Chapter 9 Connectivity ???

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Chapter 9 Connectivity???
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9.1 Connectivity

• Consider the following graphs
• G1 Deleting any edge makes it disconnected.
• G2 Cannot be disconnected by deletion of any
  edge can be disconnected by deleting its cut
  vertex
• Intuitively, G2 is more connected than G1, G3 is
  more connected thant G2, and G4 is the most
  connected one.

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9.1 Cut edges and cut vertices

• A cut edge of G is an edge such that G-e has
  more components that G.
• Theorem 9.1 Let G be a connected graph. The
  following are equivalent
• An edge e of G is a cut edge
• e is not contained in any cycle of G.
• There are two vertices u and w such that e is on
  every path connecting u and w.

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9.1 Cut edges and cut vertices
Let G be a nontrivial and loopless graph. A
vertex v of G is a cut vertex if G-v has more
components than G. Theorem 9.2 Let G be a
connected graph. The following propositions are
equivalent 1. A vertex v is a cut vertex of
G 2. There are two distinct vertices u and w such
that every path between u and w passes v 3. The
vertices of G can be partitioned into two
disjoint vertex sets U and W such that every
path between u?U and w?W passes v.
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9.1 Vertex cut and connectivity

• A vertex cut of G is a subset V of V such that
  G-V is disconnected. The connectivity , ?(G), is
  the smallest number of vertices in any vertex cut
  of G.
• A complete graph has no vertex cut. Define
  ?(Kn)n-1
• For disconnected graph G, define ?(G) 0
• G is said to be k-connected if ?(G)?k
• It is easy to see that all nontrivial connected
  graphs are 1-connected.
• ?(G)1 if and only if GK2 or G has a cut vertex.

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9.1 Edge cut and edge connectivity

• Let G be graph on n?2 vertices. An edge cut is a
  subset E of E(G) such that G-E is
  disconnected.
• The edge connectivity, ?(G), is the smallest
  number of edges in any edge cut.

• For trivial and disconnected graph G, define
  ?(G)0
• G is said to be k-edge-connected if ?(G)?k
• All nontrivial connected graphs are
  1-edge-connected.

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9.1 Edge cut and edge connectivity
Find ?(G), ?(G) and ?(G) for the following graphs

• Theorem 9.3 For any connected graph G
• ?(G)? ?(G)??(G)
• where ?(G) is the smallest vertex degree of G.

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• ?????(G)??(G),????????????????????????????

• ?????(G)? ?(G)?
• ?Ee1,e2,,ek????,?????ei?????????E,?????????
• E(u,w),(v,x)????,??u,v?????????
• ????E??????G1, G2, ??v1?V(G1),v2?V(G2),
  v1,v2???(???),????v1,v2,??E?k?????E???????,??u,x,
  ??w,v.

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9.1 Edge cut and edge connectivity

• Theorem 9.4 (Whitney) A graph G of order n(?3)
  is 2-connected if and only if any two vertices of
  G are in a common cycle.

• The theorem can be proved by induction on the
  length of the paths. See the textbook for the
  proof.
• State it in another way A graph G of order
  n(?3) is 2-connected if and only if any two
  vertices of G connected by at least 2
  vertex-disjoint paths.

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9.1 Edge cut and edge connectivity

• Theorem 9.5 (Whitney) A graph G of order n(?3) is
  2-edge-connected if and only if any two vertices
  of G are in a simple closed path.

• State it in another way A graph G of order n(?3)
  is 2-edge-connected if and only if any two
  vertices of G are connected by at least 2
  edge-disjoint paths.

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9.1 Edge-disjoint paths

• Let v and w are two vertices in a graph. A
  collection of paths from v to w are called
  edge-disjoint paths if no two paths in it share
  an edge.

Count the number of edge-disjoint paths from v to
w in the graph above. Find the edge connectivity
of the graph.
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9.1 Vertex-disjoint paths
Similarly, we can define vertex-disjoint paths.

• Find the connectivity of the graph and the number
  of vertex-disjoint paths from v to w.

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9.2 Mengers theorem

• Theorem 9.6 The maximum number of edge-disjoint
  paths connecting two distinct vertices v and w
  in connected graph G is equal to the minimum
  number of edges whose removal disconnecting v
  and w.

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9.2 Mengers theorem

• Theorem 9.6 A graph G is k-edge-connected if and
  only if any two distinct vertices of G are
  connected by at least k edge-disjoint paths.
• ProofIf there are two vertices which are
  connected by less than k edge-disjoint paths,
  then G is not k-edge-connected. On the other
  hand, if G is not k-edge-connected, there are
  edge cut that contains less than k edges, hence
  there are two vertices which are connected by
  less than k edge-disjoint paths.

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9.2 Mengers theorem
Theorem 9.7 The maximum number of vertex-disjoint
paths connecting two distinct non-adjacent
vertices v and w of a connected graph G is equal
to the minimum number of vertices whose removal
disconnecting v and w.

• Theorem 9.7 A graph of order n(?k1) is
  k-connected if and only if any two distinct
  vertices of G are connected by at least k
  vertex-disjoint paths.

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9.2 Mengers theorem

• Theorem 9.8 The maximum number of arc-disjoint
  paths from a vertex v to a vertex w in a digraph
  D is equal to the minimum number of arcs whose
  removal disconnecting v and w.

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9.3 Mengers Theorem implies the Max-flow min-cut
Theorem

• Proof Assuming the capacity of every arc is an
  integer.
• The network N can be seen as a digraph D in
  which
• the capacities represent the number of arcs
  connecting the various vertices.
• The maximum flow corresponds to the total number
  of arc-disjoint path form s to t in D
• The capacity of a minimum cut refers to the
  minimum number of arcs in a st-disconnecting set
  of D.

N
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9.3 The Max-flow min-cut theorem implies Mengers
theorem

• Lemma 9.9 Let N be a network with source s and
  sink t in which each arc has unit capacity. Then
• The value of a maximum flow in N is equal to the
  maximum number m of arc-disjoint directed
  (s,t)-paths in N and
• The capacity of a minimum cut in N is equal to
  the minimum number n of arcs whose deletion
  destroys all directed (s,t)-paths in N.

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9.3 Mengers theorem

• Theorem 9.8 (Menger) Let s and t be two vertices
  of a digraph D. Then the maximum number of
  arc-disjoint directed (s,t)-paths in D is equal
  to the minimum number of arcs whose deletion
  destroys all directed (s,t)-paths in D.

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9.3 Mengers theorem

• Theorem 9.10 (Menger, undirected version) Let s
  and t be two vertices of a graph G. Then the
  maximum number of edge-disjoint (s,t)-paths in G
  is equal to the minimum number of edges whose
  deletion destroys all (s,t)-paths in G.
• Proof Apply the directed version of Mengers
  theorem to the associated digraph D(G) of G (an
  edge becomes two directed edges). There is a
  one-one correspondence between paths in G and
  D(G). See Bondy and Murty.

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9.3 Mengers theorem

• Theorem 9.11 Let s and t be two vertices of a
  directed graph D such that s is not joined to t.
  Then the maximum number of vertex-disjoint
  (s,t)-paths in G is equal to the minimum number
  of vertices whose deletion destroys all directed
  (s,t)-paths in D.
• Proof by converting it to the arc-version of
  Mengers theorem.

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• Construct a new digraph D from D by splitting
  each vertex v?V-s,t such that v becomes an arc
  v-gtv, arcs leading to v now leading to v and
  arcs leaving v now leaving from v
• To each edge-disjoint (s,t)-path in D there
  corresponds a vertex-disjoint directed (s,t)-path
  in Dand vice verse and See Bondy and Murty.

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9.4 Mengers theorem implies Halls Theorem

• Theorem 9.12 Mengers theorem implies Halls
  theorem.

There is a complete matching from V1 to V2 if and
only if the number of vertex-disjoint paths from
v to w is equal to V1k.
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9.4 Mengers theorem implies Halls Theorem

• Proof Let G(V1,V2) be a bipartite graph. We
  have to prove that if A?N(A) for each subset
  A of V1, then there exists a complete matching
  form V1 to V2.
• We add two extra vertices v and w (see the
  graph on previous page). Using Mengers theorem
  of the vertex form, it is enough to prove that
  every vw-separating set (whose removal disconnect
  v and w) contains at least V1k vertices.
• Let S be a vw-separating set, consists of
  A?V1 and B ? V2.
• Since A?B is a vw-separating set, there can
  be no edges joining a vertex of V1-A to a vertex
  of V2-B, that is, N(V1-A)?B. It follows that
  (Halls condition)
• V1-AltN(V1-A)ltB.
• so SA?BABgtV1k, as required.

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9.4 Mengers theorem implies Halls Theorem

• ?? ?G(V1,V2)??????. ??????V1?????A, A lt
  N(A), ???V1?V2 ?????.
• ?G???????v,w, ??v?V1???????, w?V2 ???????.
  ??, ??V1?V2?????????v?w???????????V1k.
• ??V1 ????v,w?????,??,??Menger??,
  ??????v,w???????k. ?S???v,w????,
  ??V1???A?V2???B??.
• ??A?B???vw???, ???V1-A?V2-B????,
• ?? N(V1-A)?B, ??? Hall??,
• V1-AltN(V1-A)ltB.
• ??, SA?BABgtV1k.

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9.5 Reliable communication networks

• A graph representing a communication network, the
  connectivity (or edge-connectivity) becomes the
  smallest number of stations (or links) whose
  breakdown would jeopardise the system.
• The higher the connectivity and edge
  connectivity, the more reliable the network.
• Let k be a given positive integer and let G be a
  weighted graph. Determine a minimum-weight
  k-connected spanning subgraph of G.
• For k1, this is solved by Kruskals algorithm,
  for example. For kgt1, the problem is unsolved.
  

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Summary

• Vertex cut and edge cut
• Connectivity and edge-connectivity
• Menger Theorem
• Equivalence of Menger Theorem and the Max-flow
  Min-cut Theorem
• Menger Theorem implies Hall Theorem.