Abstract
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 strong 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 \(\kappa _S(D)\) (resp. \(\lambda _S(D)\)) be the maximum number of internally disjoint (resp. arc-disjoint) S-strong subgraphs in D. The strong subgraph k -connectivity is defined as
As a natural counterpart of the strong subgraph k-connectivity, we introduce the concept of strong subgraph k -arc-connectivity which is defined as
A digraph \(D=(V, A)\) is called minimally strong subgraph \((k,\ell )\)-(arc-)connected if \(\kappa _k(D)\ge \ell\) (resp. \(\lambda _k(D)\ge \ell\)) but for any arc \(e\in A\), \(\kappa _k(D-e)\le \ell -1\) (resp. \(\lambda _k(D-e)\le \ell -1\)). In this paper, we first give complexity results for \(\lambda _k(D)\), then obtain some sharp bounds for the parameters \(\kappa _k(D)\) and \(\lambda _k(D)\). Finally, minimally strong subgraph \((k,\ell )\)-connected digraphs and minimally strong subgraph \((k,\ell )\)-arc-connected digraphs are studied.
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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
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
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
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 xy, yx 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
For a spanning subgraph \(D'\) of D, we have
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].
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, x, y 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 \(D\in \mathcal{S}_{\lambda }\).
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
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 x, y 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
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:
-
(i)
For \(n\ge \kappa (D)+k,\) we have \(\kappa _k(D)\le \kappa (D)\);
-
(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 u, v 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
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
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
We can now compute the exact values of \(\lambda _k(\overleftrightarrow {K}_n)\).
Lemma 4.4
For \(2\le k\le n\), we have
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
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 k, D 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:
-
(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 \(\lambda (D)=\lambda (D^c)=0\).
-
(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
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:
-
(i)
A digraph D is minimally strong subgraph (k, 1)-connected if and only if D is a minimally strong digraph;
-
(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 k, D 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. \(\square\)
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.
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
and
We further define
and
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
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(n, k, 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 \(\ell \ge 2\).
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Acknowledgements
We are thankful to Anders Yeo for discussions related to the complexity of computing strong subgraph \((k,\ell )\)-arc-connectivity for semicomplete and symmetric digraphs
Funding
Yuefang Sun was supported by Zhejiang Provincial Natural Science Foundation (no. LY20A010013). Gregory Gutin was partially supported by Royal Society Wolfson Research Merit Award.
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Sun, Y., Gutin, G. Strong Subgraph Connectivity of Digraphs. Graphs and Combinatorics 37, 951–970 (2021). https://doi.org/10.1007/s00373-021-02294-w
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DOI: https://doi.org/10.1007/s00373-021-02294-w