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Degree sequence for karc strongly connected multiple digraphs
Journal of Inequalities and Applications volume 2017, Article number: 266 (2017)
Abstract
Let D be a digraph on \(\{v_{1},\ldots, v_{n}\}\). Then the sequence \(\{ (d^{+}(v_{1}), d^{}(v_{1})), \ldots, (d^{+}(v_{n}), d^{}(v_{n}))\}\) is called the degree sequence of D. For any given sequence of pairs of integers \(\mathbf{d}=\{(d_{1}^{+}, d_{1}^{}), \ldots, (d_{n}^{+}, d_{n}^{})\}\), if there exists a karc strongly connected digraph D such that d is the degree sequence of D, then d is realizable and D is a realization of d. In this paper, characterizations for karcconnected realizable sequences and realizable sequences with arcconnectivity exactly k are given.
Introduction
Digraphs in this paper may have loops and parallel arcs. A digraph D is called a multiple digraph (or multidigraph for short) if it has no loops. Furthermore, if D has parallel arcs neither, then D is strict. We follow [1] for undefined terminologies and notation.
For a digraph D, as in [1], \(V(D)\) and \(A(D)\) denote the vertex set and the arc set of D, respectively; and \((u,v)\) represents an arc oriented from a vertex u to a vertex v. For any two disjoint vertex sets X and Y, let \(A(X,Y) =\{(u,v) \in A(D) x \in X, y \in Y\}\). For a subset \(X \subseteq V(D)\), define
We use \(D[X]\) to denote the subdigraph of D induced by X. If F is a subdigraph of D, then for notational convenience, we often use \(\partial_{D}^{+}(F), \partial_{D}^{}(F)\) for \(\partial_{D}^{+}(V(F)), \partial _{D}^{}(V(F))\), respectively.
For a vertex u of D, define the outdegree \(d_{D}^{+}(u)\) (indegree \(d_{D}^{}(u)\), respectively) of u to be \(\vert \partial_{D}^{+}(\{u\}) \vert \) (\(\vert \partial_{D}^{}(\{u\}) \vert \), respectively). Let \(V(D)=\{v_{1},\ldots,v_{n}\}\). The sequence of integer pairs \(\{(d_{D}^{+}(v_{1}),d_{D}^{}(v_{1})), (d_{D}^{+}(v_{2}),d_{D}^{}(v_{2})), \ldots ,(d_{D}^{+}(v_{n}),d_{D}^{}(v_{n}))\}\) is called a degree sequence of D. For a given sequence \(\mathbf {d}=\{(d_{1}^{+},d_{1}^{}),\ldots,(d_{n}^{+},d_{n}^{})\} \), to determine whether there is a digraph D such that D has degree sequence d is a very essential problem in graph theory. This problem is closely linked with the other branches of combinatorial analysis such as threshold logic, integer matrices, enumeration theory, etc. The problem also has a wide range of applications in communication networks, structural reliability, stereochemistry, etc.
For a digraph D, if for any ordered pair of vertices \((u,v)\), there is a directed path from u to v, then D is said to be strongly connected. Characterizations for a digraphic sequence and a multidigraphic sequence with realizations having prescribed strong arcconnectivity have been studied, see Frank [2, 3] and Hong et al. [4]. For more in the literature on degree sequences, see surveys [5] and [6].
A sequence of integer pairs \(\mathbf {d}=\{(d_{1}^{+},d_{1}^{}),\ldots,(d_{n}^{+},d_{n}^{})\}\) is digraphic (multidigraphic, respectively) if there exists a digraph (a multidigraph, respectively) D with degree sequence d, where D is called a drealization. Let \(\langle\mathbf{d}\rangle\) be the set of all drealizations. Frank [2, 3] (see also Theorem 63.3 in [7]) showed that \(\langle\mathbf {d}\rangle\neq\emptyset\) if and only if \(\sum_{i=1}^{n} d_{i}^{+} =\sum_{i=1}^{n} d_{i}^{}\). If a multidigraphic realization of d is required, then Hong et al. [4] gave the following characterization.
Theorem 1.1
(Hong, Liu, Lai)
Let \(\mathbf {d}=\{(d_{1}^{+},d_{1}^{}),\ldots,(d_{n}^{+},d_{n}^{})\}\) be a sequence of nonnegative integer pairs. Then d is multidigraphic if and only if each of the following holds:

(i)
\(\sum_{i=1}^{n} d_{i}^{+} =\sum_{i=1}^{n} d_{i}^{}\);

(ii)
\(\textit{for }k=1,\ldots, n,d_{k}^{+} \leq\sum_{i\neq k} d_{i}^{}\).
Furthermore, for a strict digraph, there is a similar result. The following theorem, which can be found in [8–10] among others, is well known.
Theorem 1.2
(FulkersonRyser)
Let \(\mathbf {d}=\{(d_{1}^{+},d_{1}^{}),\ldots,(d_{n}^{+},d_{n}^{})\}\) be a sequence of nonnegative integer pairs with \(d_{1}^{+}\geq\cdots\geq d_{n}^{+}\). Then d is strict digraphic if and only if each of the following holds:

(i)
\(d_{i}^{+}\leq n1, d_{i}^{}\leq n1\textit{ for all }1\leq i\leq n\);

(ii)
\(\sum_{i=1}^{n} d_{i}^{+} =\sum_{i=1}^{n} d_{i}^{}\);

(iii)
\(\sum_{i=1}^{k} d_{i}^{+} \leq\sum_{i=1}^{k} \min\{k1,d_{i}^{}\} + \sum_{i=k+1}^{n} \min\{k,d_{i}^{}\}\textit{ for all }1\leq k\leq n\).
Let D be a digraph and k be an integer. If for any arc set S of D with \(\vert S \vert < k\), \(GS\) is still strongly connected, then D is said to be karc strongly connected (or karcconnected for short). Clearly, 1arc connected digraph is also a strongly connected digraph and vice versa. The arcconnectivity of D, denoted by \(\lambda(D)\), is the maximum integer k such that D is karcconnected. In [4], Hong et al. characterized the sequence of pairs of integers d so that there is a strongly connected digraph \(D\in\langle\mathbf{d}\rangle\). Also, they gave an example to point out that to characterize the case whether there is a karcconnected digraph in \(\langle\mathbf {d}\rangle\) may be very difficult. In this paper, we consider a multidigraphic version. We will give a characterization for karcconnected multidigraphs. Furthermore, we also give a characterization for multidigraphs with arcconnectivity exactly k.
In the next section, we will give some tools and methods used in this paper. In Section 3, we characterize the sequence of pairs of integers to have a karcconnected realization. In Section 4, we characterize the sequence of pairs of integers to have a realization that has arcconnectivity exactly k. In Section 5, we give a conclusion of this paper.
Methods and tools
In this section, we give a special notation used in this paper that is also the main tool. Let D be a digraph and \((u_{1},v_{1}), (u_{2}, v_{2})\) be two arcs of D. The 2switch of D is an operation to obtain a new digraph \(D'\) from \(D  \{(u_{1},v_{1}), (u_{2}, v_{2})\}\) by adding \(\{(u_{1},v_{2}), (u_{2},v_{1})\}\). The resulting digraph \(D'\) is often denoted by \(D \otimes\{(u_{1},v_{1}), (u_{2},v_{2})\}\). By this definition,
Thus, the degree sequence remains unchanged under 2switch operations. This operation will be the main tool in the arguments of this paper.
Note that in the operation of 2switch, the two arcs \((u_{1},v_{1})\), \((u_{2},v_{2})\) may have common ends. For example, if \(u_{1}=u_{2}\) or \(v_{1}=v_{2}\), then the resulting digraph is exactly the same as the original digraph. If \(v_{1}=u_{2}\) or \(v_{2}=u_{1}\), then the resulting digraph has loops. So, when this case occurs, we usually use another 2switch operation to remove the loops. For example, assume \((x,y), (y,z),(u,v)\in A(D)\) and \((D\otimes\{(x,y),(y,z)\})\otimes\{(y,y), (u,v)\}\) is just the digraph \(D\{(x,y), (y,z), (u,v)\}+\{(x,y),(u,y),(y,v)\}\). After these two 2switches, the resulting digraph still lies in \(\langle\mathbf {d}\rangle\). In this paper, we will use these operations to obtain a karcconnected digraph or a digraph with arcconnectivity exactly k from an arbitrary digraph in \(\langle\mathbf {d}\rangle\).
Let \(\mathbf {d}=\{(d_{1}^{+}, d_{1}^{}),\ldots, (d_{n}^{+}, d_{n}^{})\}\). By using the tools and the methods above, we obtain a sufficient and necessary condition of d to have a karcconnected realization (see Theorem 3.1). Furthermore, if we require the realization D to have arcconnected exactly k, then we get Theorem 4.1.
Degree sequence for karcconnected multidigraphs
In this section, we shall present a characterization for multidigraphic sequences with karcconnected realizations. We will give some notations used in this section fist.
Let D be a digraph. For a subset \(F \subseteq V(D)\), define \(\overline{F} = V(D) \setminus F\). A vertex set \(F \subseteq V(D)\) is called an outfragment (infragment, respectively) of D if \(\vert \partial_{D}^{+}(F) \vert = \lambda(D)\) (\(\vert \partial_{D}^{}(F) \vert = \lambda(D)\), respectively). Both outfragments and infragments are also called fragments of D. An outfragment (infragment, respectively) F is minimal if any proper subset of F is no longer an outfragment (infragment, respectively). Let \(\mathrm{fr}^{+}(D)\) be the number of outfragments of D and \(\mathrm{fr}^{}(D)\) be the number of infragments of D. As a vertex set F is an outfragment if and only if its complement F̅ is an infragment, \(\mathrm{fr}^{+}(D)=\mathrm{fr}^{}(D)\). Denote \(\mathrm{fr}(D)=\mathrm{fr}^{+}(D)=\mathrm{fr}^{}(D)\). It is easy to see that \(\mathrm{fr}(D)>0\) for any digraph D. This observation can be used to prove the following theorem.
Theorem 3.1
Let \(\mathbf {d}=\{(d_{1}^{+},d_{1}^{}),\ldots,(d_{n}^{+},d_{n}^{})\}\) be a sequence of integer pairs. Then d has a karcconnected realization if and only if each of the following holds:

(i)
\(\sum_{i=1}^{n} d_{i}^{+}=\sum_{i=1}^{n} d_{i}^{}\);

(ii)
for each \(1\leq j\leq n\), \(d_{j}^{+}, d_{j}^{}\geq k\);

(iii)
for each \(1\leq j\leq n\), \(d_{j}^{+}\leq\sum_{i\neq j}d_{j}^{}\).
Proof
If d has a karcconnected realization, then by Theorem 1.1, (i) and (iii) hold, and by the definition of karcconnectedness, (ii) holds. So, it suffices to prove the sufficiency. By (i), (iii) and by Theorem 1.1, \(\langle\mathbf {d}\rangle\neq\emptyset\). So we may pick a multidigraph \(D\in\langle\mathbf {d}\rangle\) such that
We shall show that D is karcconnected. Suppose this is not true. Then \(\lambda(D)< k\).
By the definition, \(\mathrm{fr}(D)>0\). Then there exist outfragments and infragments in D. Let \(F_{1}\) be a minimal outfragment of D and \(F_{2}\) be a minimal infragment contained in \(\overline{F_{1}}\). Then \(\vert \partial_{D}^{+}(F_{1}) \vert = \vert \partial_{D}^{}(F_{2}) \vert =\lambda(D)\). By (ii), \(d_{i}^{+},d_{i}^{}\geq k>\lambda(D)\), and so there must be \(u_{1}, v_{1} \in F_{1}\) and \(u_{2}, v_{2} \in F_{2}\) such that \((u_{1}, v_{1}) \in A(D[F_{1}])\) and \((u_{2},v_{2}) \in A(D[F_{2}])\). Let \(D'=D\otimes\{(u_{1},v_{1}),(u_{2},v_{2})\}\). By (1), \(D'\) is also a multidigraph in \(\langle\mathbf {d}\rangle\).
Claim 1. If F is an outfragment of \(D'\), then one of the following must hold:

(i)
\(\vert \partial_{D'}^{+}(F) \vert = \vert \partial_{D}^{+}(F) \vert \), or

(ii)
\(\partial_{D'}^{+}(F) = \partial_{D}^{+}(F) \cup\{(u_{2}, v_{1})\}\) and \(F \cap\{u_{1}, v_{1}, u_{2}, v_{2}\} = \{u_{2}, v_{2}\}\), or

(iii)
\(\partial_{D'}^{+}(F) = \partial_{D}^{+}(F) \cup\{(u_{1}, v_{2})\}\) and \(F \cap\{u_{1}, v_{1}, u_{2}, v_{2}\} = \{u_{1}, v_{1}\}\), or

(iv)
\(\partial_{D'}^{+}(F) = \partial_{D}^{+}(F)  \{(u_{2}, v_{2})\}\) and \(F \cap\{u_{1}, v_{1}, u_{2}, v_{2}\} = \{u_{2}, v_{1}\}\), or

(v)
\(\partial_{D'}^{+}(F) = \partial_{D}^{+}(F)  \{(u_{1}, v_{1})\}\) and \(F \cap\{u_{1}, v_{1}, u_{2}, v_{2}\} = \{u_{1}, v_{2}\}\).
By the definition of \(D'\), we have \(\vert \partial_{D'}^{+}(F) \vert = \vert \partial_{D}^{+}(F) \vert \) if \(\vert F \cap \{u_{1}, v_{1}, u_{2}, v_{2}\} \vert \neq2\). So, we may assume \(\vert F\cap\{u_{1},v_{1},u_{2},v_{2}\} \vert =2\). In fact, also by the definition of \(D'\), when \(F\cap\{u_{1},v_{1},u_{2},v_{2}\} \in\{ \{u_{1},u_{2}\}, \{v_{1}, v_{2}\}\}\), \(\vert \partial_{D'}^{+}(F) \vert = \vert \partial_{D}^{+}(F) \vert \) still holds. The other cases are illustrated as (ii)(v). Thus Claim 1 must hold.
Claim 2. \(\lambda(D')\geq\lambda(D)\).
By contradiction, we assume that \(D'\) has an outfragment F with \(\vert \partial_{D'}^{+}(F) \vert < \lambda(D)\). By Claim 1 and since \(\vert \partial_{D}^{+}(F) \vert \geq\lambda(D)\), we may assume that \(\{u_{1},v_{1},u_{2},v_{2}\}\cap F=\{u_{1},v_{2}\}\). Thus \(\vert \partial_{D}^{+}(F) \vert = \vert \partial_{D'}^{+}(F) \vert +1<\lambda(D)+1\). Since \(\vert \partial_{D}^{+}(F) \vert \geq\lambda(D)\), we have \(\vert \partial _{D}^{+}(F) \vert =\lambda(D)\), and so F is also an outfragment of D. Since \(u_{1}\in F_{1}\cap F\) and \(u_{2}\notin F_{1}\cup F\), by a submodular inequality, we have
which implies \(F_{1}\cap F\) is also an outfragment of D, which contradicts the minimality of \(F_{1}\). This completes the proof of Claim 2.
By choice (2)(a) of D and by Claim 2, \(\lambda (D')=\lambda(D)\). Then, by Claim 1, \(F_{1}\) is not an outfragment in \(D'\), and any outfragment F of \(D'\) is still an outfragment of D unless either \(\{u_{1},v_{1},u_{2},v_{2}\}\cap F=\{u_{1},v_{2}\}\) or \(\{u_{1},v_{1},u_{2},v_{2}\}\cap F=\{u_{1},v_{2}\}\). If there is such an F such that F is an outfragment in \(D'\) but not in D, then without loss of generality we may assume \(\{u_{1},v_{1},u_{2},v_{2}\}\cap F=\{u_{1},v_{2}\}\). Thus \(\vert \partial_{D}^{+}(F) \vert = \vert \partial_{D'}^{+}(F) \vert +1=\lambda(D)+1\). Moreover, by the minimality of \(F_{1}\) and \(F_{2}\), we have \(\vert \partial_{D}^{+}(F_{1}\cap F) \vert \geq\lambda(D)+1\). Thus, by a submodular inequality, we have
This implies \(\vert \partial_{D}^{}(\overline{F\cup F_{1}}) \vert = \vert \partial_{D}^{+}(F\cup F_{1}) \vert =\lambda(D)\). Then, by a submodular inequality again, we have
which implies \(\vert \partial_{D}^{}(\overline{F\cup F_{1}}\cap F_{2}) \vert =\lambda(D)\) contradicts to the minimality of \(F_{2}\). Hence, every outfragment of \(D'\) is also an outfragment of D. As \(F_{1}\) is an outfragment in D but not in \(D'\), \(\mathrm{fr}(D')<\mathrm{fr}(D)\), which contradicts choice (2)(b) of D. Therefore, D is karcconnected, and this completes the proof. □
By definition, the arcconnectivity of a digraph D cannot exceed \(\min\{d^{+}_{D}(v), d^{}_{D}(v): v \in V(D)\}\). A digraph D is maximally arcconnected if the arcconnectivity of D equals \(\min\{d^{+}_{D}(v), d^{}_{D}(v): v \in V(D)\}\). Applying Theorem 3.1 with \(k=\min\{d_{1}^{+}, \ldots, d_{n}^{+}, d_{1}^{}, \ldots, d_{n}^{}\}\), we have the following corollary.
Corollary 3.2
Let \(\mathbf {d}=\{(d_{1}^{+},d_{1}^{}),\ldots,(d_{n}^{+},d_{n}^{})\}\) be a multigraphical sequence. Then d is also a degree sequence of some maximally arcconnected multidigraph.
Degree sequence for multidigraphs with prescribed connectivity
In this section, we consider the degree sequence of multidigraphs with connectivity exactly k. Our method is to construct a new multidigraph in \(\langle\mathbf {d}\rangle\) from a karcconnected multidigraph by reducing the arcconnectivity step by step. Moreover, by Corollary 3.2, we may assume that \(k<\min_{1\leq i\leq n}\{d_{i}^{+}, d_{i}^{}\}\).
Theorem 4.1
Let \(n\geq6, k\geq0\) be two integers and \(\mathbf {d}=\{ (d_{1}^{+},d_{1}^{}),\ldots,(d_{n}^{+},d_{n}^{})\}\) be a sequence of pairs of integers. Denote \(\delta_{1}=\min_{1\leq i\leq n} \{d_{i}^{+}, d_{i}^{}\}\) and \(\delta_{2}=\min_{1\leq i< j\leq n}\) \(\{d_{i}^{+}+d_{j}^{+}, d_{i}^{}+d_{j}^{}\}\). Then d is a degree sequence of some multidigraph with connectivity exactly k if and only if each of the following hold.

(i)
\(\delta_{1}\geq k\);

(ii)
\(\sum_{i=1}^{n} d_{i}^{+} =\sum_{i=1}^{n} d_{i}^{}\);

(iii)
for \(j=1,\ldots, n\), \(d_{j}^{+} +\alpha\leq\sum_{i\neq j} d_{i}^{} +k\), where \(\alpha=\delta_{2}\) if \(k<\delta_{1}\) and \(\alpha=\delta_{1}\) if \(k=\delta_{1}\).
Proof
First, we consider the necessity. Assume d is the degree sequence of some multidigraph D with connectivity exactly k. By Theorem 1.2, (i) and (ii) hold. Suppose, to the contrary, that (iii) does not hold. Then there is a vertex \(v_{j}\) of D such that \(d^{+}(v_{j})=d_{j}^{+}\geq\sum_{i\neq j}d_{i}^{}+k\alpha+1\). It follows that \(d^{+}(v_{j})+d^{}(v_{j})\geq\sum_{i=1}^{n} d_{i}^{} +k+1\alpha = \vert A(D) \vert +k+1\alpha\). This implies that there are at most \(\alphak1\) arcs not incident with \(v_{j}\). On the other hand, as D has connectivity k, there exists \(X\subseteq V(D)\setminus\{v_{j}\}\) such that either \(d^{+}(X)=k\) or \(d^{}(X)=k\). Without loss of generality, we may assume the former. Then \(d^{+}(X)\geq\sum_{v_{i}\in X} d^{+}(v_{i})(\alphak1)\). If \(\vert X \vert \geq2\), then \(\sum_{v_{i}\in X}d^{+}(v_{i})\geq\delta_{2}\geq\alpha \), and thus \(d^{+}(X)\geq k+1\), a contradiction. So \(\vert X \vert =1\) and thus \(k=d^{+}(X)\geq \delta_{1}\), implying \(\alpha=k=d^{+}(X)=\sum_{v\in X}d^{+}(v)\). Then again \(d^{+}(X)\geq\sum_{v_{i}\in X}d^{+}(v_{i})(\alphak1)= k+1\), a contradiction. Hence (iii) holds.
Next, we consider the sufficiency. By Theorems 1.2 and 3.1, there is a karcconnected multidigraph \(D\in\langle\mathbf {d}\rangle\). If D has arcconnectivity k, then we are done. So we may assume that \(\lambda(D)>k\), then we will construct a multidigraph in \(\langle\mathbf {d}\rangle\) with arcconnectivity exactly k from D. First, we need some claims.
Note that
By a similarly analysis to Claim 1 in the proof of Theorem 3.1, it is easy to verify the following claim. In fact, in the tree operations in the following claim, \(d^{+}(X)\) and \(d^{}(X)\) decrease at most 1 for any \(\emptyset\neq X\subset V(D)\). The proof is easy and omitted here.
Claim 1. Each of the following holds.

(i)
For any vertex disjoint two arcs \((u,v),(x,y)\), let \(D\otimes\{(u,v),(x,y)\}\) have arcconnectivity at least \(\lambda (D)1\).

(ii)
For any \(x,y,z,w\) with \((x,y), (y,z), (z,w)\in A(D)\), \(D\{(x,y),(y,z), (z,w)\}+\{(x,z), (z,y), (y,w)\}\) has connectivity at least \(\lambda(D) 1\).

(iii)
For any \(u,v,x,y,z\) with \((u,v), (x,y), (y,z)\in A(D)\), \(D\{(u,v),(x,y),(y,z)\}+\{(u,y), (y,v), (x,z)\}\) has arcconnectivity at least \(\lambda(D)1\).
Denote by
By the definition, \(\lambda'(D)\geq\lambda(D)\) for any digraph D. Now, by Theorem 3.1, we may pick a multidigraph \(D\in \langle\mathbf {d}\rangle\) such that
Then we will construct a new digraph from D that meets our requirements.
By the choice of D, \(\lambda(D) \geq k\). If \(\lambda(D)=k\), then we are done and D is required. So we may assume that \(\lambda(D)\geq k+1\). By the definition, let \(X\subseteq V(D)\) so that \(d^{+}(X)=\lambda'(D)\).
Claim 2. For any two arcs \((x_{1},y_{1})\in\partial^{+}(X), (y_{2},x_{2})\in\partial^{}(X)\), either \(x_{1}=x_{2}\) or \(y_{1}=y_{2}\).
Suppose, to the contrary, that \(x_{1}\neq x_{2}\) and \(y_{1}\neq y_{2}\). Let \(D'=D\otimes\{(x_{1},y_{1}), (y_{2},x_{2})\}\) and then by Claim 1, \(D'\in \langle\mathbf {d}\rangle\) with arcconnectivity at least \(\lambda (D)1\geq k\) and \(\lambda'(D')\leq d^{+}(X)1=\lambda'(D)1\), a contradiction to choice (3) of D.
Claim 3. For any two arcs \((x_{1},y_{1}), (x_{2},y_{2})\in\partial ^{+}(X)\) (or \(\partial^{}(X)\)), either \(x_{1}=x_{2}\) or \(y_{1}=y_{2}\).
Suppose, to the contrary, that \(x_{1}\neq x_{2}\) and \(y_{1}\neq y_{2}\) and, without loss of generality, we may assume that \((x_{1},y_{1}), (x_{2},y_{2})\in \partial^{+}(X)\). As \(\lambda'(D)\geq k+1\geq1\), \(\partial^{}(X)\neq \emptyset\). Let \((y_{3},x_{3})\in\partial^{}(X)\). Then, by Claim 2, either \(x_{3}=x_{2}\), \(y_{3}=y_{1}\) or \(x_{3}=x_{1}\), \(y_{3}=y_{2}\). By symmetry, we may assume the former. Let \(D'=D\{(x_{1},y_{1}),(y_{1},x_{2}),(x_{2},y_{2})\}+\{(x_{1},x_{2}), (x_{2},y_{1}), (y_{1},y_{2})\}\). By Claim 1(ii), \(D'\in\langle\mathbf {d}\rangle\) and has arcconnectivity at least \(\lambda(D)1\geq k\). However, \(\lambda'(D')< \vert \partial^{+}(X) \vert =\lambda'(D)\), a contradiction to the choice of D. Claim 3 is proved.
By Claim 2 and Claim 3, it is easy to see that all arcs leaving from or interring to X are incident with a vertex, say x. We only consider the case \(x\in X\), and the other case that \(x\notin X\) can be dealt with similarly.
Claim 4. We may assume that \(X\setminus\{x\}\) is an independent set of D.
Suppose, to the contrary, that there is an edge \((x_{1},x_{2})\in A(D[X\setminus\{x\}])\), then pick \(y_{1},y_{2}\notin X\) such that \((y_{1},x), (x,y_{2})\in A(D)\). If \(y_{1}\neq y_{2}\), then let \(D'=D\{(x_{1},x_{2}), (y_{1},x), (x,y_{2})\}+\{(x_{1},x), (x,x_{2}), (y_{1},y_{2})\}\) and thus \(D'\in \langle\mathbf {d}\rangle\). By Claim 1(ii), \(\lambda(D')\geq\lambda (D)1\geq k\) and \(\lambda(D')< \lambda'(D)\), a contradiction to choice (3) of D. So \(y_{1}=y_{2}\). By the arbitrariness of \(y_{1},y_{2}\), there is \(y\notin X\) such that all arcs leaving from or interring to X are incident with y. Let \(Y=V(D)\setminus X\) and \(Y\setminus\{y\}\) is an independent set; otherwise, if there exists \((y_{1},y_{2})\in A(D[Y])\), then let \(D''=D\{(x_{1},x_{2}), (y_{1},y_{2}), (x,y), (y,x)\} +\{(x_{1},x), (x,x_{2}), (y_{1}, y), (y, y_{2})\}\), and it is easy to see that \(D''\) has arcconnectivity at least \(\lambda (D)1\geq k\) and \(\lambda'(D'')<\lambda'(D)\), a contradiction to choice (3) of D. So \(Y\setminus\{y\}\) is an independent set. Thus, we may rename \(Y, y\) as \(X, x\) and Claim 4 follows.
As \(\vert X \vert \geq3\), let \(x_{1},x_{2}\in X\setminus\{x\}\). \(m=\min\{ \vert A(DX) \vert , d^{+}(x_{1})+d^{+}(x_{2}), d^{}(x_{1})+d^{}(x_{2})\}\). We will consider a sequence of digraphs \(D_{0},D_{1},\ldots, D_{m}\), where \(D_{0}=D\), and for \(i=1,\ldots, m\), if \(D_{i1}\) is constructed, then let \(D_{i}=D_{i1} \{(x_{1},x),(x,x_{2}), (u,v)\}+\{(u,x), (x,v), (x_{1},x_{2})\}\) or \(D_{i}=D_{i1}\{(x_{2},x), (x,x_{1}), (u,v)\}+\{(u,x), (x,v), (x_{2},x_{1})\}\). By the choice of m, all \(D_{i}\)’s can be constructed although they may be not unique. It is easy to see that \(D_{i}\in\langle\mathbf {d}\rangle\).
If \(D_{m}\) has arcconnectivity at most k, then by Claim 3 there exists i such that \(D_{i}\) has arcconnectivity exactly k, and we are done. So we may assume that \(D_{m}\) is \((k+1)\)arcconnected. Then \(m= \vert A(DX) \vert <\delta_{x}\); otherwise, if \(m=d^{+}(x_{1})+d^{+}(x_{2})\), then \(\partial_{D_{m}}^{+}(\{x_{1},x_{2}\})=\emptyset\), a contradiction to the assumption that \(D_{m}\) is \((k+1)\)arcconnected. A similar contradiction is obtained when \(m=d^{}(x_{1})+d^{}(x_{2})\). Thus \(m= \vert A(DX) \vert \) and then \(V(D)\setminus\{x,x_{1},x_{2}\}\) is an independent set in \(D_{m}\).
If \(k=\delta_{1}\), then \(\alpha=\delta_{1}=k\) and by (iii), \(d_{j}^{+}\leq\sum_{i\neq j}d_{i}^{}\), and the result holds by Corollary 3.2. So we may assume that \(k<\delta_{1}\) and thus \(\alpha=\delta_{2}\). Let \(u,v\) be two vertices so that \(\delta_{2}=\min\{d^{+}(u)+d^{+}(v), d^{}(u)+d^{}(v)\}\). If \(x\in\{u,v\}\), then \(\delta_{2}< \min\{d^{+}(x), d^{}(x)\}\leq\min\{d^{}(x_{1})+d^{}(x_{2}), d^{+}(x_{1})+d^{+}(x_{2})\}\), a contradiction. So \(x\notin\{u,v\}\). Then continue to construct the sequence of digraphs \(D_{0},\ldots, D_{m}, D_{m+1},\ldots, D_{2m}\) such that for \(i=m+1,\ldots, 2m\), \(D_{i}\) is obtained from \(D_{i1}\) by replacing an arc between \(x_{1},x_{2}\) with a dipath of length 2 between \(x_{1},x_{2}\) and replacing a dipath of length 2 between u, with an arc between \(u,v\). Then, similarly to the above, we may assume that \(D_{2m}\in\langle\mathbf {d}\rangle\) is \((k+1)\)arcconnected and \(V(D)\setminus\{x,u,v\}\) is an independent set in \(D_{2m}\).
Moreover, by (ii) and (iii), for any j, \(d_{j}^{+} + d_{j}^{}+\alpha\leq \sum_{i=1}^{n} d_{i}^{}+k = \sum_{i=1}^{n} d_{i}^{+}+k\), and thus \(d_{j}^{}+\alpha\leq\sum_{i\neq j}d^{}_{i} +k\). So, by symmetry, we may assume that \(d^{+}(u)+d^{+}(v)\leq d^{}(u)+d^{}(v)\). Thus \(\delta_{2}=d^{+}(u)+d^{+}(v)\). It follows that
This implies that there is j such that \(d_{j}^{+}+\alpha\geq\sum_{i\neq j}d_{i}^{}+k+1\), a contradiction to (iii). The proof is completed. □
If Theorem 4.1(i), (ii) holds, then by Theorem 1.2 \(\langle\mathbf {d}\rangle\neq\emptyset\). Furthermore, if Theorem 4.1(iii) does not hold, then there are no digraphs in \(\langle\cdot\rangle\) that have arcconnectivity exactly k. In other words, all digraphs in \(\langle\mathbf {d}\rangle\) are \((k+1)\)arcconnected.
Corollary 4.2
Let \(\mathbf {d}=\{(d_{1}^{+}, d_{1}^{}),\ldots, (d_{n}^{+}, d_{n}^{})\}\) be a sequence of integer pairs. Denote \(\delta_{1}=\min_{1\leq i\leq n} \{ d_{i}^{+}, d_{i}^{}\}\) and \(\delta_{2}=\min_{1\leq i< j\leq n}\) \(\{d_{i}^{+}+d_{j}^{+}, d_{i}^{}+d_{j}^{}\}\). If each of the following holds, then any digraphs in \(\langle\mathbf {d}\rangle\) are karcconnected.

(i)
\(\delta_{1}\geq k\);

(ii)
\(\sum_{i=1}^{n} d_{i}^{+} =\sum_{i=1}^{n} d_{i}^{}\);

(iii)
there exists some j such that \(d_{j}^{+} +\alpha\geq\sum_{i\neq j} d_{i}^{} +k\), where \(\alpha=\delta_{2}\) if \(k<\delta_{1}\) and \(\alpha=\delta_{1}\) if \(k=\delta_{1}\).
Conclusions
In this paper, sufficient and necessary conditions for a sequences of pairs of integers have been studied. For a sequence \(\mathbf {d}=\{ (d_{1}^{+},d_{1}^{}), \ldots, (d_{n}^{+}, d_{n}^{})\}\) of pairs of integers, we give a sufficient and necessary condition of d to have a karcconnected realization by using some inequalities of these integers. As a consequence, we deduce a sufficient and necessary condition of d to have a maxarcconnected realization. Also, when \(n\geq6\), we give a sufficient and necessary condition of d to have a realization D that has arcconnectivity exactly k. These results extend a similar result from undirect graphs into directed graphs.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11326214) and the Natural Science Foundation of Fujian Province (2014J05004).
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Hong, Y., Liu, Q. Degree sequence for karc strongly connected multiple digraphs. J Inequal Appl 2017, 266 (2017). https://doi.org/10.1186/s1366001715443
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DOI: https://doi.org/10.1186/s1366001715443
Keywords
 degree sequence
 realization
 karc strongly connected