R E S E A R C H
Open Access
Degree sequence for
k
-arc strongly
connected multiple digraphs
Yanmei Hong
1and Qinghai Liu
2**Correspondence: [email protected]
2Center for Discrete Mathematics, Fuzhou University, Fuzhou, 350108, P.R. China
Full list of author information is available at the end of the article
Abstract
LetDbe a digraph on{v1,. . .,vn}. Then the sequence
{(d+(v1),d–(v1)),. . ., (d+(vn),d–(vn))}is called the degree sequence ofD. For any given
sequence of pairs of integersd={(d+1,d–1),. . ., (dn+,d–n)}, if there exists ak-arc strongly connected digraphDsuch thatdis the degree sequence ofD, thendis realizable and
Dis a realization ofd. In this paper, characterizations fork-arc-connected realizable sequences and realizable sequences with arc-connectivity exactlykare given.
Keywords: degree sequence; realization;k-arc strongly connected
1 Introduction
Digraphs in this paper may have loops and parallel arcs. A digraphDis called amultiple digraph(ormulti-digraphfor short) if it has no loops. Furthermore, ifDhas parallel arcs neither, thenDisstrict. We follow [] for undefined terminologies and notation.
For a digraphD, as in [],V(D) andA(D) denote the vertex set and the arc set ofD, respectively; and (u,v) represents an arc oriented from a vertexuto a vertexv. For any two disjoint vertex setsX andY, letA(X,Y) ={(u,v)∈A(D)|x∈X,y∈Y}. For a subset X⊆V(D), define
∂D+(X) =AF,V(D)\X and ∂D–(X) =∂D+V(D)\X.
We useD[X] to denote the subdigraph ofDinduced byX. IfFis a subdigraph ofD, then for notational convenience, we often use∂+
D(F),∂D–(F) for∂D+(V(F)),∂D–(V(F)), respectively. For a vertexuofD, define theout-degree d+
D(u) (in-degree dD–(u), respectively) ofuto be|∂D+({u})|(|∂D–({u})|, respectively). LetV(D) ={v, . . . ,vn}. The sequence of integer pairs {(d+
D(v),dD–(v)), (d+D(v),dD–(v)), . . . , (d+D(vn),d–D(vn))}is called adegree sequenceofD. For a given sequence d ={(d+
,d–), . . . , (d+n,d–n)}, to determine whether there is a digraphDsuch thatDhas 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 applica-tions in communication networks, structural reliability, stereochemistry, etc.
For a digraphD, if for any ordered pair of vertices (u,v), there is a directed path fromu tov, thenDis said to bestrongly connected. Characterizations for a digraphic sequence and a multi-digraphic sequence with realizations having prescribed strong arc-connectivity
have been studied, see Frank [, ] and Honget al.[]. For more in the literature on degree sequences, see surveys [] and [].
A sequence of integer pairs d ={(d+,d–), . . . , (d+n,dn–)}isdigraphic(multi-digraphic, re-spectively) if there exists a digraph (a multi-digraph, rere-spectively)Dwith degree sequence
d, whereDis called a d-realization. Letdbe the set of all d-realizations. Frank [, ] (see also Theorem . in []) showed thatd =∅if and only ifni=d+i =ni=di–. If a multi-digraphic realization of d is required, then Honget al.[] gave the following char-acterization.
Theorem .(Hong, Liu, Lai) Letd={(d+
,d–), . . . , (d+n,d–n)}be a sequence of non-negative integer pairs.Thendis multi-digraphic if and only if each of the following holds:
(i) ni=d+i =ni=d–i; (ii) fork= , . . . ,n,d+
k≤
i=kdi–.
Furthermore, for a strict digraph, there is a similar result. The following theorem, which can be found in [–] among others, is well known.
Theorem . (Fulkerson-Ryser) Let d ={(d+
,d–), . . . , (d+n,d–n)} be a sequence of non-negative integer pairs with d+≥ · · · ≥d+
n.Then d is strict digraphic if and only if each of the following holds:
(i) d+i ≤n– ,d–i ≤n– for all≤i≤n; (ii) ni=d+
i = n
i=di–;
(iii) ki=d+ i ≤
k
i=min{k– ,d–i}+ n
i=k+min{k,di–}for all≤k≤n.
LetDbe a digraph andkbe an integer. If for any arc setSofDwith|S|<k,G–Sis still strongly connected, thenDis said to bek-arc strongly connected(ork-arc-connected for short). Clearly, -arc connected digraph is also a strongly connected digraph and vice versa. Thearc-connectivityofD, denoted byλ(D), is the maximum integerksuch thatD isk-arc-connected. In [], Honget al.characterized the sequence of pairs of integers d so that there is a strongly connected digraphD∈ d. Also, they gave an example to point out
that to characterize the case whether there is ak-arc-connected digraph indmay be very difficult. In this paper, we consider a multi-digraphic version. We will give a characteriza-tion fork-arc-connected multi-digraphs. Furthermore, we also give a characterization for multi-digraphs with arc-connectivity exactlyk.
In the next section, we will give some tools and methods used in this paper. In Section , we characterize the sequence of pairs of integers to have ak-arc-connected realization. In Section , we characterize the sequence of pairs of integers to have a realization that has arc-connectivity exactlyk. In Section , we give a conclusion of this paper.
2 Methods and tools
In this section, we give a special notation used in this paper that is also the main tool. Let Dbe a digraph and (u,v), (u,v) be two arcs ofD. The-switchofDis an operation to obtain a new digraph D fromD–{(u,v), (u,v)} by adding{(u,v), (u,v)}. The resulting digraphDis often denoted byD⊗ {(u,v), (u,v)}. By this definition,
D⊗(u,v), (u,v)
Thus, the degree sequence remains unchanged under -switch operations. This operation will be the main tool in the arguments of this paper.
Note that in the operation of -switch, the two arcs (u,v), (u,v) may have common ends. For example, ifu=uorv=v, then the resulting digraph is exactly the same as the original digraph. Ifv=uorv=u, then the resulting digraph has loops. So, when this case occurs, we usually use another -switch operation to remove the loops. For ex-ample, assume (x,y), (y,z), (u,v)∈A(D) and (D⊗ {(x,y), (y,z)})⊗ {(y,y), (u,v)}is just the digraphD–{(x,y), (y,z), (u,v)}+{(x,y), (u,y), (y,v)}. After these two -switches, the result-ing digraph still lies ind. In this paper, we will use these operations to obtain a k-arc-connected digraph or a digraph with arc-connectivity exactlykfrom an arbitrary digraph ind.
Let d ={(d+
,d–), . . . , (dn+,d–n)}. By using the tools and the methods above, we obtain a sufficient and necessary condition of d to have ak-arc-connected realization (see The-orem .). Furthermore, if we require the realizationDto have arc-connected exactlyk, then we get Theorem ..
3 Degree sequence fork-arc-connected multi-digraphs
In this section, we shall present a characterization for multi-digraphic sequences with k-arc-connected realizations. We will give some notations used in this section fist.
LetDbe a digraph. For a subsetF⊆V(D), defineF=V(D)\F. A vertex setF⊆V(D) is called anout-fragment(in-fragment, respectively) ofDif|∂D+(F)|=λ(D) (|∂D–(F)|=λ(D), respectively). Both out-fragments and in-fragments are also called fragments ofD. An out-fragment (in-out-fragment, respectively)Fisminimalif any proper subset ofFis no longer an out-fragment (in-fragment, respectively). Let fr+(D) be the number of out-fragments ofD and fr–(D) be the number of in-fragments ofD. As a vertex setFis an out-fragment if and only if its complementFis an in-fragment, fr+(D) = fr–(D). Denote fr(D) = fr+(D) = fr–(D). It is easy to see that fr(D) > for any digraphD. This observation can be used to prove the following theorem.
Theorem . Letd={(d+
,d–), . . . , (dn+,d–n)}be a sequence of integer pairs.Thendhas a k-arc-connected realization if and only if each of the following holds:
(i) ni=d+ i =
n i=di–;
(ii) for each≤j≤n,d+ j,d–j ≥k;
(iii) for each≤j≤n,dj+≤i=jdj–.
Proof If d has ak-arc-connected realization, then by Theorem ., (i) and (iii) hold, and by the definition ofk-arc-connectedness, (ii) holds. So, it suffices to prove the sufficiency. By (i), (iii) and by Theorem .,d =∅. So we may pick a multi-digraphD∈ dsuch that
(a) the arc-connectivityλ(D) is as large as possible.
(b) subject to (a), fr(D) is as small as possible. ()
u,v∈F andu,v∈F such that (u,v)∈A(D[F]) and (u,v)∈A(D[F]). LetD= D⊗ {(u,v), (u,v)}. By (),Dis also a multi-digraph ind.
Claim. IfFis an out-fragment ofD, then one of the following must hold:
(i) |∂D+(F)|=|∂D+(F)|, or
(ii) ∂+
D(F) =∂D+(F)∪ {(u,v)}andF∩ {u,v,u,v}={u,v}, or
(iii) ∂D+(F) =∂D+(F)∪ {(u,v)}andF∩ {u,v,u,v}={u,v}, or
(iv) ∂+
D(F) =∂D+(F) –{(u,v)}andF∩ {u,v,u,v}={u,v}, or
(v) ∂D+(F) =∂D+(F) –{(u,v)}andF∩ {u,v,u,v}={u,v}.
By the definition ofD, we have|∂D+(F)|=|∂D+(F)|if|F∩ {u,v,u,v}| = . So, we may assume|F∩{u,v,u,v}|= . In fact, also by the definition ofD, whenF∩{u,v,u,v} ∈ {{u,u},{v,v}},|∂D+(F)|=|∂D+(F)|still holds. The other cases are illustrated as (ii)-(v). Thus Claim must hold.
Claim.λ(D)≥λ(D).
By contradiction, we assume that D has an out-fragment F with |∂D+(F)|<λ(D). By
Claim and since|∂D+(F)| ≥λ(D), we may assume that{u,v,u,v} ∩F={u,v}. Thus |∂+
D(F)|=|∂D+(F)|+ <λ(D) + . Since|∂D+(F)| ≥λ(D), we have|∂D+(F)|=λ(D), and soFis also an out-fragment ofD. Sinceu∈F∩Fandu∈/F∪F, by a sub-modular inequality, we have
λ(D)≤∂D+(F∩F)+∂D+(F∪F)≤∂D+(F)+∂D+(F)= λ(D),
which impliesF∩Fis also an out-fragment ofD, which contradicts the minimality ofF. This completes the proof of Claim .
By choice ()(a) ofDand by Claim ,λ(D) =λ(D). Then, by Claim ,Fis not an out-fragment inD, and any out-fragmentFofDis still an out-fragment ofDunless either {u,v,u,v} ∩F={u,v}or{u,v,u,v} ∩F={u,v}. If there is such anFsuch that F is an out-fragment inDbut not inD, then without loss of generality we may assume {u,v,u,v} ∩F={u,v}. Thus|∂D+(F)|=|∂D+(F)|+ =λ(D) + . Moreover, by the min-imality ofF andF, we have|∂D+(F∩F)| ≥λ(D) + . Thus, by a sub-modular inequality, we have
λ(D)≤∂D+(F∪F)≤∂D+(F)+∂D+(F)–∂D+(F∩F)
≤λ(D) +λ(D) + –λ(D) + =λ(D).
This implies|∂D–(F∪F)|=|∂D+(F∪F)|=λ(D). Then, by a sub-modular inequality again, we have
∂D–(F∪F∩F)≤∂D–(F)+∂D–(F∩F)–∂D–(F∪F∪F) ≤λ(D) +λ(D) –λ(D) =λ(D),
which implies|∂D–(F∪F∩F)|=λ(D) contradicts to the minimality ofF. Hence, every out-fragment ofDis also an out-fragment ofD. AsFis an out-fragment inDbut not in D, fr(D) < fr(D), which contradicts choice ()(b) ofD. Therefore,Disk-arc-connected,
By definition, the arc-connectivity of a digraph D cannot exceed min{d+D(v),d– D(v) : v∈V(D)}. A digraph Dismaximally arc-connectedif the arc-connectivity ofDequals
min{d+D(v),dD–(v) :v∈V(D)}. Applying Theorem . withk=min{d+, . . . ,d+
n,d–, . . . ,d–n}, we have the following corollary.
Corollary . Letd={(d+,d–), . . . , (d+n,d–n)}be a multi-graphical sequence.Thendis also a degree sequence of some maximally arc-connected multi-digraph.
4 Degree sequence for multi-digraphs with prescribed connectivity
In this section, we consider the degree sequence of multi-digraphs with connectivity ex-actlyk. Our method is to construct a new multi-digraph indfrom ak-arc-connected multi-digraph by reducing the arc-connectivity step by step. Moreover, by Corollary ., we may assume thatk<min≤i≤n{d+i,di–}.
Theorem . Let n≥,k≥be two integers andd={(d+,d–), . . . , (d+n,d–n)}be a sequence of pairs of integers.Denoteδ=min≤i≤n{di+,di–}andδ=min≤i<j≤n{d+i+dj+,d–i +d–j}.Then
dis a degree sequence of some multi-digraph with connectivity exactly k if and only if each of the following hold.
(i) δ≥k;
(ii) ni=d+ i =
n i=di–;
(iii) forj= , . . . ,n,d+ j +α≤
i=jdi–+k,whereα=δifk<δandα=δifk=δ.
Proof First, we consider the necessity. Assume d is the degree sequence of some multi-digraphDwith connectivity exactlyk. By Theorem ., (i) and (ii) hold. Suppose, to the contrary, that (iii) does not hold. Then there is a vertexvj ofDsuch thatd+(vj) =d+j ≥
i=jdi–+k–α+ . It follows thatd+(vj) +d–(vj)≥ n
i=di–+k+ –α=|A(D)|+k+ –α. This implies that there are at mostα–k– arcs not incident withvj. On the other hand, as Dhas connectivityk, there existsX⊆V(D)\ {vj}such that eitherd+(X) =kord–(X) =k. Without loss of generality, we may assume the former. Thend+(X)≥
vi∈Xd+(vi) – (α– k– ). If|X| ≥, thenvi∈Xd+(vi)≥δ
≥α, and thusd+(X)≥k+ , a contradiction. So |X|= and thusk=d+(X)≥δ
, implyingα=k=d+(X) =
v∈Xd+(v). Then againd+(X)≥
vi∈Xd+(vi) – (α–k– ) =k+ , a contradiction. Hence (iii) holds.
Next, we consider the sufficiency. By Theorems . and ., there is ak-arc-connected multi-digraphD∈ d. IfDhas arc-connectivityk, then we are done. So we may assume thatλ(D) >k, then we will construct a multi-digraph indwith arc-connectivity exactly kfromD. First, we need some claims.
Note that
λ(D) =mind+(X)|X⊆V(D),X,V(D)\X=∅
=mind–(X)|X⊆V(D),X,V(D)\X=∅.
By a similarly analysis to Claim in the proof of Theorem ., it is easy to verify the follow-ing claim. In fact, in the tree operations in the followfollow-ing claim,d+(X) andd–(X) decrease at most for any∅ =X⊂V(D). The proof is easy and omitted here.
Claim. Each of the following holds.
(ii) For anyx,y,z,wwith(x,y), (y,z), (z,w)∈A(D),
D–{(x,y), (y,z), (z,w)}+{(x,z), (z,y), (y,w)}has connectivity at leastλ(D) – . (iii) For anyu,v,x,y,zwith(u,v), (x,y), (y,z)∈A(D),
D–{(u,v), (x,y), (y,z)}+{(u,y), (y,v), (x,z)}has arc-connectivity at leastλ(D) – .
Denote by
λ(D) =mind+(X)|X⊆V(D),|X|,V(D)\X≥.
By the definition,λ(D)≥λ(D) for any digraphD. Now, by Theorem ., we may pick a multi-digraphD∈ dsuch that
(a)Disk-arc-connected.
(b) subject to (a),λ(D) is as small as possible. ()
Then we will construct a new digraph fromDthat meets our requirements.
By the choice ofD,λ(D)≥k. Ifλ(D) =k, then we are done andDis required. So we may assume thatλ(D)≥k+ . By the definition, letX⊆V(D) so thatd+(X) =λ(D).
Claim. For any two arcs (x,y)∈∂+(X), (y,x)∈∂–(X), eitherx=xory=y. Suppose, to the contrary, thatx=xandy=y. LetD=D⊗ {(x,y), (y,x)}and then by Claim ,D∈ dwith arc-connectivity at leastλ(D) – ≥kandλ(D)≤d+(X) – =
λ(D) – , a contradiction to choice () ofD.
Claim. For any two arcs (x,y), (x,y)∈∂+(X) (or∂–(X)), eitherx=xory=y. Suppose, to the contrary, thatx=xandy=yand, without loss of generality, we may assume that (x,y), (x,y)∈∂+(X). Asλ(D)≥k+ ≥,∂–(X)=∅. Let (y,x)∈∂–(X). Then, by Claim , eitherx=x,y=yorx=x,y=y. By symmetry, we may assume the former. LetD=D–{(x,y), (y,x), (x,y)}+{(x,x), (x,y), (y,y)}. By Claim (ii), D∈ dand has arc-connectivity at leastλ(D) – ≥k. However,λ(D) <|∂+(X)|=λ(D), a contradiction to the choice ofD. Claim is proved.
By Claim and Claim , it is easy to see that all arcs leaving from or interring toXare incident with a vertex, sayx. We only consider the casex∈X, and the other case thatx∈/X can be dealt with similarly.
Claim. We may assume thatX\ {x}is an independent set ofD.
Suppose, to the contrary, that there is an edge (x,x)∈A(D[X\ {x}]), then picky,y∈/ X such that (y,x), (x,y)∈A(D). If y =y, then let D =D–{(x,x), (y,x), (x,y)}+ {(x,x), (x,x), (y,y)}and thusD∈ d. By Claim (ii),λ(D)≥λ(D) – ≥kandλ(D) <
λ(D), a contradiction to choice () ofD. Soy=y. By the arbitrariness ofy,y, there is y∈/Xsuch that all arcs leaving from or interring toXare incident withy. LetY=V(D)\X andY \ {y}is an independent set; otherwise, if there exists (y,y)∈A(D[Y]), then let D=D–{(x,x), (y,y), (x,y), (y,x)}+{(x,x), (x,x), (y,y), (y,y)}, and it is easy to see that Dhas arc-connectivity at leastλ(D) – ≥kandλ(D) <λ(D), a contradiction to choice () ofD. SoY\ {y}is an independent set. Thus, we may renameY,yasX,xand Claim follows.
If Dm has arc-connectivity at mostk, then by Claim there existsisuch thatDihas arc-connectivity exactlyk, and we are done. So we may assume thatDm is (k+ )-arc-connected. Thenm=|A(D–X)|<δx; otherwise, ifm=d+(x) +d+(x), then∂Dm+ ({x,x}) = ∅, a contradiction to the assumption thatDmis (k+ )-arc-connected. A similar contradic-tion is obtained whenm=d–(x) +d–(x). Thusm=|A(D–X)|and thenV(D)\ {x,x,x} is an independent set inDm.
If k=δ, thenα =δ=k and by (iii), dj+≤
i=jd–i, and the result holds by Corol-lary .. So we may assume that k <δ and thus α =δ. Let u,v be two vertices so that δ =min{d+(u) +d+(v),d–(u) +d–(v)}. If x∈ {u,v}, then δ <min{d+(x),d–(x)} ≤
min{d–(x) +d–(x),d+(x) +d+(x)}, a contradiction. Sox∈ {u,/ v}. Then continue to con-struct the sequence of digraphsD, . . . ,Dm,Dm+, . . . ,Dmsuch that fori=m+ , . . . , m,Di is obtained fromDi–by replacing an arc betweenx,xwith a dipath of length between x,xand replacing a dipath of length betweenu, with an arc betweenu,v. Then, similarly to the above, we may assume thatDm∈ dis (k+ )-arc-connected andV(D)\ {x,u,v}is an independent set inDm.
Moreover, by (ii) and (iii), for anyj,d+
j +d–j +α≤ n
i=d–i +k= n
i=d+i +k, and thus dj–+α≤i=jd–i +k. So, by symmetry, we may assume thatd+(u) +d+(v)≤d–(u) +d–(v). Thusδ=d+(u) +d+(v). It follows that
w=x
d–(w) –d+(x) =ADm {u,v}
=d+(u) +d+(v) –∂Dm
{u,v} ≤α–k– .
This implies that there isjsuch thatd+ j +α≥
i=jdi–+k+ , a contradiction to (iii). The
proof is completed.
If Theorem .(i), (ii) holds, then by Theorem . d =∅. Furthermore, if Theo-rem .(iii) does not hold, then there are no digraphs in·that have arc-connectivity exactlyk. In other words, all digraphs indare (k+ )-arc-connected.
Corollary . Letd={(d+,d–
), . . . , (dn+,d–n)}be a sequence of integer pairs.Denoteδ=
min≤i≤n{d+
i,d–i}andδ=min≤i<j≤n{d+i +d+j,di–+d–j}.If each of the following holds,then any digraphs indare k-arc-connected.
(i) δ≥k;
(ii) ni=d+i =ni=di–;
(iii) there exists somejsuch thatd+ j +α≥
i=jd–i +k,whereα=δifk<δandα=δif k=δ.
5 Conclusions
In this paper, sufficient and necessary conditions for a sequences of pairs of integers have been studied. For a sequence d ={(d+
and necessary condition of d to have a realizationDthat has arc-connectivity exactlyk. These results extend a similar result from undirect graphs into directed graphs.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11326214) and the Natural Science Foundation of Fujian Province (2014J05004).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The first author has proposed the motivations of the manuscript; the second author has proved the convergence result. All authors read and approved the final manuscript.
Author details
1College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, P.R. China.2Center for Discrete Mathematics, Fuzhou University, Fuzhou, 350108, P.R. China.
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