On Frank's conjecture on k-connected orientations
OOn Frank’s conjecture on k -connectedorientations Olivier Durand de Gevigney ∗† October 30, 2018
Abstract
We disprove a conjecture of Frank [4] stating that each weakly2 k -connected graph has a k -vertex-connected orientation. For k ≥ k -vertex-connected orientation is NP-complete. Introduction An orientation of an undirected graph G is a digraph obtained from G bysubstituting an arc uv or vu for every edge uv in G . We are interestedin characterizing graphs admitting an orientation that satisfies connectivityproperties. Robbins [10] proved that a graph G admits a strongly connectedorientation if and only if G is 2-edge-connected. The following extension tohigher connectivity follows from of a result of Nash-Williams [9]: a graph G admits a k -arc-connected orientation if and only if G is 2 k -edge-connected.Little is known about vertex-connected orientations. Thomassen [12] con-jectured that if a graph has sufficiently high vertex-connectivity then it ad-mits a k -vertex-connected orientation. Conjecture 1 (Thomassen [12]) . For every positive integer k there existsan integer f ( k ) such that every f ( k ) -connected graph admits a k -connectedorientation. ∗ Laboratoire G-SCOP, CNRS, Grenoble-INP, UJF, France † This research was conducted while the author was visiting the University of Waterloo.The author was supported by a grant Explora Doc from Rhˆone-Alpes and NSERC grantNo. OGP0138432. a r X i v : . [ m a t h . C O ] D ec he case k = 2 has been proved by Jord´an [5] by showing f (2) ≤ f (2) ≤
14. However, the conjecture ofThomassen remains open for k ≥ G = ( V, E ) is called weakly k -connected if | V | > k and for all U ⊆ V and F ⊆ E such that 2 | U | + | F | < k , the graph G − U − E isconnected. It is easy to see that any graph admitting a k -connected orien-tation is weakly 2 k -connected. Note that checking the weak 2 k -connectivityof a graph can be done in polynomial time using a variation of the Max-flowMin-cut algorithm [3]. Frank [4] conjectured that this connectivity conditioncharacterizes graphs admitting a k -connected orientation. Conjecture 2 (Frank [4]) . A graph G admits a k -connected orientation ifand only if G is weakly k -connected. Berg and Jord´an [1] proved this conjecture for the special case of Euleriangraphs and k = 2. For a short proof of this result, see [7]. In this article wedisprove this conjecture for k ≥
3. For instance, the graph G in Figure 1 is acounterexample for k = 3. We also prove that deciding whether a given graphhas a k -connected orientation is NP-complete for k ≥
3. Both these resultshold also for the special case of Eulerian graphs. Hence assuming P (cid:54) = N P ,there is no good characterisation of graphs admitting a k -connected orienta-tion for k ≥
3. We mention that counterexamples can easily be derived fromour NP-completeness proof, but we give simple self-contained counterexam-ples. Furthermore, the gadgets used in the NP-completeness proof are basedon properties used in our first counterexample.This paper is organized as follows. In Section 1 we establish the necessarydefinitions and some elementary results. In Section 2 we disprove Conjecture2 for k ≥
3. For k ≥
4, we provide Eulerian counterexamples. In Section 3,for k ≥
3, we reduce the problem of
Not-All-Equal -Sat to the problemof finding a k -connected orientation of a graph. This reduction leads to anEulerian counterexample of Conjecture 2 for k = 3. Let k be a positive integer and let D = ( V, A ) be a digraph. We mentionthat digraphs may have multiple arcs. In D the indegree (respectively, theoutdegree) of a vertex v is denoted by ρ D ( v ) (respectively, by δ D ( v )). Thepair u, v ∈ V is called strongly connected if there exist a dipath from u to v and a dipath from v to u . The digraph D is called strongly connectedif every pair of vertices is strongly connected. The pair u, v ∈ V is called k -connected if, for all U ⊆ V \ { u, v } such that | U | < k , u and v are strongly2onnected in the digraph D − U . A set of vertices is called k -connected ifevery pair of vertices contained in this set is k -connected. The digraph D iscalled k -connected if | V | > k and V is k -connected.Let G = ( V, E ) be a graph. We mention that graphs may have multipleedges. In G the degree of a vertex v is denoted by d G ( v ) and the number ofedges joining v and a subset U of V − v is denoted by d G ( v, U ). The pair u, v ∈ V is called connected if there is a path joining u and v . The pair u, v ∈ V is called weakly k -connected if, for all U ⊆ V \ { u, v } and F ⊆ E such that 2 | U | + | F | < k , u and v are connected in the graph G − U − F . Aset of vertices is called weakly 2 k -connected if every pair of vertices containedin this set is weakly 2 k -connected. So G is weakly 2 k -connected if | V | > k and V is weakly 2 k -connected.The constructions in this paper are based on the following facts. Proposition 1.
Let G = ( V, E ) be a graph admitting a k -connected orien-tation D . Let v be a vertex of degree k and u (cid:54) = v be a vertex such that d G ( u, v ) = 2 . Then ρ D ( v ) = δ D ( v ) = k and the two parallel edges between u and v have opposite directions in D .Proof. By k -connectivity of D , the indegree (respectively, the outdegree) of v is at least k . Hence, since 2 k = d G ( v ) = ρ D ( v ) + δ D ( v ) we have ρ D ( v ) = δ D ( v ) = k. Now suppose for a contradiction that the two parallel edgesbetween u and v have the same direction, say from u to v . Then the set ofvertices that have an outgoing arc to v is smaller than k and deleting thisset results in a digraph that is not strongly connected, a contradiction.For U ⊂ V , a pair of dipaths of D (respectively, paths of G ) is called U -disjoint if each vertex of U is contained in at most one dipath (respectively,path). Let X and Y be two disjoint vertex sets. A k -difan from X to Y (respectively, a k -fan joining X and Y ) is a set of k pairwise U -disjointdipaths from X to Y (respectively, paths joining X and Y ) where U is definedby U = V \ ( X ∪ Y ) if | X | = | Y | = 1, U = V \ X if | X | = 1 and | Y | > U = V \ Y if | Y | = 1 and | X | > U = V if | X | > | Y | > u, v of vertices of D is k -connected if andonly if there exist a k -difan from u to v and a k -difan from v to u . Let X bea k -connected set of at least k vertices and let v be a vertex in V \ X suchthat there exist a k -difan from X to v and a k -difan from v to X ; then, it iseasy to prove that X ∪ v is k -connected.Kaneko and Ota [6] showed that a pair u, v of vertices of G is weakly2 k -connected if and only if there exist 2 edge-disjoint k -fans joining u and v . Let X be a weakly 2 k -connected set of at least k vertices and let v be a3ertex in V \ X such that there exist 2 edge-disjoint k -fans joining v and X ;then, it is easy to prove that X ∪ v is weakly 2 k -connected. Let X and Y be two disjoint weakly 2 k -connected sets each of at least k vertices such thatthere exist 2 edge-disjoint k -fans joining X and Y ; then, it is easy to provethat X ∪ Y is weakly 2 k -connected. We first disprove Conjecture 2 for k = 3 and then extend the idea of the proofto higher connectivity. We recall that G is the graph defined in Figure 1. u a t a v a w a xu b t b v b w b y AB Figure 1: G every thick and red edge represents a pair of parallel edges andblack edges represent simple edges. Proposition 2.
The graph G is weakly -connected and has no -connectedorientation.Proof. First we show that G is weakly 6-connected. Observe that there exist2 edge-disjoint 3-fans joining any pair of vertices in A \ w a . Then, note thatthere exist 2 edge-disjoint 3-fans joining w a and A \ w a . Hence A is weakly6-connected. Symmetrically B is also weakly 6-connected. There exist 2edge-disjoint 3-fans joining A and B so A ∪ B is weakly 6-connected. Thereexists 2 edge-disjoint 3-fans joining x (respectively, y ) and A ∪ B . It followsthat G is weakly 6-connected.Suppose for a contradiction that G has a 3-connected orientation D .Note that every pair of parallel edges is incident to a vertex of degree 6 andthe maximal edge multiplicity is 2 . Hence, by Proposition 1, the two edges inevery parallel pair have opposite directions in D . Thus, in D the orientationof the edges of the path u a v a w b yxw a v b u b results in a directed path from u a u b or from u b to u a . In particular both v a w b and v b w a are directed from A to B or from B to A . In both cases D − { x, y } is not strongly connected,a contradiction.We mention that G is not a minimal counterexample. Indeed the graph H obtained from G by deleting the two vertices t a and t b and adding thenew edges u a v a , v a y , yu a , u b v b , v b x and xu b is weakly 6-connected but has no3-connected orientation. (Suppose that H has a 3-connected orientation D .Then, by Proposition 1, in D the orientation of the edges of the two triangles v a yw b and v b xw a results in circuits. Considering the cut { x, y } , we see thatthose circuits must be either both clockwise or both counterclockwise, sayclockwise. Hence, by Proposition 1, in D the orientation of the path u a yxu b results in a dipath from u a to u b or from u b to u a . In the first case D −{ y, v b } is not strongly connected, in the other case D − { x, v a } is not stronglyconnected.)We now extend this construction to higher connectivity. Let k ≥ G k = ( V, E ) as follows (see Figure 2). Let a zy xw b c ABC
Figure 2: G k every thick and red edge represents a pair of parallel edges andblack edges represent simple edges. n ≥ k be an odd integer. The vertex set V is the union of the pairwisedisjoint sets A , B , C and { w, x, y, z } where | A | = | B | = n and | C | = k − A and B induces a completesimple graph. Choose arbitrarily one vertex from each of A , B and C , say a ∈ A , b ∈ B and c ∈ C and add the cycle azyxwbc . By the choice of n , wecan now add pairs of parallel edges between vertices in A ∪ B \ { a, b } and5 ∪ { w, x, y, z } such thateach vertex of A ∪ B is incident to at most one pair of parallel edges, d G k ( v, A ) = d G k ( v, B ) = 2 (cid:100) k (cid:101) for all v ∈ C − c , d G k ( c, A ) = d G k ( c, B ) = 2 (cid:100) k (cid:101) + 1 ,d G k ( w, A ) = d G k ( z, B ) = 2 k − ,d G k ( y, A ) = d G k ( x, B ) = 2 and d G k ( x, A ) = d G k ( y, B ) = 2 k − . Proposition 3.
Let k ≥ be an integer. The graph G k is Eulerian, weakly k -connected and has no k -connected orientation.Proof. Since n is odd, both of the complete graphs induced by A and B areEulerian. Hence G k , which is obtained from those graphs by adding a cycleand parallel edges, is Eulerian. Since k ≥ n ≥ k ≥ k + 2 thus both of thecomplete graphs induced by A and B are weakly 2 k -connected. Note thatthere exist 2 edge-disjoint k -fans joining A and B (one uses C ∪ { w, x, y } theother one uses C ∪ { x, y, z } ), thus A ∪ B is weakly 2 k -connected. Note alsothat, for any vertex v ∈ C ∪ { w, x, y, z } , there exist 2 edge-disjoint k -fansjoining v and A ∪ B. Hence, G k is weakly 2 k -connected.Suppose for a contradiction that G k has a k -connected orientation D .Since d G k ( w ) = d G k ( x ) = d G k ( y ) = d G k ( z ) = 2 k and by Proposition 1, theorientation of the set of simple edges of the path azyxwb results in the dipath azyxwb or the dipath bwxyza . In both cases, D − ( C ∪ { x, y } ) is not stronglyconnected, a contradiction.Note that with a slightly more elaborate construction we can obtain acounterexample such that | V | = O ( k ). In this section we prove the following result.
Theorem 1.
Let k ≥ be an integer. Deciding whether a graph has a k -connected orientation is NP-complete. This holds also for Eulerian graphs. A reorientation of a digraph D is a digraph obtained from D by reversinga subset of arcs. Obviously, the problem of finding a k -connected orientationof a graph and the problem of finding a k -connected reorientation of a di-graph are equivalent. For convenience we prove the NP-completeness of the6econd problem by giving a reduction from the problem of Not-All-Equal -Sat which is known to be NP-complete [11].Let Π be an instance of Not-All-Equal -Sat and let k ≥ D k = D k (Π) = ( V, A ) such that thereexists a k -connected reorientation of D k if an only if there is an assignmentof the variables which satisfies Π.The construction of D k associates to each variable x a circuit ∆ x and toeach pair ( C , x ) where x is a variable that appears in the clause C a special arc e Cx (see Figure 3). A reorientation of D k is called consistent if the orientationof parallel edges is preserved and, for each variable x , the orientations ofthe special arcs of type e Cx and the circuit ∆ x are either all preserved or allreversed. A consistent reorientation of D k defines a natural assignment of thevariables in which a variable x receives value true if ∆ x is preserved and false if ∆ x is reversed. We define reciprocally a natural consistent reorientation from an assignment of the variables. t Cx u (cid:48) Cx u Cx t C (cid:48) x u (cid:48) C (cid:48) x u C (cid:48) x v x t Cy u (cid:48) Cy u Cy t C (cid:48) y u (cid:48) C (cid:48) y u C (cid:48) y v y t Cz u (cid:48) Cz u Cz t C (cid:48) z u (cid:48) C (cid:48) z u C (cid:48) z v z w C w C (cid:48) ∆ x ∆ y ∆ z e Cx e Cy e Cz e C (cid:48) x e C (cid:48) y e C (cid:48) z Figure 3: Representation of the circuits and the special arcs of D (Π) whereΠ is composed of the clauses C = ( x , y , z ) and C (cid:48) = ( x , y , z ). The dashedboxes represent the clause-variable gadgets.For each clause C we construct a C -gadget (see Figure 4) that uses thespecial arcs associated to C . The purpose of the C -gadgets is to obtain thefollowing property. Proposition 4.
An assignment of the variables satisfies Π if and only if itdefines a natural consistent k -connected reorientation. For each pair ( C , x ) where C is a clause and x is a variable that appears7n C we define a ( C , x )-gadget (see Figure 5) which links the orientation of∆ x to the orientation of e Cx . We will prove the following fact. Proposition 5.
If there exists a k -connected reorientation of D k then thereexists a consistent k -connected reorientation of D k . w C u Cy u Cx u Cz e Cy e Cx e Cz L Figure 4: A clause gadget for k =3 and C = ( x , y , z ). Each red andthick edge represents a pair of par-allel arcs in opposite directions. u (cid:48)(cid:48) Cx u Cx u (cid:48)(cid:48)(cid:48) Cx u (cid:48) Cx t Cx v Cx m w C e Cx f Cx U Cx N Figure 5: A ( C , x )-gadget for k = 3and x ∈ C . Each red and thickedge represents a pair of parallelarcs in opposite directions.Let L be a set of k − C composed of the variables x , y and z we add the vertices w C , u Cx , u C y , u Cz . We add arcs such that L ∪ w C induces a complete digraph.We add the special arc w C u Cx if x ∈ C and the special arc u Cx w C if x ∈ C .This special arc is denoted by e Cx . We define similarly the special arcs e Cy and e Cz . This ends the construction of the C -gadget. Let W denote the set of allvertices of type w C .Let M be a set of k − m ∈ M . For each pair ( C , x ) where C is a clause and x is a variable thatappears in C we add the new vertices t Cx , u (cid:48) Cx , u (cid:48)(cid:48) Cx , u (cid:48)(cid:48)(cid:48) Cx , v Cx and denote U Cx = { u Cx , u (cid:48) Cx , u (cid:48)(cid:48) Cx , u (cid:48)(cid:48)(cid:48) Cx } . We add arcs such that M ∪ ( U Cx \ u Cx ) induces a completedigraph. We add pairs of parallel arcs in opposite directions between thepairs of vertices ( v Cx , t Cx ), ( t Cx , u (cid:48)(cid:48)(cid:48) Cx ), ( u (cid:48)(cid:48)(cid:48) Cx , u (cid:48) Cx ), ( u (cid:48) Cx , u (cid:48)(cid:48) Cx ) and all the pairs oftype ( t Cx , m (cid:48) ) and ( u (cid:48) Cx , m (cid:48) ) for each m (cid:48) ∈ M \ m . Note that, so far, theundirected degree of t Cx and u (cid:48) Cx is 2 k −
2. We add an arc t Cx u (cid:48) Cx if x ∈ C andan arc u (cid:48) Cx t Cx if x ∈ C . Call this arc f Cx . The definition of the ( C , x )-gadget isconcluded by the following definition of the circuit ∆ x .8or each variable x define a new vertex v x and add arcs such that v x andthe set of vertices of type t Cx and u (cid:48) Cx induce a circuit ∆ x that traverses (inarbitrary order) all the ( C , x )-gadgets such that C is a clause containing x .In this circuit connect a ( C , x )-gadget to the next ( C (cid:48) , x )-gadget by adding anarc leaving the head of f Cx and entering the tail of f C (cid:48) x (see Figure 3). Notethat now the undirected degree of t Cx and u (cid:48) Cx is 2 k .We denote by N the union of L , M and all the vertices of type v x or v Cx . To conclude the definition of D k we add edges such that N induces acomplete digraph.The proof of Proposition 5 follows from the construction of the ( C , x )-gadgets. Proof of Proposition 5.
Let D (cid:48) be a k -connected reorientation of D k and let x be a variable. Observe that all the vertices incident to ∆ x except v x are ofdegree 2 k and incident to k − x is either preserved or reversed. Let C be a clause in which x appears.In D (cid:48) − ( M ∪ t Cx ) exactly one arc enters U Cx and exactly one arc leaves U Cx (see Figure 5). One of these arcs belongs to ∆ x and the other is the specialarc e Cx . Hence, by k -connectivity of D (cid:48) , e Cx is reversed if and only if ∆ x isreversed.If there exists a pair of parallel arcs in the same direction in D (cid:48) thenreversing the orientation of one arc of this pair preserves the k -connectivity.Hence we may assume that in D (cid:48) the orientation of parallel edges is preserved.The following fact follows easily from the definition of D k . We recall that W is the set of vertices of type w C . Proposition 6.
In every consistent reorientation of D k the set V \ W is k -connected.Proof. Let D (cid:48) be a consistent reorientation of D k . Clearly N is k -connected.Let C be a clause and x be a variable that appears in C . The circuit C x contains a dipath from (respectively, to) t Cx to (respectively, from) v x that isdisjoint from M ∪ v Cx . Hence N ∪ t Cx is k -connected.We may assume without loss of generality that, in D (cid:48) − ( M ∪ t Cx ), thespecial arc e Cx enters U Cx and an arc of ∆ x leaves U Cx . Let u be a vertex of U Cx .Observe that there is a k -difan from u to M ∪ t Cx ∪ v x (the dipath to v x usesarcs of ∆ x ). Observe that there is a k -difan from M ∪ t Cx ∪ L to u (the dipathfrom L uses the arc e Cx ). Hence, since M and L are subsets of N , N ∪ t Cx ∪ U Cx is k -connected and the proposition follows.9e can now prove Proposition 4. Proof of Proposition 4.
Let Ω be an assignment of the variables and D (cid:48) thenatural consistent reorientation of D k defined by Ω. Let e Cx be a special arcassociated to a clause C and a variable x . In D (cid:48) , the arc e Cx leaves w C if andonly if x = true and x ∈ C or x = false and x ∈ C . And, in D (cid:48) , the arc e Cx enters w C if and only if x = true and x ∈ C or x = false and x ∈ C . Hence C contains a true (respectively, false ) value if and only if there exists a specialarc leaving (respectively, entering) w C in D (cid:48) . Thus a clause C is satisfied byΩ if and only if w C is left by at least one special arcand entered by at least one special arc. ( (cid:63) )Observe that, for each clause C , the only arcs incident to w C in D (cid:48) − L arespecial. Since | L | = k −
1, if D (cid:48) is k -connected then ( (cid:63) ) holds for all clausesthus Ω satisfies Π. Conversely, if Ω satisfies Π then for every clause C ( (cid:63) )holds and w C has at least k out-neighbors and at least k in-neighbors. Thusby Proposition 6 D (cid:48) is k -connectedDenote by G (cid:48) k = G (cid:48) k (Π) the underlying undirected graph of D k (Π). Wecan now prove the main theorem of this section. Proof of Theorem 1.
By Propositions 5 and 4, G (cid:48) k (Π) has a k -connected ori-entation if and only if there exists an assignment satisfying Π. Since theorder of G (cid:48) (Π) is a linear function of the size of Π and Not-All-Equal -Sat is NP-complete [11] this proves the first part of Theorem 1.Observe that in G (cid:48) k the only vertices of odd degree are of type u Cx and w C . Let l be an arbitrary vertex of L . We can add a set F of edges of type u Cx m, ml, lw C such that G (cid:48) k + F is Eulerian. Observe that for any orientationof F , Propositions 5 and 4 still hold for D k + F . This proves the second partof Theorem 1.The following fact shows that G (cid:48) k (Π) is a counterexample to Conjecture2 if Π is not satisfiable. Proposition 7.
The graph G (cid:48) k (Π) is weakly k -connected.Proof. By Proposition 6, V \ W is k -connected in D k , thus V \ W is weakly2 k -connected in G (cid:48) k . Since there exist 2 edge-disjoint k -fans from w C to V \ W for every clause C , G (cid:48) k is weakly 2 k -connected.10e now construct an Eulerian counterexample to Conjecture 2 for k = 3.Let x be a variable and C = ( x , x ) be a clause. Let H (cid:48) be the Eulerian graphobtained from G (cid:48) ( { C } ) by adding an edge u Cx m in each of the two copies ofthe ( C , x )-gadget. The next result follows from the discussion above. Proposition 8. H (cid:48) is an Eulerian weakly -connected graph that has no -connected orientation. Acknowledgement
I thank Joseph Cheriyan for inviting me to the University of Waterloo andfor his profitable discussions. I thank Zolt´an Szigeti who observed that thegraph H obtained from G by complete splitting-off on t a and on t b is asmaller counterexample. I thank both of them and Abbas Mehrabian forcareful reading of the manuscript. References [1] A. R. Berg and T. Jord´an. Two-connected orientations of Euleriangraphs.
Journal of Graph Theory , 52(3):230–242, 2006.[2] J. Cheriyan, O. Durand de Gevigney, and Z. Szigeti. Packing of rigidspanning subgraphs and spanning trees. Submitted to
Journal of Com-binatorial Theory, Series B .[3] L. R. Ford and D. R. Fulkerson.
Flows in Networks . Princeton Univ.Press, 1962.[4] A. Frank. Connectivity and network flows. In
Handbook of Combina-torics , pages 111–177. MIT Press, 1995.[5] T. Jord´an. On the existence of k edge-disjoint 2-connected spanningsubgraphs.
Journal of Combinatorial Theory, Series B , 95(2):257–262,2005.[6] A. Kaneko and K. Ota. On minimally ( n, λ )-connected graphs.
Journalof Combinatorial Theory, Series B , 80(1):156 – 171, 2000.[7] Z. Kir´aly and Z. Szigeti. Simultaneous well-balanced orientations ofgraphs.
Journal of Combinatorial Theory, Series B , 96(5):684–692, 2006.[8] K. Menger. Zur allgemeinen kurventheorie.
Fundamental Mathematics ,pages 96–115, 1927. 119] C. St. J. A. Nash-Williams. On orientations, connectivity and odd-vertex-pairings in finite graphs.
Canadian Journal Mathematics , pages555–567, 1960.[10] H. E. Robbins. A theorem on graphs, with an application to a problem oftraffic control.
The American Mathematical Monthly , 46(5):pp. 281–283,1939.[11] T. J. Schaefer. The complexity of satisfiability problems. In
ACM Sym-posium on Theory of Computing , pages 216–226, 1978.[12] C. Thomassen. Configurations in graphs of large minimum degree, con-nectivity, or chromatic number.