph of \(D'\), a contradiction.

So we now assume that each \(D_i\) contains at most one of \(u_2u_4\) and \(u_4u_2\). Without loss of generality, we may assume that \(u_2u_4\in A(D_1)\) and \(u_4u_2\in A(D_2)\). In this case, we must have that each vertex of \(\{u_i\mid 1\le i\le n, i\ne 2,4\}\) belongs to exactly one digraph from \(\{D_i\mid 1\le i\le n-2\}\) and vice versa. However, this is also impossible since the vertex set \(\{u_2, u_4, u_5\}\) cannot induce an S-strong subgraph of \(D'\), a contradiction.

Hence, we have \(\kappa _2(D')\le n-3\) in this case. For the case that \(M'\) is the digraph of Fig. 3b, we can choose \(S=\{u_2, u_3\}\) and prove that \(\kappa _S(D')\le n-3\) with a similar argument, and so \(\kappa _2(D')\le n-3\) in this case. This completes the proof of the claim.

Secondly, we consider the case that \(\overleftrightarrow {K}_n[M]\) is a union of \(\lfloor n/2\rfloor\) vertex-disjoint 2-cycles. Without loss of generality, we may assume that \(M=\{u_{2i-1}u_{2i}, u_{2i}u_{2i-1}\mid 1\le i\le \lfloor n/2\rfloor \}\). We just consider the case that \(S=\{u_1, u_3\}\) since the other cases are similar. In this case, let \(D_1\) be the subgraph of D with \(V(D_1)=\{u_1, u_3\}\) and \(A(D_1)=\{u_1u_3, u_3u_1\}\); let \(D_2\) be the subgraph of D with \(V(D_2)=\{u_1,u_2,u_3,u_4\}\) and \(A(D_2)=\{u_1u_4, u_4u_1, u_2u_4, u_4u_2, u_2u_3, u_3u_2\}\); for \(3\le i\le n-2\), let \(D_i\) be the subgraph of D with \(V(D_i)=\{u_1,u_2,u_{i+2}\}\) and \(A(D_i)=\{u_1u_{i+2}, u_3u_{i+2}, u_{i+2}u_1, u_{i+2}u_3\}\). Clearly, \(\{D_i\mid 1\le i\le n-2\}\) is a set of \(n-2\) internally disjoint S-strong subgraphs, so \(\kappa _S(D)\ge n-2\) and then \(\kappa _2(D)\ge n-2\). By (3), we have \(\kappa _2(D)\le \min \{\delta ^+(D), \delta ^-(D)\}=n-2\). Hence, \(\kappa _2(D)= n-2\). Let \(e\in A(D)\); clearly e must be incident with at least one vertex of \(\{u_i\mid 1\le i\le 2\lfloor n/2\rfloor \}\). Then we have that \(\kappa _2(D-e)\le \min \{\delta ^+(D-e), \delta ^-(D-e)\}=n-3\) by (3). Hence, D is minimally strong subgraph \((2,n-2)\)-connected.

Now let D be minimally strong subgraph \((2,n-2)\)-connected. By Theorem 4.1, we have that \(D\not \cong \overleftrightarrow {K}_n\), that is, D can be obtained from a complete digraph \(\overleftrightarrow {K}_n\) by deleting a nonempty arc set M. To end our argument, we need the following three claims. Let us start from a simple yet useful observation.

Proposition 5.4

No pair of arcs in M has a common head or tail.

Proof of Proposition 5.4. By (3) no pair of arcs in M has a common head or tail, as otherwise we would have \(\kappa _2(D)\le n-3\).

Claim 2

\(|M|\ge 3\).

Proof of Claim 2

Let \(|M|\le 2\). We may assume that \(|M|=2\) as the case of \(|M|=1\) can be considered in a similar and simpler way.

Let the arcs of M have no common vertices; without loss of generality, \(M=\{u_1u_2,u_3u_4\}\). Then \(\kappa _2(D-u_2u_1)=n-2\) as \(D-u_2u_1\) is a supergraph of \(\overleftrightarrow {K}_n\) without a union of \(\lfloor n/2\rfloor\) vertex-disjoint 2-cycles including the cycles \(u_1u_2u_1\) and \(u_3u_4u_3\). Thus, D is not minimally strong subgraph \((2,n-2)\)-connected. Let the arcs of M have no common vertex. By Proposition 5.4, without loss of generality, \(M=\{u_1u_2,u_2u_3\}\). Then \(\kappa _2(D-u_3u_1)=n-2\) as we showed in the beginning of the proof of this theorem. Thus, D is not minimally strong subgraph \((2,n-2)\)-connected. Now let the arcs of M have the same vertices, i.e., without loss of generality, \(M=\{u_1u_2,u_2u_1\}\). As above, \(\kappa _2(D-u_2u_1)=n-2\) and D is not minimally strong subgraph \((2,n-2)\)-connected.

Claim 3

If \(|M|= 3\), then \(\overleftrightarrow {K}_n[M]\) is a 3-cycle.

Proof of Claim 3

Suppose that D is minimally strong subgraph \((2,n-2)\)-connected, but \(\overleftrightarrow {K}_n[M]\) is not a 3-cycle. By Proposition 5.4, no pair of arcs in M has a common head or tail. Thus, \(\overleftrightarrow {K}_n[M]\) must be isomorphic to one of graphs in Figs. 3 and 4. If \(\overleftrightarrow {K}_n[M]\) is isomorphic to one of graphs in Fig. 3, then \(\kappa _2(D)\le n-3\) by Claim 1 and so D is not minimally strong subgraph \((2,n-2)\)-connected, a contradiction. For an arc set \(M_0\) such that \(\overleftrightarrow {K}_n[M_0]\) is a union of \(\lfloor n/2\rfloor\) vertex-disjoint 2-cycles, by the argument before, we know that \(\overleftrightarrow {K}_n-M_0\) is minimally strong subgraph \((2,n-2)\)-connected. For the case that \(\overleftrightarrow {K}_n[M]\) is isomorphic to (a) or (b) in Fig. 4, we have that \(\overleftrightarrow {K}_n-M_0\) is a proper subgraph of \(\overleftrightarrow {K}_n-M\), so \(D=\overleftrightarrow {K}_n-M\) must not be minimally strong subgraph \((2,n-2)\)-connected, this also produces a contradiction. Hence, the claim holds.

Fig. 4
figure 4

Two graphs for Claim 3

Claim 4

If \(|M|> 3\), then \(\overleftrightarrow {K}_n[M]\) is a union of \(\lfloor n/2\rfloor\) vertex-disjoint 2-cycles.

Proof of Claim 4

Suppose that D is minimally strong subgraph \((2,n-2)\)-connected, but \(\overleftrightarrow {K}_n[M]\) is not a union of \(\lfloor n/2\rfloor\) vertex-disjoint 2-cycles.

By Claim 1 and Proposition 4.3, we have that \(\overleftrightarrow {K}_n[M]\) does not contain graphs in Fig. 3 as a subgraph. Then \(\overleftrightarrow {K}_n[M]\) does not contain a path of length at least three. Hence, the underlying undirected graph of M has at least two connectivity components. By the fact that if M is a 3-cycle, then \(\overleftrightarrow {K}_n-M\) is minimally strong subgraph \((2,n-2)\)-connected, we conclude that \(\overleftrightarrow {K}_n[M]\) does not contain a cycle of length three. By Claim 1, \(\overleftrightarrow {K}_n[M]\) does not contain a path of length two. By Proposition 5.4, no pair of arcs in M has a common head or tail. Hence, each connectivity component of \(\overleftrightarrow {K}_n[M]\) must be a 2-cycle or an arc. Since D is minimally strong subgraph \((2,n-2)\)-connected, no connectivity component of \(\overleftrightarrow {K}_n[M]\) is an arc. We have arrived at a contradiction, proving Claim 4.

Hence, if a digraph D is minimally strong subgraph \((2,n-2)\)-connected, then \(D\cong \overleftrightarrow {K}_n-M\), where \(\overleftrightarrow {K}_n[M]\) is a cycle of order three or a union of \(\lfloor n/2\rfloor\) vertex-disjoint 2-cycles.

Now the claimed values of \(F(n,2,n-2)\) and \(f(n,2,n-2)\) can easily be verified.

Let \({\mathfrak {F}}(n,k,\ell )\) be the set of all minimally strong subgraph \((k,\ell )\)-connected digraphs with order n. We define

$$\begin{aligned} F(n,k,\ell )=\max \{|A(D)| \mid D\in {\mathfrak {F}}(n,k,\ell )\} \end{aligned}$$

and

$$\begin{aligned} f(n,k,\ell )=\min \{|A(D)| \mid D\in {\mathfrak {F}}(n,k,\ell )\}. \end{aligned}$$

We further define

$$\begin{aligned} Ex(n,k,\ell )=\{D\mid D\in {\mathfrak {F}}(n,k,\ell ), |A(D)|=F(n,k,\ell )\} \end{aligned}$$

and

$$\begin{aligned} ex(n,k,\ell )=\{D\mid D\in {\mathfrak {F}}(n,k,\ell ), |A(D)|=f(n,k,\ell )\}. \end{aligned}$$

Note that Theorem 5.3 implies that \(Ex(n,2,n-2)=\{\overleftrightarrow {K_n}-M\}\) where M is an arc set such that \(\overleftrightarrow {K}_n[M]\) is a directed 3-cycle, and \(ex(n,2,n-1)=\{\overleftrightarrow {K_n}-M\}\) where M is an arc set such that \(\overleftrightarrow {K}_n[M]\) is a union of \(\lfloor n/2\rfloor\) vertex-disjoint directed 2-cycles.

The following result concerns a sharp lower bound for the parameter \(f(n,k,\ell )\).

Theorem 5.5

For \(2\le k\le n\), we have

$$\begin{aligned} f(n,k,\ell )\ge n\ell . \end{aligned}$$

Moreover, the following assertions hold: (i)  If \(\ell =1\), then \(f(n,k,\ell )=n\); (ii)  If \(2\le \ell \le n-1\), then \(f(n,n,\ell )=n\ell\) for \(k=n\not \in \{4,6\}\); (iii) If n is even and \(\ell = n-2\), then \(f(n,2,\ell )=n\ell .\)

Proof

By (3), for all digraphs D and \(k \ge 2\) we have \(\kappa _k(D) \le \delta ^+(D)\) and \(\kappa _k(D) \le \delta ^-(D)\). Hence for each D with \(\kappa _k(D)=\ell\), we have that \(\delta ^+(D), \delta ^-(D)\ge \ell\), so \(|A(D)|\ge n\ell\) and then \(f(n,k,\ell )\ge n\ell .\)

For the case that \(\ell =1\), let D be a dicycle \(\overrightarrow{C_n}\). Clearly, D is minimally strong subgraph (k, 1)-connected, and we know \(|A(D)|=n\), so \(f(n,k,1)= n\).

For the case that \(k=n \not \in \{4,6\}\) and \(2\le \ell \le n-1\), let \(D\cong \overleftrightarrow {K_n}\). By Theorem 2.2, D can be decomposed into \(n-1\) Hamiltonian cycles \(H_i~(1\le i\le n-1)\). Let \(D_{\ell }\) be the spanning subgraph of D with arc sets \(A(D_{\ell })=\bigcup _{1\le i\le \ell }{A(H_i)}\). Clearly, we have \(\kappa _n(D_{\ell })\ge \ell\) for \(2\le \ell \le n-1\). Furthermore, by (3), we have \(\kappa _n(D_{\ell })\le \ell\) since the in-degree and out-degree of each vertex in \(D_{\ell }\) are both \(\ell\). Hence, \(\kappa _n(D_{\ell })= \ell\) for \(2\le \ell \le n-1\). For any \(e\in A(D_{\ell })\), we have \(\delta ^+(D_{\ell }-e)=\delta ^-(D_{\ell }-e)=\ell -1\), so \(\kappa _n(D_{\ell }-e)\le \ell -1\) by (3). Thus, \(D_{\ell }\) is minimally strong subgraph \((n,\ell )\)-connected. As \(|A(D_{\ell })|=n\ell\), we have \(f(n,n,\ell )\le n\ell\). From the lower bound that \(f(n,k,\ell )\ge n\ell\), we have \(f(n,n,\ell )= n\ell\) for the case that \(2\le \ell \le n-1, n\not \in \{4,6\}\).

Part (iii) follows directly from Theorem 5.3. \(\square\)

To prove two upper bounds on the number of arcs in a minimally strong subgraph \((k,\ell )\)-connected digraph, we will use the following result from [2].

Theorem 5.6

Every strong digraph D on n vertices has a strong spanning subgraph H with at most \(2n-2\) arcs and equality holds only if H is a symmetric digraph whose underlying undirected graph is a tree.

Proposition 5.7

We have (i\(F(n,n,\ell )\le 2\ell (n-1)\); (ii) For every k \((2\le k\le n)\), \(F(n,k,1)=2(n-1)\) and Ex(nk, 1) consists of symmetric digraphs whose underlying undirected graphs are trees.

Proof

(i) Let \(D=(V,A)\) be a minimally strong subgraph \((n,\ell )\)-connected digraph, and let \(D_1,\dots ,D_{\ell }\) be arc-disjoint strong spanning subgraphs of D. Since D is minimally strong subgraph \((n,\ell )\)-connected and \(D_1,\dots ,D_{\ell }\) are pairwise arc-disjoint, \(|A|=\sum _{i=1}^{\ell }|A(D_i)|.\) Thus, by Theorem 5.6, \(|A|\le 2\ell (n-1).\)

(ii) In the proof of Proposition 5.2 we showed that a digraph D is strong if and only if \(\kappa _k(D)\ge 1.\) Now let \(\kappa _k(D)\ge 1\) and a digraph D has a minimal number of arcs. By Theorem 5.6, we have that \(|A(D)|\le 2(n-1)\) and if \(D \in Ex(n,k,1)\) then \(|A(D)|=2(n-1)\) and D is a symmetric digraph whose underlying undirected graph is a tree.\(\square\)

We now study the minimally strong subgraph \((k,\ell )\)-arc-connected digraphs. By Proposition 4.8 and Theorem 4.5, we have the following result.

Proposition 5.8

The following assertions hold:

(i):

 A digraph D is minimally strong subgraph

(k, 1):

-arc-connected if and only if D is minimally strong digraph;

(ii):

 Let \(2\le k\le n\). If \(k\not \in \{4,6\}\), or, \(k\in \{4,6\}\) and \(k<n\), then a digraph D is minimally strong subgraph \((k,n-1)\)-arc-connected if and only if \(D\cong \overleftrightarrow {K}_n\).

The following result characterizes minimally strong subgraph \((2,n-2)\)-arc-connected digraphs. This characterization is different from the characterization of minimally strong subgraph \((2,n-2)\)-connected digraphs obtained in Theorem 5.3.

Theorem 5.9

A digraph D is minimally strong subgraph \((2,n-2)\)-arc-connected if and only if D is a digraph obtained from the complete digraph \(\overleftrightarrow {K}_n\) by deleting an arc set M such that \(\overleftrightarrow {K}_n[M]\) is a union of vertex-disjoint cycles which cover all but at most one vertex of \(\overleftrightarrow {K}_n\).

Proof

Let D be a digraph obtained from the complete digraph \(\overleftrightarrow {K}_n\) by deleting an arc set M such that \(\overleftrightarrow {K}_n[M]\) is a union of vertex-disjoint cycles which cover all but at most one vertex of \(\overleftrightarrow {K}_n\). To prove the theorem it suffices to show that (a) D is minimally strong subgraph \((2,n-2)\)-arc-connected, that is, \(\lambda _2(D)\ge n-2\) but for any arc \(e\in A(D)\), \(\lambda _2(D-e)\le n-3\), and (b) if a digraph H minimally strong subgraph \((2,n-2)\)-arc-connected then it must be constructed from \(\overleftrightarrow {K}_n\) as the digraph D above. Thus, the remainder of the proof has two parts.

Part (a). We just consider the case that \(\overleftrightarrow {K}_n[M]\) is a union of vertex-disjoint cycles which cover all vertices of \(\overleftrightarrow {K}_n\), since the argument for the other case is similar. For any \(e\in A(\overleftrightarrow {K}_n)\setminus M\), we know e must be adjacent to at least one element of M, so \(\lambda _2(D-e)\le \min \{\delta ^+(D-e), \delta ^-(D-e)\}=n-3\) by (3). Hence, it suffices to show that \(\lambda _2(D)= n-2\) in the following. We clearly have that \(\lambda _2(D)\le n-2\) by (3), so we will show that for \(S=\{x, y\}\subseteq V(D)\), there are at least \(n-2\) arc-disjoint S-strong subgraphs in D.

Case 1. x and y belong to distinct cycles of \(\overleftrightarrow {K}_n[M]\). We just consider the case that the lengths of these two cycles are both at least three, since the arguments for the other cases are similar. Assume that \(u_1x, xu_2\) belong to one cycle, and \(u_3y, yu_4\) belong to the other cycle. Note that \(u_1u_2, u_3u_4 \in A(D)\) since the lengths of these two cycles are both at least three.

Let \(D_1\) be the 2-cycle xyx; let \(D_2\) be the subgraph of D with vertex set \(\{x, y, u_1, u_2\}\) and arc set \(\{xu_1, u_1u_2, u_2x, yu_2, u_2y\}\); let \(D_3\) be the subgraph of D with vertex set \(\{x,y,u_3,u_4\}\) and arc set \(\{yu_3, u_3u_4, u_4y, xu_3, u_3x\}\); let \(D_4\) be the subgraph of D with vertex set \(\{x, y, u_1, u_4\}\) and arc set \(\{xu_4, u_4x, yu_1, u_1y, u_1u_4, u_4u_1\}\); for each vertex \(u\in V(D)\setminus \{x, y, u_1, u_2, u_3, u_4\}\), let \(D_u\) be a subgraph of D with vertex set \(\{u, x ,y\}\) and arc set \(\{ux, xu, uy, yu\}\). It is not hard to check that these \(n-2\) S-strong subgraphs are arc-disjoint.

Case 2. x and y belong to the same cycle, say \(u_1u_2 \cdots u_tu_1\), of \(\overleftrightarrow {K}_n[M]\). We just consider the case that the length of this cycle is at least three, since the argument for the remaining case is simpler.

Subcase 2.1. x and y are adjacent in the cycle. Without loss of generality, let \(x=u_1, y=u_2\). Let \(D_1\) be the subgraph of D with vertex set \(\{x, y, u_3\}\) and arc set \(\{yx, xu_3, u_3y\}\); let \(D_2\) be the subgraph of D with vertex set \(\{x, y, u_3, u_t\}\) and arc set \(\{u_3x, xu_t, u_tu_3, u_ty, yu_t\}\); for each vertex \(u\in V(D)\setminus \{x, y, u_3, u_t\}\), let \(D_u\) be a subgraph of D with vertex set \(\{u, x ,y\}\) and arc set \(\{ux, xu, uy, yu\}\). It is not hard to check that these \(n-2\) S-strong subgraphs are arc-disjoint.

Subcase 2.2. x and y are nonadjacent in the cycle. Without loss of generality, let \(x=u_1, y=u_3\). Let \(D_1\) be the 2-cycle xyx; let \(D_2\) be the subgraph of D with vertex set \(\{x, y, u_2, u_t\}\) and arc set \(\{yu_2, u_2x, xu_t, u_ty\}\); for each vertex \(u\in V(D)\setminus \{x, y, u_2, u_t\}\), let \(D_u\) be a subgraph of D with vertex set \(\{u, x ,y\}\) and arc set \(\{ux, xu, uy, yu\}\). It is not hard to check that these \(n-2\) S-strong subgraphs are arc-disjoint.

Part (b). Let H be minimally strong subgraph \((2,n-2)\)-arc-connected. By Lemma 4.4, we have that \(H\not \cong \overleftrightarrow {K}_n\), that is, H can be obtained from a complete digraph \(\overleftrightarrow {K}_n\) by deleting a nonempty arc set M. To end our argument, we need the following claim. Let us start from a simple yet useful observation, which follows by Inequality (3)\(\square\)

Proposition 5.10

No pair of arcs in M has a common head or tail.

Thus, \(\overleftrightarrow {K}_n[M]\) must be a union of vertex-disjoint cycles or paths, otherwise, there are two arcs of M such that they have a common head or tail, a contradiction with Proposition 5.10.

Claim 1

\(\overleftrightarrow {K}_n[M]\) does not contain a path of order at least two.

Proof of Claim 1

Let \(M'\supseteq M\) be a set of arcs obtained from M by adding some arcs from \(\overleftrightarrow {K}_n\) such that the digraph \(\overleftrightarrow {K}_n[M']\) contains no path of order at least two. Note that \(\overleftrightarrow {K}_n[M']\) is a supergraph of \(\overleftrightarrow {K}_n[M]\) and is a union of vertex-disjoint cycles which cover all but at most one vertex of \(\overleftrightarrow {K}_n\). By Part (a), we have that \(\lambda _2(\overleftrightarrow {K}_n[M'])=n-2\), so \(\overleftrightarrow {K}_n[M]\) is not minimally strong subgraph \((2,n-2)\)-arc-connected, a contradiction.

It follows from Claim 1 and its proof that \(\overleftrightarrow {K}_n[M]\) must be a union of vertex-disjoint cycles which cover all but at most one vertex of \(\overleftrightarrow {K}_n\), which completes the proof of Part (b).

6 Discussion

Corollaries 3.5 and 3.7 shed some light on the complexity of deciding, for fixed \(k,\ell \ge 2\), whether \(\lambda _k(D)\ge \ell\) for semicomplete and symmetric digraphs D. However, it is unclear what is the complexity above for every fixed \(k,\ell \ge 2\). If Conjecture 1 is correct, then the \(\lambda _k(D)\ge \ell\) problem can be solved in polynomial time for semicomplete digraphs. However, Conjecture 1 seems to be very difficult. It was proved in [13] that for fixed \(k, \ell \ge 2\) the problem of deciding whether \(\kappa _k(D)\ge \ell\) is polynomial-time solvable for both semicomplete and symmetric digraphs, but it appears that the approaches to prove the two results cannot be used for \(\lambda _k(D)\). Some well-known results such as the fact that the hamiltonicity problem is NP-complete for undirected 3-regular graphs, indicate that the \(\lambda _k(D)\ge \ell\) problem for symmetric digraphs may be NP-complete, too.

One of the most interesting results of this paper is the characterization of minimally strong subgraph \((2,n-2)\)-connected digraphs. As a simple consequence of the characterization, we can determine the values of \(f(n,2,n-2)\) and \(F(n,2,n-2)\). It would be interesting to determine \(f(n,k,n-2)\) and \(F(n,k,n-2)\) for every value of \(k\ge 3\). (Obtaining characterizations of all \((k,n-2)\)-connected digraphs for \(k\ge 3\) seems a very difficult problem.) It would also be interesting to find a sharp upper bound for \(F(n,k,\ell )\) for all \(k\ge 2\) and \(\ell \ge 2\).