1 Introduction
The generalized connectivity of a graph was introduced by Hager [14] in 1985 (). For a graph and a set of at least two vertices, an Steiner tree or, simply, an tree is a subgraph of which is a tree with . Two trees and are said to be edgedisjoint if . Two edgedisjoint trees and are said to be internally disjoint if . The generalized local connectivity is the maximum number of internally disjoint trees in . For an integer with , the generalized connectivity is defined as
Observe that . Li, Mao and Sun [18] introduced the following concept of generalized edgeconnectivity. The generalized local edgeconnectivity is the maximum number of edgedisjoint trees in . For an integer with , the generalized edgeconnectivity is defined as
Observe that . Generalized connectivity of graphs has become an established area in graph theory, see a recent monograph [17] by Li and Mao on generalized connectivity of undirected graphs.
To extend generalized connectivity to directed graphs, Sun, Gutin, Yeo and Zhang [23] observed that in the definition of , one can replace “an tree” by “a connected subgraph of containing .” Therefore, Sun et al. [23] defined strong subgraph connectivity by replacing “connected” with “strongly connected” (or, simply, “strong”) as follows. Let be a digraph of order , a subset of of size and . A subdigraph of is called an strong subgraph if is strong and . Two strong subgraphs and are said to be arcdisjoint if . Two arcdisjoint strong subgraphs and are said to be internally disjoint if . Let be the maximum number of internally disjoint strong subgraphs in . The strong subgraph connectivity of is defined as
By definition, if is not strong.
As a natural counterpart of the strong subgraph connectivity, Sun and Gutin [22] introduced the concept of strong subgraph arcconnectivity. Let be a digraph of order , a subset of and . Let be the maximum number of arcdisjoint strong digraphs in . The strong subgraph arcconnectivity of is defined as
By definition, if is not strong.
The strong subgraph (arc)connectivity is not only a natural extension of the concept of generalized (edge)connectivity, but also relates to important problems in graph theory. For , [23] and [22]. Hence, and could be seen as generalizations of connectivity and edgeconnectivity of undirected graphs, respectively. For , is the maximum number of arcdisjoint spanning strong subgraphs of . Moreover, since and are the number of internally disjoint and arcdisjoint strong subgraphs containing a given set , respectively, these parameters are relevant to the subdigraph packing problem, see [4, 5, 6, 7, 11].
Some basic results will be introduced in Section 2. In Section 3, we will sum up the results on algorithms and computational complexity for , , and . We will collect many upper and lower bounds for the parameters and in Section 4. Finally, in Section 5, results on minimally strong subgraph (arc)connected digraphs will be surveyed.
Additional Terminology and Notation.
For a digraph , its reverse is a digraph with same vertex set and such that if and only if . A digraph is symmetric if . In other words, a symmetric digraph can be obtained from its underlying undirected graph by replacing each edge of with the corresponding arcs of both directions, that is, A 2cycle of a strong digraph is called a bridge if is disconnected. Thus, a bridge corresponds to a bridge in the underlying undirected graph of . An orientation of a digraph is a digraph obtained from by deleting an arc in each 2cycle of . A digraph is semicomplete if for every distinct at least one of the arcs in in . We refer the readers to [2, 3, 9] for graph theoretical notation and terminology not given here.
2 Basic Results
The following proposition can be easily verified using definitions of and .
By Tillson’s decomposition theorem [26], we can determine the exact values for and .
Proposition 2.4
[22] For every fixed , a digraph is strong if and only if
3 Algorithms and Complexity
3.1 Results for and
For a fixed , it is easy to decide whether for a digraph : it holds if and only if is strong. Unfortunately, deciding whether is already NPcomplete for with , where is a fixed integer.
The wellknown Directed Linkage problem was proved to be NPcomplete even for the case that [13]. The problem is formulated as follows: for a fixed integer , given a digraph and a (terminal) sequence of distinct vertices of decide whether has vertexdisjoint paths , where starts at and ends at for all
By using the reduction from the Directed Linkage problem, we can prove the following intractability result.
Theorem 3.1
[23] Let and be fixed integers. Let be a digraph and with . The problem of deciding whether is NPcomplete.
In the above theorem, Sun et al. obtained the complexity result of the parameter for an arbitrary digraph . For , they made the following conjecture.
Conjecture 1
[23] It is NPcomplete to decide for fixed integers and and a given digraph whether .
Recently, Chudnovsky, Scott and Seymour [12] proved the following powerful result.
Theorem 3.2
[12] Let and be fixed positive integers. Then the Directed Linkage problem on a digraph whose vertex set can be partitioned into sets each inducing a semicomplete digraph and a terminal sequence of distinct vertices of , can be solved in polynomial time.
The following nontrivial lemma can be deduced from Theorem 3.2.
Lemma 3.3
[23] Let and be fixed positive integers. Let be a digraph and let be vertex disjoint subsets of , such that for all . Let and assume that every vertex in is adjacent to every other vertex in . Then we can in polynomial time decide if there exists vertex disjoint subsets of , such that and is strongly connected for each .
Using Lemma 3.3, Sun, Gutin, Yeo and Zhang proved the following result for semicomplete digraphs.
Theorem 3.4
[23] For any fixed integers , we can decide whether for a semicomplete digraph in polynomial time.
Now we turn our attention to symmetric graphs. We start with the following structural result.
Theorem 3.5
[23] For every undirected graph we have .
Theorem 3.5 immediatly implies the following positive result, which follows from the fact that can be computed in polynomial time.
Corollary 3.6
[23] For a graph , can be computed in polynomial time.
Theorem 3.5 states that when . However when , then is not always equal to , as can be seen from . Chen, Li, Liu and Mao [10] introduced the following problem, which they proved to be NPcomplete.
CLLM Problem: Given a tripartite graph with a 3partition such that , decide whether there is a partition of into disjoint 3sets such that for every and is connected.
Lemma 3.7
[10] The CLLM Problem is NPcomplete.
Now restricted to symmetric digraphs , for any fixed integer , by Lemma 3.7, the problem of deciding whether is NPcomplete for with .
Theorem 3.8
[23] For any fixed integer , given a symmetric digraph , a subset of and an integer , deciding whether , is NPcomplete.
The last theorem assumes that is fixed but is a part of input. When both and are fixed, the problem of deciding whether for a symmetric digraph , is polynomialtime solvable. We will start with the following technical lemma.
Lemma 3.9
[23] Let be fixed. Let be a graph and let be an independent set in with . For , let be any set of arcs with both endvertices in . Let a forest in be called acceptable if the digraph is strong and contains . In polynomial time, we can decide whether there exists edgedisjoint forests such that is acceptable for all and for all .
Now we can prove the following result by Lemma 3.9:
Theorem 3.10
[23] Let be fixed. For any symmetric digraph and with we can in polynomial time decide whether .
The Directed Linkage problem is polynomialtime solvable for planar digraphs [19] and digraphs of bounded directed treewidth [16]. However, it seems that we cannot use the approach in proving Theorem 3.4 directly as the structure of minimumsize strong subgraphs in these two classes of digraphs is more complicated than in semicomplete digraphs. Certainly, we cannot exclude the possibility that computing strong subgraph connectivity in planar digraphs and/or in digraphs of bounded directed treewidth is NPcomplete.
Problem 3.11
[23] What is the complexity of deciding whether for fixed integers , and and a given planar digraph ?
Problem 3.12
[23] What is the complexity of deciding whether for fixed integers , and and a digraph of bounded directed treewidth?
It would be interesting to identify large classes of digraphs for which the problem can be decided in polynomial time.
3.2 Results for and
Yeo proved that it is an NPcomplete problem to decide whether a 2regular digraph has two arcdisjoint hamiltonian cycles (see, e.g., Theorem 6.6 in [6]). (A digraph is 2regular if the outdegree and indegree of every vertex equals 2.) Thus, the problem of deciding whether is NPcomplete, where is the order of . Sun and Gutin [22] extended this result in Theorem 3.13.
Let be a digraph and let be a collection of not necessarily distinct vertices of . A weak linkage from to is a collection of arcdisjoint paths such that is an path for each . A digraph is weakly linked if it contains a weak linkage from to for every choice of (not necessarily distinct) vertices . The Directed Weak Linkage problem is the following. Given a digraph and distinct vertices ; decide whether contains arcdisjoint paths such that is an path. The problem is wellknown to be NPcomplete already for [13]. By using the reduction from the Directed Weak Linkage problem, we can prove the following intractability result.
Theorem 3.13
[22] Let and be fixed integers. Let be a digraph and with . The problem of deciding whether is NPcomplete.
BangJensen and Yeo [6] conjectured the following:
Conjecture 2
For every there is a finite set of digraphs such that arcstrong semicomplete digraph contains arcdisjoint spanning strong subgraphs unless .
BangJensen and Yeo [6] proved the conjecture for by showing that and describing the unique digraph of of order 4. This result and Theorem 4.4 imply the following:
Theorem 3.14
[22] For a semicomplete digraph , of order and an integer such that , if and only if is 2arcstrong and .
Now we turn our attention to symmetric graphs. We start from characterizing symmetric digraphs with , an analog of Theorem 3.14. To prove it we need the following result of Boesch and Tindell [8] translated from the language of mixed graphs to that of digraphs.
Theorem 3.15
A strong digraph has a strong orientation if and only if has no bridge.
Here is the characterization by Sun and Gutin.
Theorem 3.16
[22] For a strong symmetric digraph of order and an integer such that , if and only if has no bridge.
Theorems 3.14 and 3.16 imply the following complexity result, which we believe to be extendable from to any natural .
Corollary 3.17
[22] The problem of deciding whether is polynomialtime solvable if is either semicomplete or symmetric digraph of order and
Sun and Gutin gave a lower bound on for symmetric digraphs .
Theorem 3.18
Theorem 3.18 immediately implies the next result, which follows from the fact that can be computed in polynomial time.
Corollary 3.19
[22] For a symmetric digraph , can be computed in polynomial time.
Corollaries 3.17 and 3.19 shed some light on the complexity of deciding, for fixed , whether for semicomplete and symmetric digraphs . However, it is unclear what is the complexity above for every fixed . If Conjecture 2 is correct, then the problem can be solved in polynomial time for semicomplete digraphs. However, Conjecture 2 seems to be very difficult. It was proved in [23] that for fixed the problem of deciding whether is polynomialtime solvable for both semicomplete and symmetric digraphs, but it appears that the approaches to prove the two results cannot be used for . Some wellknown results such as the fact that the hamiltonicity problem is NPcomplete for undirected 3regular graphs, indicate that the problem for symmetric digraphs may be NPcomplete, too.
Problem 3.20
[22] What is the complexity of deciding whether for fixed integers and , and a semicomplete digraph ?
Problem 3.21
[22] What is the complexity of deciding whether for fixed integers and , and a symmetric digraph ?
It would be interesting to identify large classes of digraphs for which the problem can be decided in polynomial time.
4 Bounds for Strong Subgraph (Arc)Connectivity
4.1 Results for
By Propositions 2.1 and 2.2, Sun, Gutin, Yeo and Zhang obtained a sharp lower bound and a sharp upper bound for , where .
Theorem 4.1
[23] Let . For a strong digraph of order , we have
Moreover, both bounds are sharp, and the upper bound holds if and only if , and .
Sun and Gutin gave the following sharp upper bound for which improves (3) of Proposition 2.1.
Theorem 4.2
4.2 Results for
By Propositions 2.1 and 2.2, Sun and Gutin obtained a sharp lower bound and a sharp upper bound for , where .
Theorem 4.3
[22] Let . For a strong digraph of order , we have
Moreover, both bounds are sharp, and the upper bound holds if and only if , where , or, and .
They also gave the following sharp upper bound for which improves (3) of Proposition 2.1.
Theorem 4.4
Shiloach [20] proved the following:
Theorem 4.5
[20] A digraph is weakly linked if and only if is arcstrong.
Using Shiloach’s Theorem, Sun and Gutin [22] proved the following lower bound for . Such a bound does not hold for since it was shown in [23] using Thomassen’s result in [25] that for every there are digraphs with and .
Proposition 4.6
[22] Let . We have .
For a digraph , the complement digraph, denoted by , is a digraph with vertex set such that if and only if .
Given a graph parameter , the NordhausGaddum Problem is to determine sharp bounds for (1) and (2) , and characterize the extremal graphs. The NordhausGaddum type relations have received wide attention; see a recent survey paper [1] by Aouchiche and Hansen. By using Proposition 2.4, the following Theorem 4.7 concerning such type of a problem for the parameter can be obtained.
Theorem 4.7
[22]
For a digraph with order , the following assertions holds:
. Moreover, both bounds are sharp. In particular, the lower bound holds if and only if .
.
Moreover, both bounds are sharp. In particular, the lower bound
holds if and only if or .
We now discuss Cartesian products of digraphs. The Cartesian product of two digraphs and is a digraph with vertex set
and arc set
By definition, we know the Cartesian product is associative and commutative, and is strongly connected if and only if both and are strongly connected [15].
Theorem 4.8









Table . Precise values for the strong subgraph 2arcconnectivity of some special cases.
5 Minimally Strong Subgraph (Arc)Connected Digraphs
5.1 Results for Minimally Strong Subgraph Connected Digraphs
A digraph is called minimally strong subgraph connected if but for any arc , [21]. By the definition of and Theorem 4.1, we know . Let be the set of all minimally strong subgraph connected digraphs with order . We define
and
We further define
and
By the definition of a minimally strong subgraph connected digraph, we can get the following observation.
Proposition 5.1
[21] A digraph is minimally strong subgraph connected if and only if and for any arc .
A digraph is minimally strong if is strong but is not for every arc of .
Proposition 5.2
[21]
The following assertions hold:
A digraph is minimally strong subgraph connected
if and only if is minimally strong digraph;
For , a digraph is minimally strong subgraph
connected if and only if .
The following result characterizes minimally strong subgraph connected digraphs.
Theorem 5.3
[21] A digraph is minimally strong subgraph connected if and only if is a digraph obtained from the complete digraph by deleting an arc set M such that is a 3cycle or a union of vertexdisjoint 2cycles. In particular, we have , .
Note that Theorem 5.3 implies that where is an arc set such that is a directed 3cycle, and where is an arc set such that is a union of vertexdisjoint directed 2cycles.
The following result concerns a sharp lower bound for the parameter .
Theorem 5.4
[21] For , we have
Moreover, the following assertions hold:
If , then ; If , then for ; (iii) If
is even and , then
To prove two upper bounds on the number of arcs in a minimally strong subgraph connected digraph, Sun and Gutin used the following result, see e.g. [2].
Theorem 5.5
Every strong digraph on vertices has a strong spanning subgraph with at most arcs and equality holds only if is a symmetric digraph whose underlying undirected graph is a tree.
Proposition 5.6
[21] We have ; For every , and consists of symmetric digraphs whose underlying undirected graphs are trees.
The minimally strong subgraph connected digraphs was characterized in Theorem 5.3. As a simple consequence of the characterization, we can determine the values of and . It would be interesting to determine and for every value of since obtaining characterizations of all connected digraphs for seems a very difficult problem.
Problem 5.7
[21] Determine and for every value of .
It would also be interesting to find a sharp upper bound for for all and .
Problem 5.8
[21] Find a sharp upper bound for for all and .
5.2 Results for Minimally Strong Subgraph ArcConnected Digraphs
A digraph is called minimally strong subgraph arcconnected if but for any arc , . By the definition of and Theorem 4.3, we know . Let be the set of all minimally strong subgraph arcconnected digraphs with order . We define
and
We further define
and
Sun and Gutin [22] gave the following characterizations.
Proposition 5.9
[22]
The following assertions hold:
A digraph is minimally strong subgraph
arcconnected
if and only if is minimally strong digraph;
Let . If , or, and , then a digraph is minimally strong subgraph
arcconnected if and only if .
Theorem 5.10
[22] A digraph is minimally strong subgraph arcconnected if and only if is a digraph obtained from the complete digraph by deleting an arc set such that is a union of vertexdisjoint cycles which cover all but at most one vertex of .
Sun and Jin characterized the minimally strong subgraph arcconnected digraphs.
Theorem 5.11
[24] A digraph is minimally strong subgraph arcconnected if and only if is a digraph obtained from the complete digraph by deleting an arc set such that is a union of vertexdisjoint cycles which cover all but at most one vertex of .
Theorems 5.10 and 5.11 imply that the following assertions hold: For , where is an arc set such that is a union of vertexdisjoint cycles which cover all but exactly one vertex of . For , where is an arc set such that is a union of vertexdisjoint cycles which cover all vertices of .
Sun and Jin completely determined the precise value for . Note that by Theorem 4.3 and the definition of .
Theorem 5.12
[24] For any triple with such that , we have
Some results for were obtained as well.
Proposition 5.13
[24] We have ; For every , and consists of symmetric digraphs whose underlying undirected graphs are trees; (iii) for .
Note that the precise values of for each pair of and and the precise values of for were determined. Hence, similar to problems 5.7 and 5.8, the following problems are also interesting.
Problem 5.14
[24] Determine for every value of .
Problem 5.15
[24] Find a sharp upper bound for for all and .
Acknowledgements. Yuefang Sun was supported by National Natural Science Foundation of China (No.11401389) and China Scholarship Council (No.201608330111). Gregory Gutin was partially supported by Royal Society Wolfson Research Merit Award.
References
 [1] M. Aouchiche, P. Hansen, A survey of NordhausGaddum type relations, Discrete Appl. Math. 161(4/5), 2013, 466–546.
 [2] J. BangJensen and G. Gutin, Digraphs: Theory, Algorithms and Applications, 2nd Edition, Springer, London, 2009.
 [3] J. BangJensen and G. Gutin, Basic Terminology, Notation and Results, in Classes of Directed Graphs (J. BangJensen and G. Gutin, eds.), Springer, 2018.
 [4] J. BangJensen and J. Huang, Decomposing locally semicomplete digraphs into strong spanning subdigraphs, J. Combin. Theory Ser. B 102, 2012, 701–714.
 [5] J. BangJensen and M. Kriesell, Disjoint sub(di)graphs in digraphs, Electron. Notes Discrete Math. 34, 2009, 179–183.
 [6] J. BangJensen and A. Yeo, Decomposing arcstrong tournaments into strong spanning subdigraphs, Combinatorica 24(3), 2004, 331–349.
 [7] J. BangJensen and A. Yeo, Arcdisjoint spanning sub(di)graphs in Digraphs, Theoret. Comput. Sci. 438, 2012, 48–54.
 [8] F. Boesch and R. Tindell, Robbins’ theorem for mixed multigraphs, Amer. Math. Monthly 87, 1980, 716–719.
 [9] J.A. Bondy and U.S.R. Murty, Graph Theory, Springer, Berlin, 2008.
 [10] L. Chen, X. Li, M. Liu and Y. Mao, A solution to a conjecture on the generalized connectivity of graphs, J. Combin. Optim. 33(1), 2017, 275–282.
 [11] J. Cheriyan and M.R. Salavatipour, Hardness and approximation results for packing Steiner trees, Algorithmica 45, 2006, 21–43.
 [12] M. Chudnovsky, A. Scott and P.D. Seymour. Disjoint paths in unions of tournaments. arXiv:1604.02317, April 2016.
 [13] S. Fortune, J. Hopcroft and J. Wyllie, The directed subgraphs homeomorphism problem, Theoret. Comput. Sci. 10, 1980, 111–121.
 [14] M. Hager, Pendant treeconnectivity, J. Combin. Theory Ser. B 38, 1985, 179–189.
 [15] R.H. Hammack, Digraphs Products, in J. BangJensen and G. Gutin (eds.), Classes of Directed Graphs, Springer, 2018.
 [16] T. Johnson, N. Robertson, P.D. Seymour and R. Thomas, Directed TreeWidth, J. Combin. Th. Ser. B 82(1), 2001, 138–154.
 [17] X. Li and Y. Mao, Generalized Connectivity of Graphs, Springer, Switzerland, 2016.
 [18] X. Li, Y. Mao and Y. Sun, On the generalized (edge)connectivity of graphs, Australas. J. Combin. 58(2), 2014, 304–319.
 [19] A. Schrijver, Finding partially disjoint paths in a directed planar graph. SIAM J. Comput. 23(4), 1994, 780–788.
 [20] Y. Shiloach, Edgedisjoint branching in directed multigraphs, Inf. Process. Lett. 8(1), 1979, 24–27.
 [21] Y. Sun, G. Gutin, Strong subgraph connectivity bounds, arXiv:1803.00281v1 [cs.DM] 1 Mar 2018.
 [22] Y. Sun, G. Gutin, Strong subgraph arcconnectivity, arXiv:1805.01687v1 [cs.DM] 4 May 2018.
 [23] Y. Sun, G. Gutin, A. Yeo, X. Zhang, Strong subgraph connectivity, arXiv:1803.00284v1 [cs.DM] 1 Mar 2018.
 [24] Y. Sun, Z. Jin, Minimally strong subgraph arcconnected digraphs, in preparation.
 [25] C. Thomassen, Highly connected non2linked digraphs, Combinatorica 11(4), 1991, 393–395.
 [26] T.W. Tillson, A Hamiltonian decomposition of , , J. Combin. Theory Ser. B 29(1), 1980, 68–74.
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