Strong Digraphs - BUKU GRAPHS & DIGRAPH PDF

Chapter 4 Digraphs

While there are many concepts in digraphs that are analogues to concepts we have encountered in graphs, there are also concepts that are quite unique to digraphs. We consider many of these in this current chapter, beginning with the most studied type of connectedness for digraphs.

150 CHAPTER 4. DIGRAPHS in D. By Theorem 4.1,D contains both a ui−uj path and auj−ui path in D, and soD is strong.

Conversely, assume that D is a nontrivial strong digraph with V(D) = {v₁, v₂, . . . , vn}and consider the cyclic sequencev₁, v₂, . . . , vn, vn+1=v₁. Since Dis strong,Dcontains avi−vi+1pathPifori= 1,2, . . . , n. Then the sequence P₁, P₂, . . . , Pn of paths produces a closed spanning walk inD.

TheconverseD~ of a digraphDis obtained fromD by reversing the direction of every arc ofD. ThusD is strong if and only if its converseD~ is strong (see Exercise 2).

Robbins’ Theorem

Recall that an orientation of a graphGis a digraph obtained by assigning a direction to each edge of G. Herbert E. Robbins (1922–2001) studied those graphs having a strong orientation. Certainly, ifGhas a strong orientation, then Gmust be connected. Also, ifGhas a bridge, then it is impossible to produce a strong orientation ofG. Robbins [190] showed that ifGis a bridgeless connected graph, thenGalways has a strong orientation.

Theorem 4.3 (Robbins’ Theorem) A nontrivial graph G has a strong orientation if and only if Gis2-edge-connected.

Proof. We have already observed that if a graphGhas a strong orientation, then G is 2-edge-connected. Suppose that the converse is false. Then there exists a 2-edge-connected graphGthat has no strong orientation. Among the subgraphs ofG, letH be one of maximum order that has a strong orientation.

Such a subgraph exists since for eachv ∈V(G), the subgraphG[{v}] trivially has a strong orientation. Thus |V(H)|<|V(G)|, since, by assumption, G has no strong orientation.

Assign directions to the edges ofHso that the resulting digraphDis strong, but assign no directions to the edges of G−E(H). Let u ∈ V(H) and let v ∈ V(G)−V(H). Since G is 2-edge-connected, it follows by Theorem 2.43 thatGcontains two edge-disjointu−vpaths. LetPbe one of theseu−vpaths and let Qbe the v−upath that results from the otheru−v path. Further, letu₁ be the last vertex ofP that belongs toH, and letv₁ be the first vertex of Qbelonging to H. Next, letP₁ be theu₁−v subpath of P and let Q₁ be thev−v₁ subpath of Q. Direct the edges ofP₁ fromu₁ towardv, producing the directed pathP₁^′ and direct the edges of Q₁ from v toward v₁, producing the directed pathQ^′₁.

Define the digraphD^′ by

V(D^′) =V(D)∪V(P₁^′)∪V(Q^′₁) andE(D^′) =E(D)∪E(P₁^′)∪E(Q^′₁).

SinceD is strong, so too isD^′, contradicting the choice of H.

As we mentioned, Theorem 4.3 is due to Robbins. The paper in which this theorem appears is titled “A theorem on graphs, with an application to a prob- lem of traffic control” and was published in 1939 in theAmerican Mathematical Monthly, only a year after Robbins received his Ph.D. from Harvard University in topology, under the direction of Hassler Whitney. This was only Robbins’

second publication of what would become a long and impressive list. Also in 1939, at age 24, Robbins began work on the classic bookWhat is Mathematics?

with Richard Courant. Robbins classified this book as a literary work rather than a scientific work. This book discussed mathematics as it existed at that time. A few years later Robbins became interested in and devoted his research to mathematical statistics, in which he made major contributions.

Eulerian Digraphs

Eulerian and Hamiltonian graphs have natural analogues for digraphs. In both instances, these are strong digraphs. AnEulerian circuitin a connected digraphD is a circuit that contains every arc ofD (necessarily exactly once);

while anEulerian trailinD is an open trail that contains every arc ofD. A connected digraph that contains an Eulerian circuit is anEulerian digraph.

The next theorem gives a characterization of Eulerian digraphs whose statement and proof are similar to that of Theorem 3.1 (see Exercise 7).

Theorem 4.4 Let D be a nontrivial connected digraph. Then D is Eulerian if and only if odv= idv for every vertex v ofD.

With the aid of Theorem 4.4, a characterization of digraphs containing an Eulerian trail can be given (see Exercise 9).

Theorem 4.5 Let D be a nontrivial connected digraph. Then D contains an Eulerian trail if and only if D contains two verticesuandv such that

odu= idu+ 1 and idv= odv+ 1,

while odw = idw for all other vertices w of D. Furthermore, each Eulerian trail of D begins atuand ends at v.

Thus the digraphD₁of Figure 4.1 contains an Eulerian circuit,D₂contains an Eulerian u−v trail and D₃ contains neither an Eulerian circuit nor an Eulerian trail.

Suppose thatDis a connected digraph containing two verticesuandvsuch that odu−idu= idv−odv=kfor some nonnegative integerkand odw= idw for all w ∈V(D)− {u, v}. Ifk = 0, then D is Eulerian; while ifk = 1, then D contains an Eulerian u−v trail. More generally, for k ≥ 1, we have the following.

Theorem 4.6 If D is a connected digraph containing two vertices u and v such that

152 CHAPTER 4. DIGRAPHS

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D₂ D₃

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Figure 4.1: Eulerian circuits and trails in digraphs odu= idu+k and idv= odv+k

for some positive integer k andodw= idw for all other verticesw ofD, then D containsk arc-disjointu−v paths.

Proof. LetD^′ be the digraph constructed fromD by addingk new vertices w₁, w₂, . . . , wk and the 2karcs (v, wⁱ) and (wⁱ, u) for 1≤i≤k. Since odx= idxfor every vertexxofD^′, the digraphD^′is Eulerian and therefore contains an Eulerian circuitC. Suppose thatCbegins (and ends) atu. Since odu−idu=k and idv−odv =k, it follows thatC containsk arc-disjointu−v trails. By Theorem 4.1, for each suchu−v trailT, there is a u−v path each of whose arcs belongs to T. Since none of these paths contains any of the vertices wi

(1≤i≤k),Dcontainskarc-disjointu−v paths.

Hamiltonian Digraphs

A digraphD isHamiltonianifD contains a spanning cycle. Such a cycle is called aHamiltonian cycleofD. As with Hamiltonian graphs, no characterization of Hamiltonian digraphs exists. Indeed, if anything, the situation for Hamiltonian digraphs is even more complex than it is for Hamiltonian graphs.

Where there are sufficient conditions for a digraph to be Hamiltonian, these are analogues of the simpler sufficient conditions for Hamiltonian graphs.

The following result of Henri Meyniel [158] gives a sufficient condition (much like that in Theorem 3.6 for graphs) for a digraph to be Hamiltonian.

Theorem 4.7 (Meyniel’s Theorem) If D is a nontrivial strong digraph of order nsuch that

degu+ degv≥2n−1

for every pairu, v of nonadjacent vertices, thenD is Hamiltonian.

Theorem 4.7 has a large number of consequences. We consider these now, beginning with a result originally discovered by Douglas Woodall [242].

Corollary 4.8 If D is a nontrivial digraph of ordern such that odu+ idv≥n

whenever uandv are distinct vertices and(u, v)∈/ E(D), then D is Hamilto- nian.

The following theorem is due to Alain Ghouila-Houri [94]. The proof is an immediate consequence of Theorem 4.7.

Corollary 4.9 If D is a strong digraph of order n such that degv ≥ n for every vertex v of D, then D is Hamiltonian.

Corollary 4.9 also has a corollary. We provide a proof of this result.

Corollary 4.10 If D is a digraph of order nsuch that odv≥n/2 and idv≥n/2 for every vertex v ofD, thenD is Hamiltonian.

Proof. Suppose that the theorem is false. Since the theorem is clearly true forn= 2 andn= 3, there exists some integern≥4 and a digraphD of order nthat satisfies the hypothesis but which is not Hamiltonian. Let Cbe a cycle in D of maximum lengthk. It follows from Theorem 1.24 and the assumption that D is not Hamiltonian that 1 +n/2 ≤ k < n. Also, let P be a path of maximum length such that no vertex ofP lies onC. Suppose thatP is au−v path of lengthℓ≥0. Therefore,k+ℓ+ 1≤n. (See Figure 4.2.)

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Figure 4.2: A step in the proof of Corollary 4.10 Since

ℓ≤n−k−1≤n− 1 +n

−1 = n 2 −2,

it follows thatℓ≤n/2−2 and that there are at least two vertices adjacent to uwhich lie onC and at least two vertices adjacent fromvwhich lie onC.

154 CHAPTER 4. DIGRAPHS Letadenote the number of vertices onCthat are adjacent tou. Thusa≥2.

For every vertex xonC that is adjacent to u, the ℓ+ 1 vertices immediately followingxonCare not adjacent fromv, for otherwise,Dhas a cycle of length exceedingk. SinceC contains vertices adjacent fromv, there must be a vertex y on C that is adjacent tousuch that none of the ℓ+ 1 vertices immediately followingy onC are adjacent touor adjacent fromv.

For each of thea−1 vertices on C that are distinct from y and adjacent to u, the vertex immediately following it cannot be adjacent from v. Hence at least (a−1) + (ℓ+ 1) = a+ℓ vertices on C are not adjacent from v, for otherwise again,Dhas a cycle of length exceedingk. SinceP is a longest path in D containing no vertices of C, every vertex adjacent to uis either on C or onP.

Because idu≥ n/2 and the only vertices of D that can be adjacent to u belong toCorP, it follows thata+ℓ≥n/2. Therefore,vis adjacent to at most (n−1)−(a+ℓ)≤(n−1)−n/2 =n/2−1 vertices, producing a contradiction.

Line Digraphs

In Section 3.3, the line graph of a nonempty graph is defined. There is an analogous concept for digraphs. An arc e = (u, v) is adjacent to an arc f = (x, y) in a digraphD ifv=x. In theline digraphL(D) of a nonempty digraphD, the vertices ofL(D) can be put in one-to-one correspondence with the arcs ofDin such a way that vertexuis adjacent to vertexv inL(D) if the arc corresponding touinDis adjacent to the arc corresponding tov. A digraph D and its line digraph are shown in Figure 4.3, wherevi in L(D) corresponds toei in D(i= 1,2, . . . ,7).

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L(D) : D:

e3 e4

Figure 4.3: A digraph and its line digraph

While the line graph of an Eulerian graph is both Eulerian and Hamiltonian and the line graph of a Hamiltonian graph is Hamiltonian, this is not true in general for line digraphs (see Exercises 17 and 18). We do have the following result however (see Exercise 20).

Theorem 4.11 LetDbe a nontrivial connected digraph. ThenL(D)is Hamil- tonian if and only if D is Eulerian.

Exercises for Section 4.1

1. Prove Theorem 4.1: Letuandv be two vertices in a digraphD. For every u−v walk W inD, there exists au−v pathP such that every arc of P belongs toW.

2. Show that a digraph Dis strong if and only if its converseD~ is strong.

3. LetGbe a nontrivial connected graph without bridges.

(a) Show that for every edgeeofGand for every orientation ofe, there exists an orientation of the remaining edges of Gsuch that the resulting digraph is strong.

(b) Show that (a) need not be true if we begin with an orientation of two edges of G.

4. LetG be a connected graph with cut-vertices. Show that an orientation Dof G is strong if and only if the subdigraph ofDinduced by the vertices of each block of Gis strong.

5. According to Theorem 4.3, a nontrivial graphGhas a strong orientation if and only if Gis connected and contains no bridges.

(a) Prove that if G is a nontrivial connected graph with at most two bridges, then there exists an orientationDofGhaving the property that ifuandvare any two vertices ofD, there is either au−vpath or av−upath.

(b) Show that the statement (a) is false ifGcontains three bridges.

6. Prove or disprove: There exists a 4-regular graph G of order 7 and an orientationD ofGsuch that for each vertexuofD, there exists either a u−v path of length 1 or au−v path of length 2 but not both for every vertex vof Dwithv6=u.

7. Prove Theorem 4.4: Let D be a nontrivial connected digraph. Then D is Eulerian if and only if odv= idv for every vertexv of D.

8. Prove that a graph G has an Eulerian orientation if and only if G is Eulerian.

9. Prove Theorem 4.5: Let D be a nontrivial connected digraph. Then D contains an Eulerian trail if and only if D contains two verticesuandv such that odu= idu+ 1 andidv = odv+ 1, while odw= idw for all other vertices w of D. Furthermore, each Eulerian trail of D begins at u and ends at v.

10. Prove that a nontrivial connected digraph D is Eulerian if and only if E(D) can be partitioned into subsetsEi, 1≤i≤k, where the subdigraph D[Ei] induced by the setEi is a cycle for eachi.

156 CHAPTER 4. DIGRAPHS 11. Prove that ifDis a connected digraph such thatP

v∈V(D)|odv−idv|= 2t, wheret≥1, thenE(D) can be partitioned into subsetsEi, 1≤i≤t, so that the subgraphG[Eⁱ] induced byEi is an open trail for eachi.

12. Let D be a connected digraph of ordern with V(D) = {v₁, v₂, . . . , vn}. Prove that if odvi≥idvi for 1≤i≤n, thenD is Eulerian.

13. A vertexv in a digraphD is said to bereachable from a vertexuinD ifD contains a u−v path. LetD be a digraph and for each vertexuof D, letR(u) be the set of vertices reachable fromuand letr(u) =|R(u)|. Since u∈R(u) for every vertexuof D, it follows thatr(u)≥1. Prove that if r(x)6=r(y) for every two distinct verticesxand y of D, thenD contains a Hamiltonian path.

14. Corollary 4.9 states: If Dis a strong digraph of ordernsuch thatdegv ≥ n for every vertex v of D, then D is Hamiltonian. Show that if the digraphDis not required to be strong, thenDneed not be Hamiltonian.

15. Show for infinitely many positive integers n that there exists a digraph D of order nsuch that odv≥(n−1)/2 and idv≥(n−1)/2 for every vertex vof Dbut D is not Hamiltonian.

16. Prove that ifD is a nontrivial strong digraph, thenL(D) is strong.

17. Give an example of a Hamiltonian digraph whose line digraph is not Hamiltonian.

18. Give an example of an Eulerian digraph whose line digraph is not Eulerian.

19. Give an example of a digraphDsuch thatDis not Eulerian andL(L(D)) is Hamiltonian.

20. Prove Theorem 4.11:LetDbe a nontrivial connected digraph. ThenL(D) is Hamiltonian if and only if D is Eulerian.

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