Chapter 2
Trees and Connectivity
In nearly every concept, problem and theorem that we encounter, we are pri- marily concerned with connected graphs. There are various ways of measuring connectedness for graphs and in many instances, it will be important for us to know the degree of connectedness of the graphs being considered. While a graph may contain au−vpath for every pairu, vof vertices and consequently is connected, a graph can satisfy this requirement in minimal ways or in much stronger ways.
56 CHAPTER 2. TREES AND CONNECTIVITY
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Figure 2.1: Cut-vertices in graphs
Proof. LetGbe a nontrivial connected graph and letP be a longest path in G. Suppose thatP is au−vpath. We show thatuandvare not cut-vertices.
Assume, to the contrary, thatuis a cut-vertex ofG. ThenG−uis disconnected and so contains two or more components. Letwbe the vertex adjacent touon P and letP′be thew−vsubpath ofP. Necessarily,P′belongs to a component, sayG1, ofG−u. Let G2 be another component of G−u. Then G2 contains some vertexxthat is adjacent tou. This produces anx−vpath that is longer thanP, which is impossible. Similarly,v is not a cut-vertex ofG.
Ifv is a cut-vertex of a graphG, then there exist paths inG that cannot avoidv.
Theorem 2.2 A vertexv in a graphGis a cut-vertex ofGif and only if there are two vertices uand w distinct from v such that v lies on every u−w path inG.
Proof. We may assume thatGis connected, for otherwise we can consider a component ofGcontainingv. Ifvis a cut-vertex in a connected graphG, then, of course,G−v contains two or more components. Ifuandw are vertices in distinct components of G−v, then uand w are not connected in G−v. On the other hand, uandw are necessarily connected inG. Thusv lies on every u−wpath inG.
For the converse, suppose that there are two verticesuandwdistinct from v such thatv lies on everyu−w path in G. Then there is no u−w path in G−v. Thusuandware not connected inG−v, and soG−vis disconnected.
Therefore,v is a cut-vertex ofG.
Thus some connected graphs may contain a vertex the removal of which separates the graph into two or more connected graphs. These vertices are, of courses, cut-vertices. As we saw, a connected graph need not contain any cut-vertices.
A nontrivial connected graph containing no cut-vertices is anonseparable graph. In particular, the cyclesCn,n≥3, and the complete graphsKn,n≥2, are nonseparable graphs. In fact,K2 andK3are the only nonseparable graphs of order 3 or less. Not only do nonseparable graphs of order 3 or more contain cycles, they contain cycles possessing a rather interesting property.
Theorem 2.3 LetGbe a nonseparable graph of order3 or more. Then every two vertices of Glie on a common cycle ofG.
Proof. Assume, to the contrary, that there are pairs of vertices ofGthat do not lie on a common cycle. Among all such pairs, letu, v be a pair for which d(u, v) is minimum. If d(u, v) = 1, then uv ∈ E(G). Since the order of G is 3 or more, either degu ≥2 or degv ≥2, say the former. Let w be a vertex different fromvthat is adjacent tou. SinceG−uis connected,G−ucontains a w−v pathP. Then the pathP together with the path (w, u, v) produce a cycle containinguandv. Hence we may assume thatd(u, v) =k≥2.
Let P = (u = v0, v1, . . . , vk−1, vk = v) be a u−v geodesic in G. Since d(u, vk−1) =k−1< k, there is a cycleCcontaininguandvk−1. By assumption, vis not onC. Sincevk−1is not a cut-vertex ofGanduandvare distinct from vk−1, it follows from Theorem 2.2 that there is av−upath Qthat does not containvk−1. Sinceuis on C, there is a first vertexxofQthat is on C. Let Q′ be thev−xsubpath ofQand letP′ be avk−1−xpath onC that contains u. (Ifx6=u, then the pathP′ is unique.) However, the cycle C′ produced by proceeding fromv to its neighborvk−1, along P′ tox, and then alongQ′ tov contains bothuandv, a contradiction.
This theorem has several consequences (see Exercises 1-3). For two distinct verticesuandv in a graphG, twou−v paths areinternally disjointif they have onlyuandv in common.
Corollary 2.4 A connected graphGof order3or more is nonseparable if and only if for every two distinct vertices u and v in G, there are two internally disjoint u−v paths.
Corollary 2.5 Let u and w be two distinct vertices in a nonseparable graph G. If H is obtained from G by adding a new vertex v and joining v touand w, thenH is nonseparable.
Corollary 2.6 IfU andW are disjoint sets of vertices in a nonseparable graph G of order 4 or more with |U|=|W|= 2, then Gcontains two disjoint paths connecting the vertices ofU and the vertices of W.
Blocks
LetGbe a nontrivial connected graph. AblockofGis a maximal nonsepa- rable subgraph ofG, that is, a block ofGis a nonseparable subgraph ofGthat is not a proper subgraph of any nonseparable subgraph ofG. Every two distinct blocks of Ghave at most one vertex in common; and if they have a vertex in common, then this vertex is a cut-vertex ofG. A block of G containing exactly one cut-vertex ofG is called anend-block of G. IfB and B′ are two blocks containing the cut-vertexv, whileu∈V(B) andw∈V(B′) foru, w6=v, then
58 CHAPTER 2. TREES AND CONNECTIVITY everyu−wpath inGmust contain the vertexv. This is basically Theorem 2.2.
A graph G and its five blocks Bi, 1 ≤ i ≤ 5, are shown in Figure 2.2. The end-blocks ofGareB1,B2 andB5. A connected graph with cut-vertices must contain two or more end-blocks.
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Figure 2.2: The blocks of a graph
Theorem 2.7 Every connected graph containing cut-vertices has at least two end-blocks.
Proof. If Gcontains only one cut-vertex, then every block of G is an end- block. Hence we may assume thatGcontains two or more cut-vertices. Among all pairs of cut-vertices of G, let u, v be a pair for which d(u, v) is maximum and letP be au−v geodesic, say
P= (u=u0, u1, . . . , uk =v), wherek≥1.
Thenu1 belongs to a blockB and uk−1 belongs to a block B′, where possibly B = B′. In fact, possibly u1 = v. Since u is a cut-vertex of G, it follows thatubelongs to one or more blocks different fromB. Let B0be one of these.
Similarly, letB0′ be a block different fromB′that containsv. We claim thatB0 is an end-block ofG. If B0 is not an end-block, thenB0 contains a cut-vertex xdifferent fromu. Since everyx−u1path must pass throughu, it follows that
d(x, v) =d(x, u) +d(u, v)> d(u, v),
which is impossible. Similarly,B0′ is an end-block different fromB0.
The following result, which can be proved in a similar manner, is often useful as well.
Theorem 2.8 Let Gbe a connected graph with at least one cut-vertex. Then G contains a cut-vertex v with the property that, with at most one exception, all blocks of Gcontainingv are end-blocks.
Proof. IfG has only one cut-vertex, then every block of G is an end-block and contains the cut-vertex. Hence we may assume thatGcontains two or more cut-vertices. Among the cut-vertices ofG, letuandv be two for whichd(u, v) is maximum and let P = (u=u0, u1, . . . , uk =v),k≥1, be au−v geodesic.
Then uk−1 belongs to a block B containing v. LetB′ be a block containing v that is different fromB. If B′ is not an end-block, then B′ contains a cut- vertex wdifferent fromv. LetP′ be av−w geodesic in G. Then the path P followed byP′ produces au−wgeodesic whose length exceeds that ofP. This is a contradiction. Thus every block containingv that is different fromBis an end-block.
Another interesting property of blocks of graphs was observed by Frank Harary and Robert Z. Norman [113].
Theorem 2.9 The center of every connected graph G lies in a single block of G.
Proof. Suppose thatGis a connected graph whose center Cen(G) does not lie within a single block of G. Then G has a cut-vertex v such that G−v contains components G1 and G2, each of which contains vertices of Cen(G).
Letube a vertex such thatd(u, v) =e(v), and letP1 be av−ugeodesic. At least one of G1 and G2, sayG2, contains no vertices of P1. Letwbe a vertex of Cen(G) belonging toG2, and letP2 be aw−v geodesic. The pathsP1 and P2together form a u−w pathP3, which is necessarily au−wpath of length d(u, w). However, then e(w) > e(v), which contradicts the fact that w is a central vertex. Thus Cen(G) lies in a single block ofG.
If a graph G has components G1, G2, . . . , Gk and a nonempty connected graph H has blocks B1, B2, . . . , Bℓ, then {V(G1), V(G2), . . ., V(Gk)} is a partition ofV(G) and{E(B1),E(B2),. . .,E(Bℓ)}is a partition ofE(H).
Suppose, for a cut-vertexv of a connected graphG, that the disconnected graphG−v haskcomponentsG1, G2, . . . , Gk (k≥2). The induced subgraphs
Bi=G[V(Gi)∪ {v}]
are connected and referred to as the branches of G at v. If a subgraphGi
contains no cut-vertices of G, then the branch Bi is a block of G, in fact, an end-block ofG.
A connected graphGcontaining three cut-verticesu, vandwand six blocks is shown in Figure 2.3. Four of these blocks are end-blocks. The graphGhas four branches at v, all of which are shown in Figure 2.3. Two of the four branches atv are end-blocks ofG.
60 CHAPTER 2. TREES AND CONNECTIVITY
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Figure 2.3: The four branches of a graphGat a cut-vertexv
Exercises for Section 2.1
1. Prove Corollary 2.4: A connected graphGof order3 or more is nonsepa- rable if and only if for every two distinct verticesuandv in G, there are two internally disjointu−v paths.
2. Prove Corollary 2.5: Letuandwbe two distinct vertices in a nonseparable graph G. If H is obtained from G by adding a new vertexv and joining v touandw, then H is nonseparable.
3. Prove Corollary 2.6: IfU andW are disjoint sets of vertices in a nonsep- arable graph Gof order 4 or more with|U|=|W|= 2, then Gcontains two disjoint paths connecting the vertices ofU and the vertices of W. 4. (a) Letkbe the maximum length of a cycle in a nonseparable graphG.
Prove that if C and C′ are any twok-cycles in G, thenC and C′ have at least two vertices in common.
(b) Show that the result in (a) cannot be improved.
5. Prove that if v is a cut-vertex of a connected graph G, thenv is not a cut-vertex ofG.
6. Letuandvbe distinct vertices of a nonseparable graphGof ordern≥3.
IfP is a given u−v path ofG, does there always exist a u−v pathQ such that P andQare internally disjointu−v paths?
7. (a) An elementof a graphGis a vertex or an edge of G. Prove that a connected graph Gof order at least 3 is nonseparable if and only if every pair of elements of Glie on a common cycle ofG.
(b) Let Gand H be graphs withV(G) ={v1, v2, . . . , vn} and V(H) = {u1, u2, . . . , un},n≥3. Two verticesui anduj are adjacent inH if and only if vi and vj belong to a common cycle inG. Characterize those graphsGfor whichH is complete.
8. Prove that ifGis a graph of ordern≥3 with the property that degu+ degv ≥ n for every pair u, v of nonadjacent vertices of G, then G is nonseparable.
9. Prove or disprove: IfB is a block of order 3 or more in a connected graph G, then there is a cycle inB that contains all the vertices ofB.
10. A connected graphGcontains k blocks andℓ cut-vertices. What is the relationship between kandℓ?
11. Prove or disprove: IfGis a connected graph with cut-vertices anduand v are antipodal vertices ofG, then no block ofGcontains bothuandv. 12. (a) Show that for every positive integerk, there exists a connected graph
Gand a non-cut-vertexuofGsuch that rad(G−u) = rad(G) +k.
(b) Prove for every nontrivial connected graph G and every non-cut- vertex vofGthat rad(G−v)≥rad(G)−1.
(c) LetGbe a nontrivial connected graph with rad(G) =r. Among all connected induced subgraphs of Ghaving radiusr, let H be one of minimum order. Prove that rad(H−v) = r−1 for every non-cut- vertex vofH.