Arc-disjoint strong spanning subdigraphs in compositions and products of digraphs
aa r X i v : . [ c s . D M ] D ec Arc-disjoint strong spanning subdigraphsin compositions and products of digraphs
Yuefang Sun , ∗ Gregory Gutin , † Jiangdong Ai Department of Mathematics, Shaoxing UniversityZhejiang 312000, P. R. China, [email protected] Department of Computer ScienceRoyal Holloway, University of LondonEgham, Surrey, TW20 0EX, [email protected], [email protected]
Abstract
A digraph D = ( V, A ) has a good decomposition if A has two dis-joint sets A and A such that both ( V, A ) and ( V, A ) are strong. Let T be a digraph with t vertices u , . . . , u t and let H , . . . H t be digraphssuch that H i has vertices u i,j i , ≤ j i ≤ n i . Then the composition Q = T [ H , . . . , H t ] is a digraph with vertex set { u i,j i | ≤ i ≤ t, ≤ j i ≤ n i } and arc set A ( Q ) = ∪ ti =1 A ( H i ) ∪{ u ij i u pq p | u i u p ∈ A ( T ) , ≤ j i ≤ n i , ≤ q p ≤ n p } . For digraph compositions Q = T [ H , . . . H t ], we obtain sufficientconditions for Q to have a good decomposition and a characterizationof Q with a good decomposition when T is a strong semicompletedigraph and each H i is an arbitrary digraph with at least two vertices.For digraph products, we prove the following: (a) if k ≥ G is a strong digraph which has a collection of arc-disjointcycles covering all vertices, then the Cartesian product digraph G (cid:3) k (the k th powers with respect to Cartesian product) has a good decom-position; (b) for any strong digraphs G, H , the strong product G ⊠ H has a good decomposition. Keywords: strong spanning subdigraph; decomposition into strongspanning subdigraphs; semicomplete digraph; digraph composition;Cartesian product; strong product.
AMS subject classification (2010) : 05C20, 05C70, 05C76, 05C85.
We refer the readers to [1, 2, 6] for graph theoretical notation and ter-minology not given here. A digraph D = ( V, A ) is strongly connected (or ∗ Yuefang Sun was supported by National Natural Science Foundation of China (No.11401389). † Corresponding author. Gregory Gutin was partially supported by Royal Society Wolf-son Research Merit Award. trong ) if there exists a path from x to y and a path from y to x in D forevery pair of distinct vertices x, y of D . A digraph D is k -arc-strong if D − X is strong for every subset X ⊆ A of size at most k − out-branching B + s (respectively, in-branching B − s ) in a digraph D =( V, A ) is a connected spanning subdigraph of D in which each vertex x = s has precisely one arc entering (leaving) it and s has no arcs entering (leaving)it. The vertex s is the root of B + s (respectively, B − s ). Edmonds [9] character-ized digraphs with have k arc-disjoint out-branchings rooted at a specifiedvertex s. Furthermore, there exists a polynomial algorithm for finding k arc-disjoint out-branchings from a given root s if they exist (see p. 346 of [1]).However, if we ask for the existence of a pair of arc-disjoint branchings B + s , B − s such that the first is an out-branching rooted at s and the latter is anin-branching rooted at s , then the problem becomes NP-complete (see Sec-tion 9.6 of [1]). In connection with this problem, Thomassen [12] posed thefollowing conjecture: There exists an integer N so that every N -arc-strongdigraph D contains a pair of arc-disjoint in- and out-branchings.Bang-Jensen and Yeo generalized the above conjecture as follows. Adigraph D = ( V, A ) has a good decomposition if A has two disjoint sets A and A such that both ( V, A ) and ( V, A ) are strong [4]. Conjecture 1.1 [5] There exists an integer N so that every N -arc-strongdigraph D contains a pair of arc-disjoint strong spanning subdigraphs. For a general digraph D , it is a hard problem to decide whether D has adecomposition into two strong spanning subdigraphs. Theorem 1.1 [5] It is NP-complete to decide whether a digraph containsa pair of arc-disjoint strong spanning subdigraphs.
Clearly, every digraph with a good decomposition is 2-arc-strong. Bang-Jensen and Yeo characterized the semicomplete digraphs with a good de-composition.
Theorem 1.2 [5] A 2-arc-strong semicomplete digraph D has a pair ofarc-disjoint strong spanning subdigraphs if and only if D is not isomorphicto S , where S is obtained from the complete digraph with four vertices bydeleting a cycle of length four. Furthermore, a good decomposition of D canbe obtained in polynomial time when it exists. The following result extends Theorem 1.2 to locally semicomplete di-graphs.
Theorem 1.3 [4] A 2-arc-strong locally semicomplete digraph D has a pairof arc-disjoint strong spanning subdigraphs if and only if D is not the secondpower of an even cycle. Every strong digraph has an out- and in-branching. T be a digraph with t vertices u , . . . , u t and let H , . . . H t be digraphssuch that H i has vertices u i,j i , ≤ j i ≤ n i . Then the composition Q = T [ H , . . . , H t ] is a digraph with vertex set { u i,j i | ≤ i ≤ t, ≤ j i ≤ n i } andarc set A ( Q ) = ∪ ti =1 A ( H i ) ∪ { u ij i u pq p | u i u p ∈ A ( T ) , ≤ j i ≤ n i , ≤ q p ≤ n p } . In this paper, we continue research on good decompositions in classes ofdigraphs and consider digraph compositions and products.In Section 2, for digraph compositions Q = T [ H , . . . H t ], we obtain suf-ficient conditions for Q to have a good decomposition (Theorem 2.2) and acharacterization of Q with a good decomposition when T is a strong semi-complete digraph and each H i is an arbitrary digraph with at least twovertices (Theorem 2.3). Remarkably, in Theorem 2.3 as in Theorem 1.2,there are only a finite number of exceptional digraphs, which for Theorem2.3 is three. Thus, as Theorems 1.2 and 1.3, Theorem 2.3 confirms Conjec-ture 1.1 for a special class of digraphs.In Section 3, for digraph products, we prove the following: (a) if k ≥ G is a strong digraph which arcs can be partitioned intocycles, then the Cartesian product digraph G (cid:3) k (the k th powers with respectto Cartesian product) has a good decomposition (Theorem 3.4); (b) for anystrong digraphs G, H , the strong product G ⊠ H has a good decomposition(Theorem 3.7). Necessary definitions of the digraph products are given inSection 3.Simple examinations of our constructive proofs show that all our decom-positions can be found in polynomial time.We conclude the paper in Section 4, where we pose a number of openproblems. Let H ′ i denote H i with all arcs deleted, where 1 ≤ i ≤ t and let Q ′ = T [ H ′ , . . . , H ′ t ] . Compositions of digraphs is a useful concept in digraph theory, see e.g. [1].In particular, they are used in the Bang-Jensen-Huang characterization ofquasi-transitive digraphs and its structural and algorithmic applications forquasi-transitive digraphs and their extensions, see e.g. [1, 2, 8].Let us start from a simple observation, which will be useful in the proofsof the theorems of this section.
Lemma 2.1
Let Q = D [ H , . . . , H t ] . If an induced subdigraph Q ∗ of Q ′ = D [ H ′ , . . . , H ′ t ] with at least one vertex in each H i has a good decomposition,then so have Q ′ and Q. Proof:
For every 1 ≤ i ≤ t, let H ( m i ) i be the subdigraph of H ′ i inducedby { u i, , u i, , . . . , u i,m i } , where 1 ≤ m i ≤ n i . Without loss of generality, let Q ∗ = D [ H ( m )1 , . . . , H ( m t ) t ] and let Q ∗ have a decomposition into arc-disjointstrong spanning subdigraphs D , D . To extend this decomposition to Q ′ , ≤ i ≤ t and j = 1 ,
2, add to D j the vertices u i,m i +1 , . . . , u i,n i andlet them have the same in- and out-neighbors as u i, . ✷ The following theorem gives sufficient conditions for a digraph composi-tion to have a good decomposition. As in Theorem 1.2, S will denote thedigraph obtained from the complete digraph of order 4 by deleting a cycleof length 4. Theorem 2.2
Let Q = T [ H , . . . , H t ] , where t ≥ . Then Q has a gooddecomposition if at least one of the following conditions holds:(a) T is a 2-arc-strong semicomplete digraph and H , . . . , H t are arbitrarydigraphs, but Q is not isomorphic to S ; (b) T has a Hamiltonian cycle and either t is even and n i ≥ for every i = 1 , . . . , t or t is odd and n i ≥ for every i = 1 , . . . , t apart from one i for which n i ≥ , or t is odd, n i ≥ for every i = 1 , . . . , t and at least twodistinct subdigraphs H i have arcs.(c) If T and all H i are strong digraphs of orders at least 2. Proof: Part (a) If T is not isomorphic to S then we are done by Theorem1.2 and Lemma 2.1. Now assume that T is isomorphic to S , but Q is notisomorphic to S . Let the vertices of T be u , u , u , u and its arcs u u , u u , u u , u u , u u , u u , u u , u u . Since Q is not isomorphic to S , at least one of H , H , H , H has at leasttwo vertices. Without loss of generality, let H have at least two vertices.Consider the subdigraph Q ∗ of Q ′ induced by { u , , u , , u , , u , , u , } . Then Q ∗ has two arc-disjoint strong spanning subdigraphs: D with arcs { u , u , , u , u , , u , u , , u , u , , u , u , } and D with arcs { u , u , , u , u , , u , u , , u , u , , u , u , , u , u , } . It remains to apply Lemma 2.1 to obtain a good decomposition of Q. Part (b)
Without loss of generality, assume that u u . . . u t u is a Hamil-tonian cycle of T. Let U = ∪ ti =1 { u i, , u i, } . Case 1: t is even and n i ≥ for every i = 1 , . . . , t. The following arcsets induce arc-disjoint strong spanning subdigraphs D , D of Q ′ [ U ] : { u i,j u i +1 ,j | ≤ i ≤ t − , ≤ j ≤ } ∪ { u t, u , , u t, u , } (1) { u i,j u i +1 , ( j +1 mod | ≤ i ≤ t − , ≤ j ≤ } ∪ { u t, u , , u t, u , } . (2)It remains to apply Lemma 2.1. 4 ase 2: t is odd, n i ≥ for every i = 1 , . . . , t and at least two distinctsubdigraphs H i have arcs. Let e p , e q be arcs in two distinct subdigraphs H p and H q . We may assume that both end-vertices of e p and e q are in U. Observe that while D (with arcs listed in (1)) is strong, D (with arcs listedin (2)) forms two arc-disjoint cycles C and Z. We may assume that the tail(head) of e p ( e q ) is in C and and the head (tail) of e p ( e q ) is in Z (otherwise,relabel vertices in { u p, , u p, } and/or { u q, u q } ). Thus, adding e p and e q to D makes it strong. To obtain two arc-disjoint strong spanning subdigraphsof Q from D , D , let every vertex u i,j for j ≥ ≤ i ≤ t have thesame out- and in-neighbors as u i, in Q ′ . Case 3: t is odd and n i ≥ for every i = 1 , . . . , t apart from one i for which n i ≥ . Without loss of generality, assume that n ≥ n i ≥ ≤ i ≤ t. First we consider the subcase in which t = 3, n = 2 , and n = n = 3 . Then Q ′ has two arc-disjoint spanning subdigraphs D and D with arc sets { u , u , , u , u , , u , u , , u , u , , u , u , , u , u , , u , u , , u , u , , u , u , } , { u , u , , u , u , , u , u , , u , u , , u , u , , u , u , , u , u , , u , u , , u , u , } , respectively. It is not hard to see that D and D are strong by constructingclosed walks through all vertices.Now we extend the previous subcase to that in which n = 2 and n i = 3 for all 2 ≤ i ≤ t. First replace index 3 in every vertex of theform u ,i by t in the two arc sets of the previous subcase. Then replaceevery arc of the form u ,i u t,j in D by the path u ,i u ,i . . . u t − ,i u t,j . In D , we replace u , u t, by the path u , u , u , u , . . . u t − , u t, , replace u , u t, by the path u , u , u , u , . . . u t − , u t, , replace u , u t, by the path u , u , u , u , . . . u t − , u t, , and finally add the path u , u , u , u , . . . u t − , .Finally, we extend the previous subcase to the general one using Lemma2.1. Part (c)
For j = 1 ,
2, let T j be the subdigraph of Q induced by vertex set { u i,j | ≤ i ≤ t } . Clearly, T ∼ = T ∼ = T and T and T are strong.Let Q be the spanning subdigraph of Q with arc set A ( Q ) = A ( T ) ∪ ( S ti =1 A ( H i )). Observe that Q is strong since T and each H i are strong,and T has a common vertex with each H i , where 1 ≤ i ≤ t .Let Q be the spanning subdigraph of Q with arc set A ( Q ) = A ( Q ) \ A ( Q ). To see that Q is strong, we only need to find a strong subdigraphin Q which contains x and y for each pair of distinct vertices x and y in Q . We will consider two cases. Case 1: x ∈ V ( T ) . Without loss of generality, we assume that x = u , and y ∈ { u , , u , , u , } . We first consider the subcase that y = u , .Observe that there is at least one arc entering and one arc leaving u , ( u , )in T , and so there are two arcs, say a and b ( c and d ), with oppositedirections between x ( y ) and T in Q . Then by adding the arcs a, b, c, d ,and the vertices x, y to T , we obtain a strong subdigraph T ′ of Q which5ontains both x and y , as desired. For the case that y ∈ { u , , u , } , wejust add the arcs a, b , and the vertex x to T , and then obtain a strongsubdigraph T ′′ of Q which contains both x and y . Case 2: x V ( T ) . Without loss of generality, we assume that x = u , and y ∈ { u , , u , , u , , u , , u , } (if u , and u , exist). By Case 1 andthe fact that T ∼ = T is strong, we are done if y ∈ { u , , u , } . For thecase that y = u , , by adding the arcs c, d and the vertex y to T , we canobtain a strong subdigraph T ′′′ of Q which contains both x and y . Witha similar argument, we can get the desired strong subdigraph for the casethat y ∈ { u , , u , } .Hence, we complete the argument and conclude that Q has a good de-composition. ✷ We will use Theorem 2.2 to prove the following characterization for cer-tain compositions T [ H , . . . , H t ], where T is a strong semicomplete digraph.In the characterization, K p will stand for the digraph of order p with noarcs. Also, −→ C k and −→ P k will denote the cycle and path with k vertices,respectively. Theorem 2.3
Let T be a strong semicomplete digraph on t ≥ vertices andlet H , . . . , H t be arbitrary digraphs, each with at least two vertices. Then Q = T [ H , . . . , H t ] has a good decomposition if and only if Q is not isomor-phic to one of the following three digraphs: −→ C [ K , K , K ] , −→ C [ −→ P , K , K ] . −→ C [ K , K , K ] . Proof:
Let us first prove the ‘only if’ part of the theorem, i.e. −→ C [ K , K , K ] , −→ C [ −→ P , K , K ] and −→ C [ K , K , K ] do not have good decompositions. ByLemma 2.1, it suffices to show that neither −→ C [ −→ P , K , K ] nor −→ C [ K , K , K ]has a good decomposition. The proof is by reductio ad absurdum.Suppose that Q = −→ C [ −→ P , K , K ] has a decomposition into two strongspanning subdigraphs Q , Q . Since Q has 13 arcs, without loss of generality,we may assume that Q is a Hamiltonian cycle of Q. Since the arc of H cannot be in a Hamiltonian cycle of Q , without loss of generality, let Q = u , u , u , u , u , u , u , . Then the remaining arcs of Q form two disjointcycles u , u , u , u , and u , u , u , u , and a single arc between them, acontradiction to the assumption that Q is strong.Suppose that Q = −→ C [ K , K , K ] has a decomposition into two strongspanning subdigraphs Q , Q . Since Q has 16 arcs and has no Hamiltoniancycle, each of Q , Q has 8 arcs. Since Q has only cycles of lengths 3 and6 and Q is strong, without loss of generality, we may assume that Q consists of a cycle u , u , u , u , u , u , u , and a path u , u , u , . Then Q consists of two cycles u , u , u , u , and u , u , u , u , and a path u , u , u , . Observe that Q is not strong, a contradiction.Now we will show the ‘if’ part of the theorem by reductio ad absurdumas well. Assume that Q is not isomorphic to either of the three digraphs,but has no good decomposition. 6y Camion’s Theorem [7], T has a Hamiltonian cycle C = u u . . . u t u .Thus, Conditions (b) of Theorem 2.2 are applicable. By the conditions, t must be odd and for at least two distinct indexes p, q ∈ { , , . . . , t } , we have n p = n q = 2 . Suppose t ≥ . Then there will be arcs between H i and H i +2 in Q forevery i = 1 , , . . . , t − . Recall Case 2 of Part (b) of the proof of Theorem2.2. The arcs between H i and H i +2 arcs can be used to make D stronginstead of arcs e p and e q used in Case 2 of Part (b) of the proof of Theorem2.2. Thus, Q has a good decomposition, a contradiction. Hence, t = 3 and,without loss of generality, n = n = 2 and n ≥ . Suppose that T has opposite arcs. One of these arcs will not be on theHamiltonian cycle C of T and will correspond to four or more arcs in Q. Now recall Case 2 of Part (b) of the proof of Theorem 2.2. Two of theabove-mentioned arcs can be used to make D strong instead of arcs e p and e q used in Case 2 of Part (b) of the proof of Theorem 2.2. Thus, Q has agood decomposition, a contradiction. Hence, T = −→ C . Suppose that n ≥ . To get a contradiction, by Lemma 2.1 it sufficesto show that Q = −→ C [ K , K , K ] has a decomposition into two strong span-ning subdigraphs D , D , where D consists of a cycle u , u , u , u , u , u , u , and two paths u , u , u , and u , u , u , and D consists of two cycles u , u , u , u , and u , u , u , u , and two paths u , u , u , and u , u , u , . Thus, n ≤ . Now consider the case of n = n = 2 and n = 3 . Since Q is notisomorphic to −→ C [ K , K , K ] , it has an arc in either H or H or H , and byConditions (b) of Theorem 2.2, only one of H , H , H has an arc a. Withoutloss of generality, assume that if H has an arc then a = u , u , , if H hasan arc then a = u , u , and if H has an arc then a = u , u , . Then Q hasa decomposition into two spanning subdigraphs D , D , where D consistsof a cycle u , u , u , u , u , u , u , and a path u , u , u , and D consistsof two cycles u , u , u , u , and u , u , u , u , , a path u , u , u , and arc a . Observe that both D and D are strong, a contradiction.It remains to consider the case of n = n = n = 2 . Since Q is notisomorphic to −→ C [ K , K , K ], at least one of H , H and H has an arc.By Conditions (b) of Theorem 2.2, only one of H , H and H has an arc.Without loss of generality, assume that H has an arc. Suppose that H has two arcs. Then H = −→ C . Then we can use the arcs of H to make D strong instead of arcs e p and e q used in Case 2 of Part (b) of the proof ofTheorem 2.2. Thus, Q has a good decomposition, a contradiction. Hence,if H has an arc, it must have just one arc. This concludes our proof. ✷ The
Cartesian product G (cid:3) H of two digraphs G and H is a digraph withvertex set V ( G (cid:3) H ) = V ( G ) × V ( H ) = { ( x, x ′ ) | x ∈ V ( G ) , x ′ ∈ V ( H ) } and arc set A ( G (cid:3) H ) = { ( x, x ′ )( y, y ′ ) | xy ∈ A ( G ) , x ′ = y ′ , or x = y, x ′ y ′ ∈ A ( H ) } . By definition, we know the Cartesian product is associative and7ommutative, and G (cid:3) H is strongly connected if and only if both G and H are strongly connected [10]. We define the n th powers with respect toCartesian product as D (cid:3) n = D (cid:3) D (cid:3) · · · (cid:3) D . Gu u u v v v v H G ( v ) G ( v ) G ( v ) G ( v ) H ( u ) H ( u ) H ( u ) 1 1 12 2 2(a) (b) (c) Figure 1: Two digraphs G , H and their Cartesian product.In the argument of this section, we will use the following terminologyand notation. Let G and H be two digraphs with V ( G ) = { u i | ≤ i ≤ n } and V ( H ) = { v j | ≤ j ≤ m } . For simplicity, we let u i,j = ( u i , v j ) for1 ≤ i ≤ n, ≤ j ≤ m . We use G ( v j ) to denote the subdigraph of G (cid:3) H induced by vertex set { u i,j | ≤ i ≤ n } where 1 ≤ j ≤ m , and use H ( u i )to denote the subdigraph of G (cid:3) H induced by vertex set { u i,j | ≤ j ≤ m } where 1 ≤ i ≤ n . Clearly, we have G ( v j ) ∼ = G and H ( u i ) ∼ = H . (Forexample, as shown in Figure 1, G ( v j ) ∼ = G for 1 ≤ j ≤ H ( u i ) ∼ = H for 1 ≤ i ≤ ≤ j = j ≤ m , u i,j and u i,j belong to the samedigraph H ( u i ) where u i ∈ V ( G ); we call u i,j the vertex corresponding to u i,j in G ( v j ); for 1 ≤ i = i ≤ n , we call u i ,j the vertex correspondingto u i ,j in H ( u i ). Similarly, we can define the subdigraph corresponding tosome other subdigraph. For example, in Fig. 1(c), let P ( P ) be the pathlabelled 1 (2) in H ( u ) ( H ( u )), then P is called the path corresponding to P in H ( u ). Lemma 3.1
For any integer n ≥ , the product digraph D = −→ C n (cid:3) −→ C n canbe decomposed into two arc-disjoint Hamiltonian cycles. Proof:
Let G = H ∼ = −→ C n ; moreover G = u u . . . u n u and H = v v . . . v n v . Let P i = G ( v i ) − u n − i,i u n +1 − i,i for 1 ≤ i ≤ n − P n = G ( v n ) − u n,n u ,n .Let D ′ be the subdigraph of D which is a union of n paths P i and the fol-lowing n arcs: { u n − i,i u n − i,i +1 | ≤ i ≤ n − } ∪ { u n,n u n, } . Let D ′′ be aspanning subdigraph of D with A ( D ′′ ) = A ( D ) \ A ( D ′ ). It is not hard tocheck that both D ′ and D ′′ are Hamiltonian cycles of D ; this completes theproof. ✷ Note that deciding whether a digraph D has a collection of arc-disjointcycle covering all vertices of D , can be done in polynomial time using networkflows. Indeed, assign lower bound 1 and upper bound min { d − ( x ) , d + ( x ) } toevery vertex x in D and lower bound 0 and upper bound 1 to every arc of D .Observe that the resulting network has a feasible flow if and only if D hasa collection of arc-disjoint cycle covering all vertices of D . Observe that the8xistence of a flow in a network with lower and upper bounds on vertices andarcs can be decided in polynomial time, see e.g. Chapter 4 in [1]. Moreover,we can compute such a flow in polynomial time (if it exists) and obtainthe corresponding collection of cycles in D. The following lemma may be ofindependent interest.
Lemma 3.2
Let G be a strong digraph of order at least two which has acollection of arc-disjoint cycle covering all its vertices. Then the productdigraph D = G (cid:3) G can be decomposed into two arc-disjoint strong spanningsubdigraphs. Moreover, these two arc-disjoint strong spanning subdigraphscan be found in polynomial time. Proof:
By the arguments in the paragraph before this lemma, we mayassume that we are given a collection ( P , P , P , · · · , P p ) of arc-disjointcycle covering all vertices of G . For each h = 0 , , , · · · , p , let G h denotethe digraph with vertices S hi =0 V ( P i ) and arcs S hi =0 A ( P i ). Now we willprove the lemma by induction on the number of cycles in the collection.For the base step, by Lemma 3.1, we have that G (cid:3) G = P (cid:3) P can bedecomposed into two arc-disjoint strong spanning subdigraphs.For the inductive step, we assume that G h (cid:3) G h (0 ≤ h ≤ p −
1) can be de-composed into two arc-disjoint strong spanning subdigraphs D ′ h and D ′′ h . Wewill construct two arc-disjoint strong spanning subdigraphs in G h +1 (cid:3) G h +1 .If V ( G h ) ⊆ V ( P h +1 ), then P h +1 is a Hamiltonian cycle of G h +1 , and weare done by Lemma 3.1. If V ( P h +1 ) ⊆ V ( G h ), then G h is a strong spanningsubdigraph of G h +1 , and we are also done.In the following argument, we assume that V ( G h ) \ V ( P h +1 ) = ∅ and V ( P h +1 ) \ V ( G h ) = ∅ . Without loss of generality, for the first copies of G h and P h +1 in G h (cid:3) G h and P h +1 (cid:3) P h +1 , let V ( G h ) = { u i | ≤ i ≤ t } , V ( P h +1 ) = { u i | s ≤ i ≤ ℓ } . We have 1 < s ≤ t < ℓ . For the second copiesof G h and P h +1 in G h (cid:3) G h and P h +1 (cid:3) P h +1 , we will use v i ’s rather than u i ’s.By Lemma 3.1, in G h +1 (cid:3) G h +1 , the subdigraph P h +1 (cid:3) P h +1 can be de-composed into two arc-disjoint strong spanning subdigraphs D ′ h and D ′′ h .Observe that V ( G h (cid:3) G h ) ∩ V ( P h +1 (cid:3) P h +1 ) ⊇ { u t,t } and A ( G h (cid:3) G h ) ∩ A ( P h +1 (cid:3) P h +1 ) = ∅ . For 1 ≤ j ≤ s −
1, let G h,j be the subdigraph of G ( v j ) corresponding to P h +1 . For t + 1 ≤ j ≤ ℓ , let G h,j be the subdigraph of G ( v j ) correspondingto G h . For 1 ≤ i ≤ s −
1, let H h,i be the subdigraph of H ( u i ) correspondingto P h +1 . For t +1 ≤ i ≤ ℓ , let H h,i be the subdigraph of H ( u i ) correspondingto G h .Now let D ′ h +1 be a union of the following strong digraphs: D ′ h , D ′ h , H h,i and G h,j for all t + 1 ≤ i, j ≤ ℓ . Observe that D ′ h +1 is a strong spanningsubdigraph of G h +1 (cid:3) G h +1 since D ′ h has at least one common vertex witheach of D ′ h , H h,i and G h,j for all t + 1 ≤ i, j ≤ ℓ . Let D ′′ h +1 be a spanningsubdigraph of G h +1 (cid:3) G h +1 with A ( D ′′ h +1 ) = A ( G h +1 (cid:3) G h +1 ) \ A ( D ′ h +1 ). Ob-serve that D ′′ h +1 is the union of D ′′ h , D ′′ h , H h,i and G h,j for all 1 ≤ i, j ≤ s − D ′′ h has at least one common vertex with each of D ′′ h , H h,i and G h,j forall 1 ≤ i, j ≤ s −
1, thus D ′′ h +1 is strong.9ence, we complete the inductive step and conclude that D = G (cid:3) G canbe decomposed into two arc-disjoint strong spanning subdigraphs. More-over, by the above argument, these subdigraphs can be found in polynomialtime. ✷ Lemma 3.3
For any two strong digraphs G and H , if G contains a pairof arc-disjoint strong spanning subdigraphs, then the product digraph D = G (cid:3) H can be decomposed into two arc-disjoint strong spanning subdigraphs. Proof:
Let V ( G ) = { u i | ≤ i ≤ n } , V ( H ) = { v j | ≤ j ≤ m } , and G contain two arc-disjoint strong spanning subdigraphs G and G . For1 ≤ j ≤ m , let G ,j be the subdigraph of G ( v j ) corresponding to G . Let D ′ be a union of H ( u ) and G ,j for all 1 ≤ j ≤ m , and D ′′ be a subdigraph of D with V ( D ′′ ) = V ( D ) and A ( D ′′ ) = A ( D ) \ A ( D ′ ). It is not hard to verifythat both D ′ and D ′′ are strong spanning subdigraphs of D . This completesthe proof. ✷ By the definition of D (cid:3) k , associativity of the Cartesian product, andLemmas 3.2 and 3.3, we can obtain the following result on G (cid:3) k for anyinteger k ≥ Theorem 3.4
Let G be a strong digraph of order at least two which has acollection of arc-disjoint cycle covering all its vertices and let k ≥ be aninteger. Then the product digraph D = G (cid:3) k can be decomposed into twoarc-disjoint strong spanning subdigraphs. Moreover, for any fixed integer k ,these two subdigraphs can be found in polynomial time. The strong product G ⊠ H of two digraphs G and H is a digraph withvertex set V ( G ⊠ H ) = V ( G ) × V ( H ) = { ( x, x ′ ) | x ∈ V ( G ) , x ′ ∈ V ( H ) } and arc set A ( G ⊠ H ) = { ( x, x ′ )( y, y ′ ) | xy ∈ A ( G ) , x ′ = y ′ , or x = y, x ′ y ′ ∈ A ( H ) , or xy ∈ A ( G ) , x ′ y ′ ∈ A ( H ) } . By definition, G (cid:3) H is a spanningsubdigraph of G ⊠ H , and G ⊠ H is strongly connected if and only if both G and H are strongly connected [10]. In the following argument, we will stilluse the terminology and notation introduced earlier in this section, since G (cid:3) H is a spanning subdigraph of G ⊠ H . Lemma 3.5
For any two integers n, m ≥ , the product digraph D = −→ C n ⊠ −→ C m can be decomposed into two arc-disjoint strong spanning subdigraphs. Proof:
Let −→ C n = u u . . . u n u and −→ C m = v v . . . v m . Let D ′ be the span-ning subdigraph of D which is the union of G ( v j ) for 1 ≤ j ≤ m and thefollowing additional m arcs: { u n,j u ,j +1 | ≤ j ≤ m − } ∪ { u ,m u , } .Observe that D ′ is strong. Let D ′′ be a spanning subdigraph of D with A ( D ′′ ) = A ( D ) \ A ( D ′ ). To see that D ′′ is strong, observe that it contains H ( u i ) for 1 ≤ i ≤ n and arcs { u i, u i +1 , | ≤ i ≤ n − } ∪ { u n,m u , } . ✷ We will use the following decomposition of strong digraphs.An ear decomposition of a digraph D is a sequence P = ( P , P , P , · · · , P t ),where P is a cycle or a vertex and each P i is a path, or a cycle with the10ollowing properties:( a ) P i and P j are arc-disjoint when i = j .( b ) For each i = 0 , , , · · · , t : let D i denote the digraph with vertices S ij =0 V ( P j ) and arcs S ij =0 A ( P j ). If P i is a cycle, then it has preciselyone vertex in common with V ( D i − ). Otherwise the end vertices of P i aredistinct vertices of V ( D i − ) and no other vertex of P i belongs to V ( D i − ).( c ) S tj =0 A ( P j ) = A ( D ).The following result is well-known, see e.g. [1]. Theorem 3.6
Let D be a digraph with at least two vertices. Then D isstrong if and only if it has an ear decomposition. Furthermore, if D is strong,every cycle can be used as a starting cycle P for an ear decomposition of D , and there is a linear-time algorithm to find such an ear decomposition. Theorem 3.7
For any strong digraphs G and H with orders at least 2, theproduct digraph D = G ⊠ H can be decomposed into two arc-disjoint strongspanning subdigraphs. Moreover, these two arc-disjoint strong spanning sub-digraphs can be found in polynomial time. Proof:
By Theorem 3.7 G has an ear decomposition P = ( P , P , P , · · · , P p )and H has an ear decomposition Q = ( Q , Q , Q , · · · , Q q ), such that P isa cycle of G and Q is a cycle of H by Theorem 3.6. Let G i denote thesubdigraph of G with vertices S ij =0 V ( P j ) and arcs S ij =0 A ( P j ) and let H i denote the subdigraph of H with vertices S ij =0 V ( Q j ) and arcs S ij =0 A ( Q j ) . We will prove the theorem by induction on r ∈ { , , . . . , p + q } . For thebase step, by Lemma 3.5, we have that P ⊠ Q can be decomposed into twoarc-disjoint strong spanning subdigraphs. For the inductive step, we assumethat r = h + g < p + q ( h ≤ p, g ≤ q ) and G h ⊠ H g can be decomposed intotwo arc-disjoint strong spanning subdigraphs D ′ and D ′′ .Since strong product is a commutative operation, without loss of gener-ality it suffices to prove that G h +1 ⊠ H g ( h < p ) can be decomposed intotwo arc-disjoint strong spanning subdigraphs. Let V ( G h ) = { u , u , . . . , u ℓ } , V ( H g ) = { v , v , . . . , v m } and v v s ∈ A ( H g ) . Let P h +1 ,j be the subdigraphof G ( v j ) corresponding to P h +1 for 1 ≤ j ≤ m. We will consider two cases.
Case 1: P h +1 is a cycle. Let P h +1 = u ℓ u ℓ +1 . . . u n u ℓ . Observe that every P h +1 ,j for 1 ≤ j ≤ m shares vertex u ℓ,j with D ′ . Thus, the union U of D ′ and P h +1 ,j for 1 ≤ j ≤ m is a strong spanning subdigraph of G h +1 ⊠ H g . Let V ( U ) = V ( G h +1 ⊠ H g ) and A ( U ) = A ( G h +1 ⊠ H g ) \ A ( U ) . Observe that A ( U ) contains A ( D ′′ ), A ( H ( u i )) for ℓ + 1 ≤ i ≤ n and { u i, u i +1 ,s | ℓ ≤ i ≤ n − } ∪ { u n, u ℓ,s } . Thus, U is strong. Case 2: P h +1 is a path. Let P h +1 = u ℓ u ℓ +1 . . . u n − u t , where t < ℓ. Let U be the union of D ′ and P h +1 ,j for 1 ≤ j ≤ m. Observe that U is a spanning subdigraph of G h +1 ⊠ H g and strong since every P h +1 ,j for1 ≤ j ≤ m shares its end-vertices with D ′ . Let V ( U ) = V ( G h +1 ⊠ H g )and A ( U ) = A ( G h +1 ⊠ H g ) \ A ( U ) . Observe that A ( U ) contains A ( D ′′ ), A ( H ( u i )) for ℓ + 1 ≤ i ≤ n − { u i, u i +1 ,s | ℓ ≤ i ≤ n − } ∪ { u n − , u t,s } . Thus, U is strong. 11ence, we complete the inductive step and conclude that D = G ⊠ H can be decomposed into two arc-disjoint strong spanning subdigraphs. Fur-thermore, by Theorem 3.6, the proof of Lemma 3.5, and the argument ofthis theorem, we can conclude that these two strong spanning subdigraphscan be found in polynomial time. ✷ The lexicographic product G ◦ H of two digraphs G and H is a digraph withvertex set V ( G ◦ H ) = V ( G ) × V ( H ) = { ( x, x ′ ) | x ∈ V ( G ) , x ′ ∈ V ( H ) } andarc set A ( G ◦ H ) = { ( x, x ′ )( y, y ′ ) | xy ∈ A ( G ) , or x = y and x ′ y ′ ∈ A ( H ) } [10]. By definition, G ⊠ H is a spanning subdigraph of G ◦ H , so the followingresult holds by Theorem 3.7: For any strong connected digraphs G and H with orders at least 2, the product digraph D = G ◦ H can be decomposedinto two arc-disjoint strong spanning subdigraphs. Moreover, these two arc-disjoint strong spanning subdigraphs can be found in polynomial time. Infact, we can get a more general result.A digraph is Hamiltonian decomposable if it has a family of Hamiltoniandicycles such that every arc of the digraph belongs to exactly one of thedicycles. Ng [11] gives the most complete result among digraph products.
Theorem 3.8 [11] If G and H are Hamiltonian decomposable digraphs,and | V ( G ) | is odd, then G ◦ H is Hamiltonian decomposable. Theorem 3.8 implies that if G and H are Hamiltonian decomposabledigraphs, and | V ( G ) | is odd, then G ◦ H can be decomposed into two arc-disjoint strong spanning subdigraphs. It is not hard to extend this result asfollows: for any strong digraphs G and H of orders at least 2, if H contains ℓ ≥ D = G ◦ H can be decomposed into ℓ + 1 arc-disjoint strong spanningsubdigraphs. We have characterized digraphs T [ H , . . . , H t ] , where T is strong semi-complete and every H i is arbitrary with at least two vertices, which have agood decomposition. It is a natural open problem to extend the character-ization to all such digraphs, where some H i ’s can have just one vertex. Ofcourse, the extended characterization would generalize also Theorem 1.2.A digraph Q is quasi-transitive , if for any triple x, y, z of distinct verticesof Q , if xy and yz are arcs of Q then either xz or zx or both are arcs of Q. For a recent survey on quasi-transitive digraphs and their generalizations,see a chapter [8] by Galeana-S´anchez and Hern´andez-Cruz. Bang-Jensenand Huang [3] proved that a quasi-transitive digraph is strong if and onlyif Q = T [ H , . . . , H t ] , where T is a strong semicomplete digraph and each H i is a non-strong quasi-transitive digraph or has just one vertex. Thus, aspecial case of the above problem is to characterize strong quasi-transitivedigraphs with a good decomposition. This would generalize Theorem 1.2 aswell. 12e believe that these characterizations will confirm Conjecture 1.1 forthe classes of quasi-transitive digraphs and digraphs T [ H , . . . , H t ] , where T is strong semicomplete. In the absence of the characterizations, it would stillbe interesting to confirm the conjecture at least for quasi-transitive digraphs.In Lemma 3.2, we show that G (cid:3) H contains a pair of arc-disjoint strongspanning subdigraphs when G ∼ = H . However, the following result impliesLemma 3.2 cannot be extended to the case that G = H , since it is not hardto show that the Cartesian product digraph of any two cycles has a pairof arc-disjoint strong spanning subdigraphs if and only if it has a pair ofarc-disjoint Hamiltonian cycles. Theorem 4.1 [13] The Cartesian product −→ C p (cid:3) −→ C q is Hamiltonian if andonly if there are non-negative integers d , d for which d + d = gcd( p, q ) ≥ and gcd( p, d ) = gcd( q, d ) = 1 . However, Lemma 3.2 could hold for the case that G = H if we add otherconditions. As shown in Lemma 3.3, we know G (cid:3) H contains a pair of arc-disjoint strong spanning subdigraphs when one of G and H contains a pair ofarc-disjoint strong spanning subdigraphs. So the following open question isinteresting: for any two strong digraphs G and H , neither of which containa pair of arc-disjoint strong spanning subdigraphs, under what conditionthe product digraph G (cid:3) H contains a pair of arc-disjoint strong spanningsubdigraphs?Furthermore, we may also consider the following more challenging ques-tion: under what conditions the product digraph G (cid:3) H ( G ⊠ H ) has more(than two) arc-disjoint strong spanning subdigraphs? References [1] J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Ap-plications, 2nd Edition, Springer, London, 2009.[2] J. Bang-Jensen and G. Gutin, Basic Terminology, Notation and Results,in
Classes of Directed Graphs (J. Bang-Jensen and G. Gutin, eds.),Springer, 2018.[3] J. Bang-Jensen and J. Huang, Quasi-transitive digraphs, J. Graph The-ory, 20(2), 1995, 141–161.[4] J. Bang-Jensen and J. Huang, Decomposing locally semicomplete di-graphs into strong spanning subdigraphs, J. Combin. Theory Ser. B,102, 2012, 701–714.[5] J. Bang-Jensen and A. Yeo, Decomposing k -arc-strong tournamentsinto strong spanning subdigraphs, Combinatorica 24(3), 2004, 331–349.[6] J.A. Bondy and U.S.R. Murty, Graph Theory, Springer, Berlin, 2008.[7] P. Camion, Chemins et circuits hamiltoniens des graphes complets,Comptes Rendus de l’Acad´emie des Sciences de Paris, 249, 1959, 2151–2152. 138] H. Galeana-S´anchez and C. Hern´andez-Cruz, Quasi-transitive digraphsand their extensions, in Classes of Directed Graphs (J. Bang-Jensen andG. Gutin, eds.), Springer, 2018.[9] J. Edmonds, Edge-disjoint branchings, in
Combinatorial Algorithms (B.Rustin ed.), Academic Press, 1973, 91–96.[10] R.H. Hammack, Digraphs Products, in