Extremal Results for Directed Tree Connectivity

Abstract

For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\ge 2$, an (S, r)-tree is an out-tree T rooted at r with $S\subseteq V(T)$. Two (S, r)-trees $T_1$ and $T_2$ are said to be arc-disjoint if $A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint (S, r)-trees $T_1$ and $T_2$ are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let $\kappa _{S,r}(D)$ and $\lambda _{S,r}(D)$ be the maximum number of internally disjoint and arc-disjoint (S, r)-trees in D, respectively. The generalized k-vertex-strong connectivity of D is defined as

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

Similarly, the generalized k-arc-strong connectivity of D is defined as

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

The generalized k-vertex-strong connectivity and generalized k-arc-strong connectivity are also called directed tree connectivity which could be seen as a generalization of classical connectivity of digraphs and a natural extension of undirected tree connectivity. A digraph $D=(V(D), A(D))$ is called minimally generalized $(k, \ell )$-vertex (respectively, arc)-strongly connected if $\kappa _k(D)\ge \ell$ (respectively, $\lambda _k(D)\ge \ell$) but for any arc $e\in A(D)$, $\kappa _k(D-e)\le \ell -1$ (respectively, $\lambda _k(D-e)\le \ell -1$). In this paper, we study the minimally generalized $(k, \ell )$-vertex (respectively, arc)-strongly connected digraphs. We compute the minimum and maximum sizes of these digraphs and give characterizations of such digraphs for some pairs of k and $\ell$.

1 Introduction

We refer the readers to [1] for graph theoretical notation and terminology not given here. Note that all digraphs considered in this paper have no parallel arcs or loops.

An out-tree (respectively, in-tree) is an oriented tree in which every vertex except one, called the root, has in-degree (respectively, out-degree) one. An out-branching (respectively, in-branching) of D is a spanning out-tree (respectively, in-tree) in D. For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\ge 2$, a directed (S, r)-Steiner tree or, simply, an (S, r)-tree is an out-tree T rooted at r with $S\subseteq V(T)$ [2]. Two (S, r)-trees $T_1$ and $T_2$ are said to be arc-disjoint if $A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint (S, r)-trees $T_1$ and $T_2$ are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$.

Sun and Yeo [12] introduced the concept of directed tree connectivity which is related to directed Steiner tree packing problems [2, 12] and extends the concept of tree connectivity of undirected graphs (see, e.g., [3, 6, 7, 13]) to digraphs (Another extension of undirected tree connectivity, the strong subgraph connectivity of digraphs, was discussed in [10, 11]). Let $\kappa _{S,r}(D)$ and $\lambda _{S,r}(D)$ be the maximum number of internally disjoint and arc-disjoint (S, r)-trees in D, respectively. The generalized k-vertex-strong connectivity of D is defined as

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

Similarly, the generalized k-arc-strong connectivity of D is defined as

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

By definition, $\kappa _2(D)=\kappa (D)$ and $\lambda _2(D)=\lambda (D)$. Therefore, these two parameters could be seen as generalizations of vertex-strong connectivity and arc-strong connectivity of a digraph, respectively. The generalized k-vertex-strong connectivity and generalized k-arc-strong connectivity are also called directed tree connectivity. The complexity for $\kappa _{S,r}(D)$ and $\lambda _{S,r}(D)$ on general digraphs [2, 12], Eulerian digraphs [12] and symmetric digraphs [12] have been completely determined. Furthermore, in [12], some sharp bounds and precise values for $\kappa _k(D)$ and $\lambda _k(D)$ were obtained.

Now we introduce more concepts and parameters related to directed tree connectivity. A digraph $D=(V(D), A(D))$ is called minimally generalized $(k, \ell )$-vertex (respectively, arc)-strongly connected if $\kappa _k(D)\ge \ell$ (respectively, $\lambda _k(D)\ge \ell$) but for any arc $e\in A(D)$, $\kappa _k(D-e)\le \ell -1$ (respectively, $\lambda _k(D-e)\le \ell -1$). By the definition of $\kappa _k(D)$ (respectively, $\lambda _k(D)$) and Theorem 2.3, we clearly have $2\le k\le n, 1\le \ell \le n-1$.

Let ${\mathfrak {F}}(n,k,\ell )$ be the set of all minimally generalized $(k, \ell )$-vertex-strongly 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

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

Similarly, let ${\mathfrak {G}}(n,k,\ell )$ be the set of all minimally generalized $(k, \ell )$-arc-strongly connected digraphs with order n. We define

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

and

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

We further define

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

and

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

By definition, we directly have $f(n,k,\ell )\le F(n,k,\ell )$ and $g(n,k,\ell )\le G(n,k,\ell )$.

In this paper, we will study the minimally generalized $(k, \ell )$-vertex-strongly connected digraphs and minimally generalized $(k, \ell )$-arc-strongly connected digraphs. We will first give characterizations of such digraphs for some pairs of k and $\ell$ (Theorem 2.8), and then obtain exact values or sharp bounds for the functions $f(n,k,\ell )$, $F(n,k,\ell )$, $g(n,k,\ell )$ and $G(n,k,\ell )$ (Theorem 3.4). Some open problems will also be posed.

Additional Terminology and Notation For a digraph D, its reverse $D^{\mathrm{rev}}$ is a digraph with the same vertex set 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, $D=\overleftrightarrow {G}.$ A digraph D is minimally strong if D is strong but $D-e$ is not for every arc e of D.

2 Characterizations

The following proposition can be verified using definitions of $\kappa _k(D)$ and $\lambda _{k}(D)$.

Proposition 2.1

[12] Let D be a digraph of order n, and let $2\le k\le n$ 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)

$$\begin{aligned}&\kappa _k(D')\le \kappa _k(D), \lambda _k(D')\le \lambda _k(D) \text{ where } D' \text{ is } \text{ a } \text{ spanning } \text{ subdigraph } \text{ of } \text{ D } \end{aligned}$$

(2)

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

(3)

$$\begin{aligned}&\text{ D } \text{ is } \text{ strong } \text{ if } \text{ and } \text{ only } \text{ if } \lambda _k(D)\ge 1\text{. } \end{aligned}$$

(4)

We will use the following Tillson’s decomposition theorem.

Theorem 2.2

(Tillson’s decomposition theorem) [14] The arcs of $\overleftrightarrow {K}_n$ can be decomposed into Hamiltonian cycles if and only if $n\ne 4,6$.

Sun and Yeo got the following sharp bounds for $\kappa _k(D)$ and $\lambda _k(D)$.

Theorem 2.3

[12] Let D be a strong digraph of order n, and let $2\le k\le n$ be an integer. Then,

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

(5)

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

(6)

Moreover, all bounds are sharp, and the upper bounds hold if and only if $D\cong \overleftrightarrow {K}_n$.

By Proposition 2.1(4) and the fact that $\kappa _k(D)\ge 1$ if and only if $\lambda _k(D)\ge 1$, the following result directly holds:

Proposition 2.4

A digraph D is strong if and only if $\kappa _k(D)\ge 1$ for $2\le k\le n$.

In the rest of this paper, we use ${\mathcal {D}}_{n, 1}$ (respectively, ${\mathcal {D}}_{n, 2})$ to denote the set of digraphs 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 (respectively, $n-1$) vertices of $\overleftrightarrow {K}_n$.

Lemma 2.5

If $D\in {\mathcal {D}}_{n, 1}\cup {\mathcal {D}}_{n, 2}$, then D contains $n-2$ arc-disjoint out-branchings rooted at any vertex r.

Proof

Let $V(D)=\{u_i\mid 1\le i\le n\}$. Firstly, we consider the case that $D\in {\mathcal {D}}_{n, 1}$ such that $\overleftrightarrow {K}_n[M]$ is a union of vertex-disjoint cycles $C_1, C_2, \ldots , C_p$ which cover all vertices of $\overleftrightarrow {K}_n$. Without loss of generality, let $u_1$ be the root and belong to the cycle $C_1:= u_1, u_2, \ldots , u_s, u_1$ $(s\ge 2)$. For each $3\le i\le s$ (if exists), let $T_{i-2}$ be an out-branching of D with arc set $\{u_1u_i, u_iu_2, u_2u_{i+1}, u_iv\mid v\in V(D){\setminus } \{u_1, u_2, u_i, u_{i+1}\}\}$ if $s\ge 4$; otherwise, we have $s=3$, and then let $T_1$ be an out-branching of D with arc set $\{u_1u_3, u_3v\mid v\in V(D){\setminus } \{u_1, u_3\}\}$. Now consider the vertex $u_i$ for $s+1\le i\le n$, without loss of generality, assume that $u_i$ belongs to the cycle $C_2:= u_i, u_{i+1}, \ldots , u_{i+t-1}, u_i$ $(t\ge 2)$, where we set $u_{i+t}=u_i$ in the cycle. Let $T_{i-2}$ be an out-branching of D with arc set $\{u_1u_i, u_iu_2, u_2u_{i+1}, u_iv\mid v\in V(D){\setminus } \{u_1, u_2, u_i, u_{i+1}\}\}$. Hence, we get $n-2$ out-branchings rooted at $u_1$ and it can be checked these out-branchings are pairwise arc-disjoint. For example, let $D\in {\mathcal {D}}_{5, 1}$ such that $\overleftrightarrow {K}_5[M]$ is a union of vertex-disjoint cycles $C_1, C_2$, where $C_1:= u_1, u_2, u_3, u_1$ and $C_2:= u_4, u_5, u_4$. Let $T_1, T_2$ and $T_3$ be out-branchings rooted at $u_1$ with arc sets $\{u_1u_3, u_3u_2, u_3u_4, u_3u_5\}$, $\{u_1u_4, u_4u_2, u_2u_5, u_4u_3\}$ and $\{u_1u_5, u_5u_2, u_2u_4, u_5u_3\}$, respectively. Observe that these out-branchings are pairwise arc-disjoint.

Secondly, let $D\in {\mathcal {D}}_{n, 2}$ such that $\overleftrightarrow {K}_n[M]$ is a union of vertex-disjoint cycles $C_1, C_2, \ldots , C_p$ which cover all but at most one vertex, say $u_n$, of $\overleftrightarrow {K}_n$. It suffices to consider the case that $u_n$ is the root since the argument for the remaining case is similar to the above paragraph. Without loss of generality, assume that the arc $u_1u_2$ belongs to one of the above cycles, say $C_1:= u_1, u_2, \ldots , u_s, u_1$ $(s\ge 2)$. Let $T_1$ be an out-branching with arc set $\{u_nu_1, u_nu_2, u_1v\mid v\in V(D){\setminus } \{u_1, u_2, u_n\}\}$. For each $3\le i\le s$ (if exists), let $T_{i-1}$ be the out-branching of D with arc set $\{u_nu_i, u_iu_2, u_2u_{i+1}, u_iv\mid v\in V(D){\setminus } \{u_2, u_i, u_{i+1}, u_n\}\}$ (note that if $i=s$, then we set $u_{i+1}=u_{s+1}=u_1$ in this branching). Now consider the vertex $u_i$ for $s+1\le i\le n$, without loss of generality, assume that $u_i$ belongs to the cycle $C_2:= u_i, u_{i+1}, \ldots , u_{i+t-1}, u_i$ $(t\ge 2)$. Let $T_{i-1}$ be an out-branching of D with arc set $\{u_nu_i, u_iu_2, u_2u_{i+1}, u_iv\mid v\in V(D){\setminus } \{u_2, u_i, u_{i+1}, u_n\}\}$. Hence, we get $n-2$ out-branchings rooted at $u_n$ and it can be checked that these out-branchings are pairwise arc-disjoint.

Lemma 2.6

Let $D\in {\mathcal {D}}_{n, 1}\cup {\mathcal {D}}_{n, 2}$. For any $S\subset V(D)$ with $|S|=n-1$, D contains $n-2$ internally disjoint (S, r)-trees, where $r\in S$.

Proof

Let $V(D)=\{u_i\mid 1\le i\le n\}$.

Case 1: $D\in {\mathcal {D}}_{n, 1}$ such that $\overleftrightarrow {K}_n[M]$ is a union of vertex-disjoint cycles $C_1, C_2, \ldots , C_p$ which cover all vertices of $\overleftrightarrow {K}_n$. Without loss of generality, assume that $S=V(D){\setminus } \{u_2\}$ and $C_1:= u_1, u_2, \ldots , u_s, u_1$ $(s\ge 2)$.

Subcase 1.1: $s\ge 3$. Let $D'=D[S]-\{u_1u_3\}$. Observe that $D'\in {\mathcal {D}}_{n-1, 1}$. By Lemma 2.5, $D'$ contains $n-3$ arc-disjoint out-branchings $T_i~(1\le i\le n-3)$ rooted at any vertex $r\in S$. If $u_1$ is the root, then let $T_{n-2}$ be a tree with arc set $\{u_1u_3, u_3u_2, u_2v\mid v\in V(D){\setminus }\{u_1, u_2, u_3\}\}$; if $u_i~(i\not = 1,2)$ is the root, then let $T_{n-2}$ be a tree with arc set $\{u_3u_2, u_2v\mid v\in V(D){\setminus }\{u_3, u_2\}\}$ (respectively, $\{u_iu_2, u_1u_3, u_2v\mid v\in V(D){\setminus }\{u_i, u_2, u_3\}\}$) when $i=3$ (respectively, $i>3$). It can be checked that the above $n-2$ trees are pairwise internally disjoint (S, r)-trees for any $r\in S$ in each case.

Subcase 1.2: $s=2$. Without loss of generality, assume that $u_3, u_4$ belong to the cycle $C_2:= u_3, u_4, \ldots , u_t, u_3$ $(t\ge 4)$. Let $T_1$ be a tree with arc set $\{u_1u_4, u_4u_3, u_3v\mid V(D){\setminus }\{u_1, u_2, u_3, u_4\}\}$; let $T_2$ be a tree with arc set $\{u_1u_3, u_3u_2, u_2v\mid V(D){\setminus }\{u_1, u_2, u_3\}\}$. For $5\le i\le n$, let $T_{i-2}$ be a tree with arc set $\{u_1u_i, u_iu_4, u_4u_{i+1}, u_iv\mid v\in V(D){\setminus } \{u_1, u_2, u_3, u_4, u_i\} \}$. Note that here $u_iu_{i+1}$ belongs to one of the cycles $C_2, C_3, \ldots , C_p$. It can be checked that the above $n-2$ trees are pairwise internally disjoint (S, r)-trees for any $r\in S$.

Case 2: $D\in {\mathcal {D}}_{n, 2}$ such that $\overleftrightarrow {K}_n[M]$ is a union of vertex-disjoint cycles $C_1, C_2, \ldots , C_p$ which cover all but at most one vertex, say $u_n$, of $\overleftrightarrow {K}_n$.

Subcase 2.1: $S=V(D){\setminus } \{u_n\}$. Observe that $D'=D[S]\in {\mathcal {D}}_{n-1, 1}$, by Lemma 2.5, $D'$ contains $n-3$ pairwise arc-disjoint out-branchings $T_i~(1\le i\le n-3)$ rooted at any vertex $r\in S$. Let $T_{n-2}$ be a tree with arc set $\{ru_n, u_nv\mid v\in V(D){\setminus } \{r, u_n\}\}$ where $r\in S$. It can be checked that the above $n-2$ trees are pairwise internally disjoint (S, r)-trees for any $r\in S$.

Subcase 2.2: $u_n\in S$. Without loss of generality, assume that $u_2 \not \in S$ (that is, $S=V(D){\setminus } \{u_2\}$) and $C_1:= u_1, u_2, \ldots , u_s, u_1$ $(s\ge 2)$. We first consider the case that $s\ge 3$. Let $D'=D[S]-\{u_1u_3\}$. Observe that $D'\in {\mathcal {D}}_{n-1, 2}$, by Lemma 2.5, $D'$ contains $n-3$ arc-disjoint out-branchings $T_i~(1\le i\le n-3)$ rooted at any vertex $r\in S$. With a similar argument to that of Subcase 1.1, we can get $n-2$ pairwise internally disjoint (S, r)-trees for any $r\in S$. It remains to consider the case that $s=2$. Let $D''=D[S]-\{u_1u_n, u_nu_1\}$. Observe that $D''\in {\mathcal {D}}_{n-1, 1}$, by Lemma 2.5, $D''$ contains $n-3$ pairwise arc-disjoint out-branchings $T_i~(1\le i\le n-3)$ rooted at any vertex $r\in S$. If $u_n$ is the root, then let $T_{n-2}$ be a tree with arc set $\{u_nu_1, u_nu_2, u_2v\mid v\in V(D){\setminus } \{u_1, u_2, u_n\}\}$; otherwise, let $T_{n-2}$ be a tree with arc set $\{u_iu_n, u_nv\mid v\in V(D){\setminus } \{u_1, u_i, u_n\}\}$, where $u_i~(i\not = 2, n)$ is the root. It can be checked that the above $n-2$ trees are pairwise internally disjoint (S, r)-trees for any $r\in S$.

Lemma 2.7

Let $D\in {\mathcal {D}}_{n, 1}\cup {\mathcal {D}}_{n, 2}$. For any $S\subset V(D)$ with $|S|=2$, D contains $n-2$ internally disjoint (S, r)-trees, where $r\in S$.

Proof

Let $S=\{r,v\}\subset V(D)$ and r be the root. If $rv\not \in M$, then let $T_1$ be the arc rv and $T_u$ be an out-tree with arc set $\{ru, uv\mid uv\not \in M\}$. Otherwise, let $T_u$ be an out-tree with arc set $\{ru, uv\mid u\in V(D){\setminus } \{r, v\}\}$. Observe that in both cases we get $n-2$ internally disjoint (S, r)-trees, as desired.

We will now characterize minimally generalized $(k, \ell )$-vertex (respectively, arc)-strongly connected digraphs for some pairs of k and $\ell$.

Theorem 2.8

The following assertions hold:

• (a)  For any integer $2\le k\le n$, a digraph D is minimally generalized (k, 1)-vertex (respectively, arc)-strongly connected if and only if D is minimally strong.

• (b)  For any integer $2\le k\le n$, a digraph D is minimally generalized $(k, n-1)$-vertex (respectively, arc)-strongly connected if and only if $D\cong \overleftrightarrow {K}_n$.

• (c)  For any integer $2\le k\le n$, a digraph D is minimally generalized $(k, n-2)$-arc-strongly connected if and only if $D\in {\mathcal {D}}_{n, 1}\cup {\mathcal {D}}_{n, 2}$; moreover, $ex'(n,k,n-2)={\mathcal {D}}_{n, 1}$ and $Ex'(n,k,n-2)={\mathcal {D}}_{n, 2}$.

• (d)  For $k\in \{2, n-1, n\}$, a digraph D is minimally generalized $(k, n-2)$-vertex-strongly connected if and only if $D\in {\mathcal {D}}_{n, 1}\cup {\mathcal {D}}_{n, 2}$; moreover, $ex(n, k, n-2)={\mathcal {D}}_{n, 1}$ and $Ex(n, k, n-2)={\mathcal {D}}_{n, 2}$.

Proof

By Theorem 2.3, Propositions 2.1(4) and 2.4, and the well-known fact that every strong digraph has an out-branching rooted at any vertex, we have (a) and (b).

In the following argument, we just prove (c) since the argument for (d) is similar and simpler (by Lemmas 2.5, 2.6 and 2.7). If $D\in {\mathcal {D}}_{n, 1}\cup {\mathcal {D}}_{n, 2}$, then by Proposition 2.1 and Lemma 2.5, we have $\lambda _k(D)\ge \lambda _n(D)\ge n-2$ for $2\le k\le n$. For any $e\in A(D)$, $\min \{\delta ^+(D-e), \delta ^-(D-e)\}=n-3$, so $\lambda _k(D-e)\le n-3$ by Proposition 2.1(3). Thus, D is minimally generalized $(k, n-2)$-arc-strongly connected.

Let D be minimally generalized $(k, n-2)$-arc-strongly connected. By Theorem 2.3, we have $D\not \cong \overleftrightarrow {K}_n$, 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 proposition. Let us start from a simple yet useful observation, which follows from Proposition 2.1.

Observation 2.9

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 Observation 2.9.

Proposition 2.10

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

Proof

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 Proposition 2.1 and Lemma 2.5, we have $\lambda _k(\overleftrightarrow {K}_n[M'])\ge n-2$, so $\overleftrightarrow {K}_n[M]$ is not minimally generalized $(k, n-2)$-arc-strongly connected, a contradiction.

It follows from Proposition 2.10 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.

3 The Functions $f(n,k,\ell )$, $g(n,k,\ell )$, $F(n,k,\ell )$ and $G(n,k,\ell )$

By definition, we can get the following proposition.

Proposition 3.1

The following assertions hold:

• (a)  A digraph D is minimally generalized $(k, \ell )$-vertex-strongly connected if and only if $\kappa _k(D)= \ell$ and $\kappa _k(D-e)= \ell -1$ for any arc $e\in A(D)$.

• (b)  A digraph D is minimally generalized $(k, \ell )$-arc-strongly connected if and only if $\lambda _k(D)= \ell$ and $\lambda _k(D-e)= \ell -1$ for any arc $e\in A(D)$.

Proof

Part (a). The direction “if” is clear by definition, it suffices to prove the direction “only if.” Let D be a minimally generalized $(k, \ell )$-vertex-strongly 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, r)-trees, where $r\in S$ is a root. As e must belong to exactly one element of ${\mathcal {D}}$, we are done. The argument for

Part (b) is similar.

For $2\le k\le n$ and $1\le \ell \le n-1$, let $s(n, k, \ell )$ ($t (n, k, \ell )$, respectively) be the minimum size of a strong digraph D with order n and $\kappa _k(D)=\ell$ ($\lambda _k(D)=\ell$, respectively).

Lemma 3.2

For any $2\le k\le n$ and $1\le \ell \le n-1$,

$$\begin{aligned} s(n, k, \ell )=f(n, k, \ell ), t(n, k, \ell )=g(n, k, \ell ). \end{aligned}$$

Proof

Let ${\mathfrak {D}}(n,k,\ell )$ be the set of all strong digraphs D with order n and $\kappa _k(D)=\ell$. Let

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

Recall that ${\mathfrak {F}}(n,k,\ell )$ is the set of all minimally generalized $(k, \ell )$-vertex-strongly connected digraphs with order n, and

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

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

By Proposition 3.1, ${\mathfrak {F}}(n,k,\ell )$ is the set of all strong digraphs D with order n such that $\kappa _k(D)= \ell$ and $\kappa _k(D-e)= \ell -1$ for any arc $e\in A(D)$. Hence, ${\mathfrak {F}}(n,k,\ell ) \subseteq {\mathfrak {D}}(n,k,\ell )$ and so $s(n, k, \ell )\le f(n,k,\ell )$.

Let $D\in {\mathfrak {D}}'(n,k,\ell )$. Then, $\kappa _k(D)=\ell$ and $\kappa _k(D-e)\le \ell -1$ for any arc $e\in A(D)$, that is, $D\in {\mathfrak {F}}(n,k,\ell )$. This means that ${\mathfrak {D}}'(n,k,\ell )\subseteq {\mathfrak {F}}(n,k,\ell )$ and so $s(n, k, \ell )\ge f(n,k,\ell )$. Hence, $s(n, k, \ell )=f(n, k, \ell )$. The equality $t(n, k, \ell )=g(n, k, \ell )$ can be proved similarly.

We still need the following result, see, e.g., Corollary 5.3.6 of [1].

Theorem 3.3

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.

We will now prove our second main result:

Theorem 3.4

The following assertions hold:

• (a)

  • $f(n,k,\ell )\ge n\ell$ for any two integers $2\le k\le n$ and $1\le \ell \le n-1$; moreover, the bound can be attained if $\ell =1$, or, $2\le \ell \le n-1$ and $k=n$.

  • $f(n,k,\ell )\le a(\ell )+2\ell (n-\ell )$ for the case $n\ge k+\ell$, where

    $$\begin{aligned} a(\ell )=\left\{ \begin{array}{ll} 2{\ell \atopwithdelims ()2}, &{}\quad \ell \ge 2\text{; }\\ 0, &{}\quad \ell =1\text{. } \end{array} \right. \end{aligned}$$

    Especially, $f(n,k,\ell )\le 2\ell (n-\ell )$ when $n\ge k+2\ell$. Moreover, both bounds are sharp.

• (b)  $g(n,k,\ell )= n\ell$ for any two integers $2\le k\le n$ and $1\le \ell \le n-1$.

• (c)

  • $F(n,k,\ell )\ge n\ell$ for any two integers $2\le k\le n$ and $1\le \ell \le n-1$; especially, $F(n,k,\ell )\ge 2\ell (n-\ell )$ when $n\ge k+2\ell$. Moreover, both bounds are sharp.

  • $F(n, k, \ell )=(\ell +1)(n-1)$ if $k\in \{2, n-1, n\}$ and $\ell =n-2$, or, $2\le k\le n$ and $\ell \in \{1, n-1\}$.

• (d)

  • $G(n,k,\ell )\ge n\ell$ for any two integers $2\le k\le n$ and $1\le \ell \le n-1$; especially, $G(n,k,\ell )\ge 2\ell (n-\ell )$ when $n\ge k+2\ell$. Moreover, both bounds are sharp.

  • $G(n, k, \ell )=(\ell +1)(n-1)$ if $2\le k\le n$ and $\ell \in \{1, n-2, n-1\}$.

Proof

Part (a). We first prove the lower bound. By Proposition 2.1(3), for any digraph 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 $f(n,k,\ell )\ge n\ell .$

We now prove the sharpness of the lower bound. For the case that $\ell =1$, let D be a dicycle $\overrightarrow{C_n}$. Clearly, D is minimally generalized (k, 1)-vertex-strongly 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 set $A(D_{\ell })=\bigcup _{1\le i\le \ell }{A(H_i)}$. Clearly, we have $\kappa _n(D_{\ell })\ge \ell$ for $2\le \ell \le n-1$. Furthermore, by Proposition 2.1(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 Proposition 2.1(3). Thus, $D_{\ell }$ is minimally generalized $(n, \ell )$-vertex-strongly 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 $2\le \ell \le n-1, n\not \in \{4,6\}$. For the case that $n=6$, let $C_1$ be the cycle $u_1, u_2, u_3, u_4, u_5, u_6, u_1$, $C_2={C_1}^{\mathrm{rev}}$, $C_3$ be the cycle $u_1, u_3, u_5, u_2, u_6, u_4, u_1$, $C_4={C_3}^{\mathrm{rev}}$. For $1\le \ell \le 4$, let $D_{\ell }$ be the digraph with vertex set $\{u_i\mid 1\le i\le 6\}$ and arc set $\bigcup _{1\le i\le \ell }{A(C_i)}$, let $D_{5}=\overleftrightarrow {K}_6$. It can be checked that $D_{\ell }$ is minimally generalized $(n, \ell )$-vertex-strongly 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 $2\le \ell \le n-1, n=6$. For the case that $n=4$, let $C_1$ be the cycle $u_1, u_2, u_3, u_4, u_1$, $C_2={C_1}^{\mathrm{rev}}$. For $1\le \ell \le 2$, let $D_{\ell }$ be the digraph with vertex set $\{u_i\mid 1\le i\le 4\}$ and arc set $\bigcup _{1\le i\le \ell }{A(C_i)}$, let $D_{3}=\overleftrightarrow {K}_4$. With a similar but simpler argument, we can deduce that $f(n,n,\ell )= n\ell$ for the case that $2\le \ell \le n-1, n=4$. Hence, $f(n,k,\ell )= n\ell$ when $2\le \ell \le n-1$ and $k=n$.

To prove the upper bound, we need to construct the following two digraphs $H_1$ and $H_2$:

Let $H_1$ be a symmetric digraph whose underlying undirected graph is $K_{\ell }\bigvee {\overline{K}}_{n-\ell }$ ($n\ge k+\ell$), i.e., the graph obtained from disjoint graphs $K_{\ell }$ and ${\overline{K}}_{n-\ell }$ by adding all edges between the vertices in $K_{\ell }$ and ${\overline{K}}_{n-\ell }$. Let $V(H_1)=W_1\cup U_1$ such that $W_1=V(K_{\ell })=\{w_i\mid 1\le i\le \ell \}$ and $U_1=V({\overline{K}}_{n-\ell })=\{u_j\mid 1\le j\le n-\ell \}$.

Let $H_2=\overleftrightarrow {K}_{\ell , n-\ell }$ ($n\ge k+2\ell$), the complete bipartite digraphs with two parts $W_2$ and $U_2$, where $W_2=\{w_i\mid 1\le i\le \ell \}$ and $U_2=\{u_j\mid 1\le j\le n-\ell \}$.

Proposition 3.5

 The following assertions hold:

• (i)  For $n\ge k+\ell$, $\kappa _k(H_1)=\ell$.

• (ii)  For $n\ge k+2\ell$, $H_2$ is minimally generalized $(k, \ell )$-vertex(arc)-strongly connected.

Proof

For (i), let $S_1$ be any k-subset of vertices of $V(H_1)$ such that $|S_1\cap W_1|=s$ ($s\le \ell$) and $|S_1\cap U_1|=k-s$ ($k-s\le n-\ell$ since $n\ge k+\ell$). Without loss of generality, let $w_i\in S_1$ for $1\le i\le s$ and $u_j\in S_1$ for $1\le j\le k-s$. For $1\le i\le s$, let $T'_i$ be a tree with edge set

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

For $s+1\le j\le \ell$, let $T'_j$ be a tree with edge set

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

This is reasonable since $(k-s)+s=k\le n-\ell$. It is not hard to obtain an $(S_1, r)$-tree $D'_i$ from $T'_i$ by adding appropriate directions to edges of $T'_i$ for any $r\in S_1$. Observe that $\{D'_i\mid 1\le i\le s\}\cup \{D'_j\mid s+1\le j\le \ell \}$ is a set of $\ell$ internally disjoint $(S_1, r)$-trees, so $\kappa _{S_1, r}(H_1)\ge \ell$, and then $\kappa _k(H_1)\ge \ell$. Combining this with the bound that $\kappa _k(H_1)\le \min \{\delta ^+(H_1), \delta ^-(H_1)\}=\ell$, we have $\kappa _k(H_1)=\ell$.

For (ii), Let $S_2$ be any k-subset of vertices of $V(H_2)$ such that $|S_2\cap W_2|=s$ ($s\le \ell$) and $|S_2\cap U_2|=k-s$ ($k-s< n-\ell$ since $n\ge k+2\ell$). Without loss of generality, let $w_i\in S_2$ for $1\le i\le s$ and $u_j\in S_2$ for $1\le j\le k-s$. For $1\le i\le s$, let $T''_i$ be a tree with edge set

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

For $s+1\le j\le \ell$, let $T''_j$ be a tree with edge set

$$\begin{aligned} \{w_ju_1, w_ju_2, \dots , w_ju_{k-s}, w_ju_{k+j-s}, w_ju_1, w_ju_2, \dots , w_ju_{k-s}\}. \end{aligned}$$

This is reasonable since $(k-s)+s+(\ell -s)=k+\ell -s\le n-\ell$. It is not hard to obtain an $(S_2, r)$-tree $D''_i$ from $T''_i$ by adding appropriate directions to edges of $T''_i$ for any $r\in S_2$. Observe that $\{D''_i\mid 1\le i\le s\}\cup \{D''_j\mid s+1\le j\le \ell \}$ is a set of $\ell$ internally disjoint $(S_2, r)$-trees, so $\kappa _{S_2, r}(H_2)\ge \ell$, and then $\kappa _k(H_2)\ge \ell$. Combining this with the bound that $\kappa _k(H_2)\le \min \{\delta ^+(H_2), \delta ^-(H_2)\}=\ell$, we have $\kappa _k(H_2)=\ell$. Observe that $\kappa _k(H_2-e)\le \min \{\delta ^+(H_2-e), \delta ^-(H_2-e)\}=\ell -1$. Hence, $H_2$ is minimally generalized $(k, \ell )$-vertex-strongly connected. Since the above $\ell$ internally disjoint trees are also arc-disjoint, it can be similarly proved that $H_2$ is minimally generalized $(k, \ell )$-arc-strongly connected.

Clearly, we have $|A(H_1)|=a(\ell )+2\ell (n-\ell )$ and $|A(H_2)|=2\ell (n-\ell )$. So $f(n, k, \ell )= s(n, k, \ell )\le a(\ell )+2\ell (n-\ell )$ when $n\ge k+\ell$ by Lemma 3.2 and Proposition 3.5(i); especially, $f(n, k, \ell )\le 2\ell (n-\ell )$ when $n\ge k+2\ell$ by Proposition 3.5(ii). By Theorem 2.8(b), when $\ell =n-1$, we have $f(n, k, \ell )= n(n-1)= a(\ell )+2\ell (n-\ell )$. Recall that when $\ell =1$, we have $f(n, k, \ell )= 2(n-1)$. Therefore, the above two bounds are sharp.

Part (b). By Proposition 2.1(3), for any digraph D and $k \ge 2$ we have $\lambda _k(D) \le \delta ^+(D)$ and $\lambda _k(D) \le \delta ^-(D)$. Hence, for each D with $\lambda _k(D)=\ell$, we have that $\delta ^+(D), \delta ^-(D)\ge \ell$, so $|A(D)|\ge n\ell$ and then $g(n,k,\ell )\ge n\ell .$ Now consider the graph $D_{\ell }$ in

Part (a). By Proposition 2.1(1), we have $\lambda _k(D_{\ell })\ge \lambda _n(D_{\ell })= \kappa _n(D_{\ell }) \ge \ell$ for $2\le k\le n, 1\le \ell \le n-1$. Furthermore, by Proposition 2.1(3), we have $\lambda _k(D_{\ell })\le \ell$ since the in-degree and out-degree of each vertex in $D_{\ell }$ are both $\ell$. Hence, $\lambda _k(D_{\ell })= \ell$ for $2\le k\le n, 1\le \ell \le n-1$. For any $e\in A(D_{\ell })$, we have $\delta ^+(D_{\ell }-e)=\delta ^-(D_{\ell }-e)=\ell -1$, so $\lambda _k(D_{\ell }-e)\le \ell -1$ by Proposition 2.1(3). Thus, $D_{\ell }$ is minimally generalized $(k, \ell )$-arc-strongly connected. As $|A(D_{\ell })|=n\ell$, we have $g(n,k,\ell )\le n\ell$. From the lower bound that $g(n,k,\ell )\ge n\ell$, we have $g(n,k,\ell )= n\ell$.

Parts (c) and (d). By Theorem 2.8, the following assertions hold: $F(n, k, \ell )=(\ell +1)(n-1)$ if $k\in \{2, n-1, n\}$ and $\ell =n-2$, or, $2\le k\le n$ and $\ell =n-1$; $G(n, k, \ell )=(\ell +1)(n-1)$ if $2\le k\le n$ and $\ell \in \{n-2, n-1\}$. Let D be a minimally generalized (k, 1)-vertex (respectively, arc)-strongly connected digraph. By Theorems 2.8 and 3.3, we have $|A(D)|\le 2(n-1)$ and the bound can be attained when D is a symmetric digraph whose underlying undirected graph is a tree. Furthermore, we have $F(n, k ,1)=G(n, k, 1)=2(n-1)$.

By the assertions (a) and (b), and the fact that $f(n,k,\ell )\le F(n,k,\ell )$, $g(n,k,\ell )\le G(n,k,\ell )$, we directly have $F(n,k,\ell )\ge n\ell$ and $G(n,k,\ell )\ge n\ell$. Moreover, both lower bounds can be attained when $\ell =n-1$. Furthermore, we have $F(n, k, \ell )\ge 2\ell (n-\ell )$ and $G(n, k, \ell )\ge 2\ell (n-\ell )$ when $n\ge k+2\ell$ by Proposition 3.5(ii), and these two bounds can be attained for the case that $\ell =1$.

4 Discussion

For $k\in \{2, n-1, n\}$, the minimally generalized $(k, n-2)$-vertex-strongly connected digraphs are characterized in Theorem 2.8. It is natural to extend this result to the case of a general k, like that of the minimally generalized $(k, n-2)$-arc-strongly connected digraphs in Theorem 2.8.

Problem 4.1

Characterize the minimally generalized $(k, n-2)$-vertex-strongly connected digraphs for $2\le k\le n$.

Recall that in the proof of Theorem 2.8, we use the monotone property of $\lambda _k$, that is, $\lambda _{k+1}(D)\le \lambda _{k}(D)$ for every $2\le k\le n-1$ (Proposition 2.1(1)). However, this property does not hold for the parameter $\kappa _k$ as shown in [12], so we need to find other approach to solve Problem 4.1.

In Theorem 3.4, we give sharp lower bounds for $F(n,k,\ell )$ and $G(n,k,\ell )$, but we still cannot give nice upper bounds for these two functions. So it would be interesting to study the following question.

Problem 4.2

Find sharp upper bounds for $F(n,k,\ell )$ and $G(n,k,\ell )$ for all $k\ge 2$ and $\ell \ge 2$.

Let $|D_{\ell }^+|$ (respectively, $|D_{\ell }^-|)$ denote the number of vertices of out-degree $\ell$ (respectively, in-degree $\ell$) of a digraph D. By Propositions 2.1(3) and 3.1, we have $\delta ^+(D), \delta ^-(D)\ge \ell$ for a minimally generalized $(k, \ell )$-vertex-strongly connected digraph D. It would be interesting to bound $|D_{\ell }^+|$ (respectively, $|D_{\ell }^-|)$. Note that similar questions for minimally (strongly) connected (di)graphs were discussed in the literature, see, e.g., [4, 5, 8, 9].

Problem 4.3

Does $|D_{\ell }^+|$ (respectively, $|D_{\ell }^-|$) $>0$ hold for every minimally generalized $(k, \ell )$-vertex-strongly connected digraph D?

Here is a stronger question.

Problem 4.4

Does $|D_{\ell }^+|$ (respectively, $|D_{\ell }^-|$) $\ge \ell +1$ hold for every minimally generalized $(k, \ell )$-vertex-strongly connected digraph D?

Similar to Problems 4.3 and 4.4, the following problems are also of interest.

Problem 4.5

Does $|D_{\ell }^+|$ (respectively, $|D_{\ell }^-|$) $>0$ hold for every minimally generalized $(k, \ell )$-arc-strongly connected digraph D?

Problem 4.6

Does $|D_{\ell }^+|$ (respectively, $|D_{\ell }^-|$) $\ge \ell +1$ hold for every minimally generalized $(k, \ell )$-arc-strongly connected digraph D?

A digraph D is called minimally $\ell$-vertex-strongly connected if D is $\ell$-vertex-strongly connected, but $D-e$ is not for any arc $e\in A(D)$. By definition, a 1-vertex-strongly connected digraph is also a minimally strongly connected digraph. Mader obtained the following result on minimally $\ell$-vertex-strongly connected digraphs:

Theorem 4.7

[9] For every minimally $\ell$-vertex-strongly connected digraph D, $|D_{\ell }^+|\,($respectively, $|D_{\ell }^-|)\,\ge \ell +1$ holds.

By Theorems 2.8 and 4.7, we have the following supports for the above four problems: Problems 4.5 and 4.6 are true for any pair of k and $\ell$ with $2\le k\le n$ and $\ell \in \{1, n-2, n-1\}$; Problems 4.3 and 4.4 are true for any pair of k and $\ell$ satisfying: $2\le k\le n$ and $\ell \in \{1, n-1\}$, or, $k\in \{2, n-1, n\}$ and $\ell =n-2$.