12. Theconnection numberc(G) of a connected graphGof ordern≥2 is the smallest integer k with 2≤k≤nsuch that everyinduced subgraph of order k in G is connected. State and prove a theorem that gives a relationship between κ(G) andc(G) for a graphGof ordern.
13. For an even integerk≥2, show that the minimum size of ak-connected graph of orderniskn/2.
14. Prove or disprove: LetGbe a nontrivial graph. For every vertexv ofG, κ(G−v) =κ(G) orκ(G−v) =κ(G)−1.
15. (a) Prove that ifGis a k-connected graph ande is an edge of G, then G−eis (k−1)-connected.
(b) Prove that ifGis a k-edge-connected graph and eis an edge ofG, thenG−eis (k−1)-edge-connected.
16. (a) Show that ifGis a 0-regular graph, thenκ(G) =λ(G).
(b) Show that ifGis a 1-regular graph, thenκ(G) =λ(G).
(c) Show that ifGis a 2-regular graph, thenκ(G) =λ(G).
(d) By (a) – (c) and Theorem 2.31, ifG isr-regular, where 0≤r≤3, then κ(G) =λ(G). Find the minimum positive integerr for which there exists an r-regular graphGsuch thatκ(G)6=λ(G).
(e) Find the minimum positive integer r for which there exists an r- regular graphGsuch thatλ(G)≥κ(G) + 2.
17. For a graphG, defineκ(G) = max{κ(H)}andλ(G) = max{λ(H)}, where each maximum is taken over all subgraphsH of G. How are κ(G) and λ(G) related toκ(G) andλ(G), respectively, and to each other?
18. Let G1 and G2 be two k-connected graphs, where k ≥ 2, and let G be the set of all graphs obtained by adding k edges between G1 and G2. Determine max{κ(G) :G∈ G}.
98 CHAPTER 2. TREES AND CONNECTIVITY of distinguished mathematicians, including Georg Cantor, Camille Jordan and Giuseppe Peano. Some mathematicians, including Felix Hausdorff and Ludwig Bieberbach, felt that it was unlikely that this problem would ever be solved. De- spite being an undergraduate with limited mathematical background, Menger solved the problem and presented his solution to Hahn. This led Menger to work on curve and dimension theory. After completing his studies at Vienna, Menger left to broaden his mathematics in Amsterdam.
In 1927 Menger returned to the University of Vienna to accept the po- sition of Chair of Geometry. It was during that year that he published the paper “Zur allgemeinen Kurventheorie” (which containedMenger’s Theorem).
Menger himself referred to this result as the “n-arc theorem” and proved it as a lemma for a theorem in curve theory.
In the spring of 1930, Karl Menger traveled to Budapest and met many Hungarian mathematicians, including D´enes K¨onig. Menger had read some of K¨onig’s papers. During his visit, Menger learned that K¨onig was working on a book that would contain what was known about graph theory at that time. Menger was pleased to hear this and mentioned his theorem to K¨onig, which had only been published three years earlier. K¨onig was not aware of Menger’s work and, in fact, didn’t believe that the theorem was true. Indeed, the very evening of their meeting, K¨onig set out to construct a counterexample.
When the two met again the next day, K¨onig greeted Menger with “A sleepless night!”. K¨onig then asked Menger to describe his proof, which he did. After that, K¨onig said that he would add a final section to his book on the theorem.
As a result, K¨onig added a chapter to his 1936 book Theorie der endlichen und unendliehen Graphen, which would become the first book written on graph theory [139]. This was a major reason why Menger’s theorem became so widely known among those interested in graph theory.
Before stating and presenting a proof of Menger’s theorem, some additional terminology is needed. For two nonadjacent vertices u and v in a graph G, a u−v separating set is a set S ⊆ V(G)− {u, v} such that uand v lie in different components ofG−S. Au−v separating set of minimum cardinality is called aminimum u−v separating set.
For two distinct verticesuandvin a graphG, a collection ofu−v paths is internally disjoint if every two paths in the collection have onlyuandv in common. Menger’s theorem states that the concepts of internally disjointu−v paths andu−v separating sets are linked. In the graphGof Figure 2.30, there is a set S ={w1, w2, w3} of vertices of G that separate the vertices uand v. No set with fewer than three vertices separatesuandv. According to Menger’s theorem, there are three internally disjointu−v paths inG.
Theorem 2.36 (Menger’s Theorem)Let uand v be nonadjacent vertices in a graphG. The minimum number of vertices in au−vseparating set equals the maximum number of internally disjoint u−v paths in G.
Proof. We proceed by induction on the size of graphs. The theorem is cer- tainly true for every empty graph. Assume that the theorem holds for all graphs
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Figure 2.30: A graph illustrating Menger’s theorem
of size less thanm, wherem≥1, and letGbe a graph of sizem. Moreover, let uandv be two nonadjacent vertices ofG. Ifuand v belong to different com- ponents of G, then the result follows. So we may assume thatuandv belong to the same component of G. Suppose that a minimumu−v separating set consists ofk≥1 vertices. ThenGcontains at mostk internally disjointu−v paths. We show, in fact, thatGcontainskinternally disjointu−vpaths. Since this is obviously true if k= 1, we may assume that k≥ 2. We now consider three cases.
Case 1. Some minimumu−v separating set X in G contains a vertex x that is adjacent to both uandv. ThenX− {x} is a minimumu−v separating set in G−xconsisting of k−1 vertices. Since the size ofG−x is less than m, it follows by the induction hypothesis thatG−xcontainsk−1 internally disjointu−v paths. These paths together with the pathP = (u, x, v) produce kinternally disjointu−vpaths in G.
Case2. For every minimumu−vseparating setSinG, either every vertex inS is adjacent touand not tov or every vertex inS is adjacent tovand not tou. Necessarily then,d(u, v)≥3. LetP = (u, x, y, . . . , v) be au−vgeodesic in G, wheree=xy. Every minimumu−v separating set inG−econtains at leastk−1 vertices. We show, in fact, that every minimumu−v separating set in G−econtainskvertices.
Suppose that there is some minimumu−vseparating set inG−ewithk−1 vertices, sayZ ={z1, z2, . . . , zk−1}. ThenZ∪ {x}is au−v separating set in Gand therefore a minimumu−vseparating set inG. Sincexis adjacent tou (and not tov), it follows that every vertexzi (1 ≤i≤k−1) is also adjacent touand not adjacent tov.
SinceZ∪ {y}is also a minimumu−v separating set inGand each vertex zi (1≤i≤k−1) is adjacent toubut not tov, it follows thaty is adjacent to u. This, however, contradicts the assumption thatP is a u−v geodesic. Thus k is the minimum number of vertices in au−v separating set in G−e. Since the size ofG−eis less thanm, it follows by the induction hypothesis that there arekinternally disjointu−vpaths in G−eand inGas well.
100 CHAPTER 2. TREES AND CONNECTIVITY Case3. There exists a minimum u−v separating set W inGin which no vertex is adjacent to bothuandvand containing at least one vertex not adjacent touand at least one vertex not adjacent to v. LetW ={w1, w2, . . . , wk}. Let Gu be the subgraph ofG consisting of, for eachi with 1 ≤i ≤k, all u−wi
paths inGin whichwi∈W is the only vertex of the path belonging toW. Let G′ube the graph constructed fromGu by adding a new vertexv′and joiningv′ to each vertexwi for 1≤i≤k. The graphsGv andG′v are defined similarly.
SinceW contains a vertex that is not adjacent touand a vertex that is not adjacent to v, the sizes of both G′u andG′v are less than m. SoG′u containsk internally disjointu−v′ pathsAi (1≤i≤k), whereAicontainswi. Also,G′v
contains k internally disjoint u′−v paths Bi (1 ≤i ≤k), where Bi contains wi. LetA′i be theu−wi subpath ofAi and letBi′ be thewi−v subpath ofBi
for 1≤i≤k. Thek paths constructed fromA′i andBi′ for eachi(1≤i≤k) are internally disjointu−v paths inG.
Whitney’s Theorem
We mentioned that Karl Menger visited D´enes K¨onig in Budapest, Hungary, early in 1930, which led to his theorem being included in K¨onig’s book. Later in 1930, Menger went to the United States and spent the period from September of 1930 to February of 1931 as a visiting lecturer at Harvard University. It was at one of his seminar talks that he presented his theorem. During that period, Hassler Whitney (1907–1989) was doing research for his doctoral thesis in graph theory at Harvard. For a short period after receiving his Ph.D. in 1932 from Harvard, Whitney worked on graph theory, making important contributions, but thereafter turned to topology when the area was just being calledtopology.
Whitney was a faculty member at Harvard until 1952, when he went to the Institute for Advanced Study in Princeton. Whitney was at Harvard during the period that the United States was involved in World War II. He had great interest and ability in applied mathematics. Because of this, he was brought in as a consultant to the Applied Mathematics Group at Columbia University.
That part of the Applied Mathematics Panel was primarily the responsibility of Whitney. He developed mathematical principles to discover best techniques for aerial gunnery and was involved in making improvements to weapons systems.
Although research was a large part of Whitney’s professional life, he con- tributed to mathematics in many ways. During 1944-1949 he was editor of the American Journal of Mathematicsand during 1949-1954 he was editor ofMath- ematical Reviews. During 1953-1956, he chaired the National Science Founda- tion mathematical panel. On the personal side, Whitney was an avid mountain climber. In fact, the Whitney-Gilman Ridge on Cannon Cliff in Franconia, New Hampshire, was named for his cousin and him, who were the first to climb it (on August 3, 1939).
If G is a k-connected graph (k ≥ 1) and v is a vertex of G, then G−v is (k−1)-connected. In fact, if e = uv is an edge of G, then G−e is also
(k−1)-connected (see Exercise 15 in Section 2.4). With the aid of Menger’s theorem, a useful characterization ofk-connected graphs, due to Whitney [238], was established. Since nonseparable graphs of order 3 or more are 2-connected, this gives a generalization of Corollary 2.4.
Theorem 2.37 (Whitney’s Theorem) A nontrivial graphGisk-connected for some integer k≥2if and only if for each pairu, v of distinct vertices ofG, there are at least kinternally disjoint u−v paths in G.
Proof. First, suppose thatGis ak-connected graph, wherek≥2, and let u andv be two distinct vertices ofG. Assume first thatuandvare not adjacent.
LetU be a minimumu−v separating set. Then k≤κ(G)≤ |U|.
By Menger’s theorem,Gcontains at leastkinternally disjointu−vpaths.
Next, assume thatuandvare adjacent, wheree=uv. As observed earlier, G−eis (k−1)-connected. LetW be a minimumu−vseparating set inG−e. So
k−1≤κ(G−e)≤ |W|.
By Menger’s theorem, G−e contains at least k−1 internally disjoint u−v paths, implying thatGcontains at leastkinternally disjointu−v paths.
For the converse, assume thatGcontains at leastkinternally disjointu−v paths for every pairu, vof distinct vertices ofG. IfGis complete, thenG=Kn, wheren≥k+ 1, and soκ(G) =n−1≥k. HenceGisk-connected. Thus we may assume that Gis not complete.
LetU be a minimum vertex-cut ofG. Then |U| =κ(G). Let xand y be vertices in distinct components ofG−U. ThusU is anx−y separating set of G. Since there are at leastk internally disjointx−y paths inG, it follows by Menger’s theorem that
k≤ |U|=κ(G).
and soGisk-connected.
The following three results are consequences of Theorem 2.37 (see Exer- cises 3-5).
Corollary 2.38 Let Gbe ak-connected graph,k≥1, and let S be any set of kvertices ofG. If a graphH is obtained fromGby adding a new vertexwand joiningw to the vertices ofS, then H is alsok-connected.
Corollary 2.39 If G is a k-connected graph, k ≥2, and u, v1, v2, . . . , vt are t+ 1distinct vertices ofG, where2≤t≤k, thenGcontains au−vi path for each i(1≤i≤t), every two paths of which have onlyuin common.
102 CHAPTER 2. TREES AND CONNECTIVITY Corollary 2.40 A graphGof ordern≥2kisk-connected if and only if for ev- ery two disjoint setsV1 andV2 ofkdistinct vertices each, there existkpairwise disjoint paths connectingV1 andV2.
By Theorem 2.3, every two vertices in a 2-connected graph lie on a common cycle of the graph. Gabriel Dirac [61] generalized this tok-connected graphs.
Theorem 2.41 If G is a k-connected graph, k ≥2, then every k vertices of Glie on a common cycle ofG.
Proof. LetS ={v1, v2, . . . , vk}be a set of kvertices of G. Among all cycles in G, let C be one containing a maximum number ℓ of vertices of S. Then ℓ ≤k. Ifℓ =k, then the result follows, so we may assume that ℓ < k. Since G is k-connected, G is 2-connected and so by Theorem 2.3, ℓ ≥ 2. We may further assume thatv1, v2, . . . , vℓlie onC. Letube a vertex ofSthat does not lie onC. We consider two cases.
Case1. The cycleC contains exactlyℓvertices, sayC= (v1, v2, . . . , vℓ, v1).
By Corollary 2.39, Gcontains au−vi pathPi for eachiwith 1≤i≤ℓsuch that every two of the pathsP1, P2, . . . , Pℓ have onlyuin common. Replacing the edge v1v2 on C byP1 and P2 produces a cycle containing at least ℓ+ 1 vertices ofS. This is a contradiction.
Case2. The cycleC contains at leastℓ+ 1 vertices. Letv0 be a vertex on C that does not belong toS. Since 2< ℓ+ 1≤k, it follows by Corollary 2.39 that G contains a u−vi path Pi for each i with 0 ≤ i ≤ ℓ such that every two of the pathsP0, P1, . . . , Pℓ have onlyuin common. For eachi(0≤i≤ℓ), let ui be the first vertex of Pi that belongs to C and let Pi′ be the u−ui
subpath ofPi. Suppose that the vertices ui (0≤i≤ℓ) are encountered in the orderu0, u1, . . . , uℓas we proceed aboutC in some direction. For someiwith 0 ≤ i ≤ ℓ and uℓ+1 = u0, there is a ui−ui+1 path P on C, none of whose internal vertices belong to S. Replacing P on C by Pi′ and Pi′+1 produces a cycle containing at leastℓ+ 1 vertices ofS. Again, this is a contradiction.
There are analogues to Menger’s theorem (Theorem 2.36) and to Whitney’s theorem (Theorem 2.37) in terms of edge-cuts. For two distinct verticesuand v in a graph G, an edge-cut X of G is a u−v separating set if u and v lie in different components of G−X. We begin with the edge analogue of Theorem 2.36.
Theorem 2.42 For distinct vertices u and v in a graph G, the minimum cardinality of a u−v separating setX ⊆E(G)equals the maximum number of pairwise edge-disjoint u−v paths inG.
Proof. It is convenient to actually prove a stronger result here by allowingG to be a multigraph.
If u and v are vertices in different components of a multigraph G, then the theorem is immediate. Hence we may assume that the multigraphs under
consideration are connected. If the minimum number of edges that separateu andvis 1, thenGcontains a bridgeeso thatuandvlie in different components of G−e. Thus everyu−v path in Gcontainse, so the maximum number of pairwise edge-disjointu−v paths inGis also 1.
Suppose that the statement is false. Then there is a smallest integerk≥2 such that there exist multigraphs containing two verticesuandv for which the minimum number of edges that separateuandv isk but there do not existk pairwise edge-disjoint u−v paths. Among all such multigraphs, let Gbe one of minimum size.
If everyu−v path ofGhas length 1 or 2, then since the minimum number of edges of G that separateu and v is k, it follows that there are k pairwise edge-disjointu−vpaths inG, which produces a contradiction. ThusGcontains a u−v pathP of length 3 or more.
Lete1 be an edge ofP that is incident with neitherunorv. Consider the multigraph G−e1. Since the size of G−e1 is less than the size of G, it is impossible forG−e1 to contain kedges that separateuandv. Thuse1 must belong to every set ofk edges that separateuandv. LetS ={e1, e2, . . . , ek} be one such set.
We now subdivide each edge ofS, that is, each edgeei=uiviofSis replaced by a new vertex wi and the two new edgesuiwi and wivi for 1≤i≤k. The k vertices wi (1≤i≤k) are now identified, producing a new vertexw and a new multigraphH. (See Figure 2.31 for a possible situation.) Observe thatw is a cut-vertex inH and everyu−v path ofH containsw.
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Figure 2.31: A step in the proof of Theorem 2.42
Denote byHu the submultigraph of H consisting of allu−w paths of H and denote by Hv the submultigraph consisting of allv−w paths ofH. The minimum number of edges separating uand w in Hu is k and the minimum number of edges separatingv andw inHv is k. Since each ofHu and Hv has smaller size thanG, it follows thatHu containsk pairwise edge-disjointu−w paths and thatHv containsk pairwise edge-disjointw−vpaths.
Fori= 1,2, . . . , k, we can pair off au−wpath inHu and aw−v path in Hv to produce au−v path inH containing the two edgesuiwand wvi. This produceskpairwise edge-disjointu−vpaths inH. The process of subdividing the edges uivi of G and identifying the vertices wi to obtain w can now be
104 CHAPTER 2. TREES AND CONNECTIVITY reversed to produce kpairwise edge-disjoint u−v paths inG. This, however, produces a contradiction.
Since the theorem has been proved for multigraphs, it is valid for graphs.
With the aid of Theorem 2.42, it is now possible to present an edge analogue of Theorem 2.37 (see Exercise 9).
Theorem 2.43 A nontrivial graph G is k-edge-connected if and only if G containskpairwise edge-disjointu−vpaths for each pairu, vof distinct vertices of G.
Exercises for Section 2.5
1. Show that the converse of Theorem 2.41 is not true in general.
2. Prove that a graphGof ordern≥k+ 1≥3 isk-connected if and only if for each setSofkdistinct vertices ofGand for each two-vertex subsetT ofS, there is a cycle ofGthat contains both vertices ofT but no vertices ofS−T.
3. Prove Corollary 2.38: Let G be a k-connected graph, k≥1, and letS be any set ofkvertices ofG. If a graphHis obtained fromGby adding a new vertexwand joining wto the vertices ofS, then H is alsok-connected.
4. Prove Corollary 2.39: If Gis a k-connected graph, k≥2, andu,v1, v2, . . .,vt are t+ 1distinct vertices of G, where 2≤t≤k, then Gcontains au−vi path for eachi (1≤i≤t), every two paths of which have only u in common.
5. Prove Corollary 2.40: A graph G of order n≥2k is k-connected if and only if for every two disjoint sets V1 andV2 of k distinct vertices each, there existk pairwise disjoint paths connectingV1 andV2.
6. LetGbe a k-connected graph and letv be a vertex ofG. For a positive integer t, define Gt to be the graph obtained from G by adding t new vertices u1, u2, . . . , utand all edges of the formuiw, where 1≤i≤tand for whichvw∈E(G). Show thatGtisk-connected.
7. Show that ifGis ak-connected graph with nonempty disjoint subsetsS1 andS2ofV(G), then there existkinternally disjoint pathsP1, P2, . . . , Pk
such that each path Pi is au−v path for someu∈S1 and somev∈S2 fori= 1,2, . . . , kand |S1∩V(Pi)|=|S2∩V(Pi)|= 1.
8. Let G be a k-connected graph, k ≥ 3, and let v, v1, v2, . . . , vk−1 be k vertices of G. Show thatGhas a cycle containing all ofv1, v2, . . . , vk−1
but not v and k−1 internally disjoint v−ui paths Pi (1 ≤i ≤k−1) such that for eachi, the vertexui is the only vertex ofPi onC.
9. Prove Theorem 2.43: A nontrivial graph G is k-edge-connected if and only if Gcontains k pairwise edge-disjoint u−v paths for each pair u, v of distinct vertices ofG.
10. Prove or disprove: IfG is ak-edge-connected graph and v, v1, v2, . . . , vk
arek+ 1 vertices of G, then for i= 1,2, . . . , k, there exist v−vi paths Pi such that each pathPi contains exactly one vertex of{v1, v2, . . . , vk}, namelyvi, and fori6=j,Pi andPj are edge-disjoint.
11. Prove or disprove: If Gis ak-edge-connected graph with nonempty dis- joint subsetsS1 and S2 of V(G), then there exist k edge-disjoint paths P1, P2, . . . , Pk such that for eachi,Pi is au−vpath for someu∈S1and somev∈S2 fori= 1,2, . . . , kand|S1∩V(Pi)|=|S2∩V(Pi)|= 1.
12. Show thatκ(Qn) =λ(Qn) =nfor all positive integern.
13. Assume thatGis a graphin the proof of Theorem 2.42. Does the proof go through? If not, where does it fail?
14. LetGbe a graph of ordernwithκ(G)≥1. Prove that n≥k(G)[diam(G)−1] + 2.
15. By a chorded cycle is meant a cycle C (of length at least 4) together with an edge that joins two nonconsecutive vertices of C. Prove that every 3-connected graph contains a chorded cycle but that this need not be the case for a 2-connected graph.
16. (a) Show that for every two verticesuandv of a 3-connected graphG, there exist two internally disjoint u−vpaths of different lengths in G.
(b) Show that the result in (a) is not true in general ifGis 2-connected.
17. (a) Prove that ifGis a connected graph of ordernand kis an integer with 1 ≤ k ≤ n−3 such that the maximum number of internally disjointu−vpaths inGiskfor every pairu, vof nonadjacent vertices ofG, thenGcontains a vertex-cut with exactlyk+ 1 vertices.
(b) Let G be a k-connected graph of diameter k, where k ≥ 2. Prove thatGcontainsk+1 distinct verticesv, v1, v2, . . . , vkandkinternally disjoint v−vi pathsPi (1≤i≤k) such thatPi has lengthi.