Connectivity and Paths

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Connectivity and Paths

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Connectivity

• A separating set of a graph G is a set
  such that G-S has more than one component.
  
• The connectivity of G, is the minimum size
  of a vertex set S such that G-S is disconnected
  or has only one vertex.
• A graph G is k-connected if its connectivity is
  at least k.

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Example

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Hypercube

• The K-dimensional cube is the simple graph
  whose vertices are the k-tuples with entries in
  and whose edges are the pairs of k-tuples
  that differ in exactly one position.

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• The neighbors of one vertex in form a
  separating set, so . To prove
  , we show that every separating set has size at
  least .
• Prove by induction on .
• Basis step For , is a complete
  graph with vertices and has connectivity
  .

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An example

• Induction step Let S be a vertex cut in
• Case 1 If Q-S is connected and Q-S is
  connected, then , for .
• Case 2 If Q-S is disconnected, which means S has
  at least k-1 vertex in Q. And, S must also
  contain a vertex of . We have .

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Harary graphs

• Given k ltn, place n vertices around a circle. If
  k is even, form by making each vertex
  adjacent to the nearest k/2 vertices in each
  direction around the circle.

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Harary graphs

• If k is odd and n is even, form by making
  each vertex adjacent to the nearest (k-1)/2
  vertices in each direction around the circle and
  to the diametrically opposite vertex.

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Harary graphs

• If k and n are both odd, index the vertices by
  the integers modulo n. Construct form
  by adding the edges
  for

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Harary graphs

• Theorem. , and hence the minimum
  number of edges is a k-connected graph on n
  vertices is

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Harary graphs

• Proof. (Only the even case k 2r. Pigeonhole)
• Since , it suffices to prove
• Clockwise u,v paths and counterclockwise u,v
  paths.
• Let A and B be the sets of internal vertices on
  these two paths.
• One of A, B has fewer that k/2 vertices.
• Thus, we can find a u,v path in G-S via the set A
  or B in which S has fewer than k/2 vertices.

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Harary graphs
u
v
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Edge-Connectivity

• A disconnecting set of a graph G is a set
  such that G-F has more than one component.
• The edge-connectivity of G, is the minimum
  size of a disconnecting set.
• A graph G is k-edge-connected if every
  disconnecting set has at least k edges.

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Edge-Connectivity

• An edge cut is an edge set of the form
  where is a nonempty proper subset of and
  denotes

Disconnecting set
Edge cut
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Theorem

• If G is a simple graph, then
• Proof , trivial.
• Case 1 if every vertex of is adjacent to every
  vertex of , then

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Theorem

• Case 2 , with
• consist of all neighbors of in and
  all vertices of with neighbors in .?
  is a separating set
• picking the red edges yields T distinct
  edges.?

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Example
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Theorem

• If G is a 3-regular graph, then
• Proof

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Theorem

• If G is a 3-regular graph, then
• Proof

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Theorem

• If G is a 3-regular graph, then
• Proof

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Theorem

• If G is a 3-regular graph, then
• Proof

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Definition

• A Bond is a minimal nonempty edge cut.
• Here minimal means that no proper nonempty
  subset is also an edge cut.

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Proposition

• If is a connected graph, then an edge cut is
  a bound if and only if has exactly two
  components.
• Proof?
• is a subset of . is
  connected.

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Proposition

• If is a connected graph, then an edge cut is
  a bound if and only if has exactly two
  components.
• Proof? Suppose has more than two
  component. and are proper subsets of
  , so is not a bound.

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Definition

• A Block of a graph is a maximal connected
  subgraph of G that has no cut-vertex.
• A connected graph with on cut-vertex need not be
  2-connected, since it can be or .

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Proposition

• Two blocks in a graph share at most one vertex.
• ProofSuppose for a contradiction. and
  have at least two common vertices.
• Since the blocks have at least two common
  vertices, deleting one singe vertex, what remains
  is connected. A contradiction.

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Definition

• The block-cutpoint graph of a graph G is a
  bipartite graph H in which one partite set
  consists of the cut-vertices of G, and the other
  has a vertex for each block of G. We
  include as an edge of H if and only if
  .

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Algorithm

• Computing the blocks of a graph.

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Algorithm

• Computing the blocks of a graph.

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Algorithm

• Computing the blocks of a graph.

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Algorithm

• Computing the blocks of a graph.

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Definition

• Two paths from u to v are internally disjoint if
  they have no common internal vertex.

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Theorem

• G is 2-connected if and only if for each for each
  pair there exist internally disjoint u,v
  paths in G.
• Proof? Since for every pair u,v, G has
  internally disjoint u,v paths, deletion of one
  vertex cannot make any vertex unreachable from
  any other.

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Theorem

• Prove by induction on
• Basis step.
• The graph G-uv is connected.
• Induction step.
• Let w be the vertex before v on a
  shortest u,v path

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Theorem

• Case 1 if , done.
• Case 2 G-w is connected and contains a u,v path
  R. If R avoids P or Q, done.
• Case 3 Let z be the last vertex of R.

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Expansion Lemma

• If G is a k-connected, and G is obtained from G
  by adding a new vertex y with at least k
  neighbors in G, then G is k-connected.
• Case 1 if , then
• Case 2 if and , then
• Case 3 and lie in a single
  component of , then

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Theorem

• For a graph G with at least three vertices, the
  following condition are equivalent.
• G is connected and no cut-vertex.
• For , there are internally
  disjoint x, y paths.
• For , there is a cycle through x
  and y.
• , and every pair of edges in G lies
  on a common cycle.

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Definition

• In a graph G, subdivision of an edge uv is the
  operation of replacing uv with a path u, w, v
  through a new vertex w.

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Corollary

• If G is a 2-connected, then the graph G obtained
  by subdividing an edge of G is 2-connected.
• Proof It suffices to find a cycle through
  arbitrary edges e,f of G. Since G is
  2-connected, any two edges of G lie on a common
  cycle.
• Case 1 if a cycle through them in G uses uv,
  then replace the edge uv with a path u,w,v.
• Case 2 if and
  , then
• Case 3 if , then

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Definition

• An ear of a graph G is a maximal path whose
  internal vertices have degree 2 in G.
• An ear decomposition of G is a decomposition
  such that is a cycle and
  for is an ear of .

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Theorem

• A graph is 2-connected if and only if it has an
  ear decomposition.
• Proof ? Since cycles are 2-connected, it
  suffices to show that adding an ear preserves
  2-connectedness. Trivial.
• ?

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Theorem
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Definition

• An close ear in a graph G is a cycle C such that
  all vertices of C expect one have degree 2 in G
• An close-ear decomposition of G is a
  decomposition such that is a
  cycle and for is either an (open)
  ear or a closed ear in .

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Theorem

• A graph is 2-edge-connected if and only if it has
  an closed-ear decomposition.
• Proof ? G is 2-edge-connected if and only
  if every edge lies on a cycle.
• Case 1 when adding a closed ear, Trivial.
• Case 2 when adding a open ear ,

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Theorem

• Proof ?

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Theorem

• Proof ?

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Connectivity of Digraphs

• A separating set of a digraph D is a set
  such that D-S is not strongly
  connected.
• The connectivity of G, is the minimum size
  of a vertex set S such that D-S is not strong or
  has only one vertex.
• A graph G is k-connected if its connectivity is
  at least k.

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Edge-Connectivity of Digraphs

• For vertex sets S, T in a digraph D, let S,T
  denote the set of edges with tail in S and head
  in T.
• An edge cut is an edge set of the form
  for some . A diagraph is
  k-edge-connected if every edge cut has at least k
  edges.
• The minimum size of an edge cut is the
  edge-connected

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Proposition

• Adding a directed ear to a strong digraph
  produces a larger strong digraph.

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Theorem

• A graph has a strong orientation if and only if
  it is 2-edge-connected.
• Proof?If G has a cut-edge xy oriented from x
  to y in an orientation D, then y cannot reach x
  in D.
• ? 1.) G has a closed-ear
  decomposition.
• 2.) Orient the initial cycle
  consistently to
  obtain a strong diagraph.
• 3.) Directing new ear
  consistently.

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Definition

• Given , a set
  is an x,y separator or x, y-cut if G-S has no x,
  y-path.
• Let be the minimum size of an
  x,y-cut.
• Let be the maximum size of a set of
  pairwise internally disjoint x, y-paths.
• For , an X, Y-path is a graph having
  first vertex in X, last vertex in Y, and no other
  vertex in

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Remark

• An x, y-cut must contain an internal vertex of
  every x, y-path, and no vertex can cut two
  internally disjoint x,y-paths. Therefore, always

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Example

• Although , it takes four edges to
  break all w, z-paths, and there are four pairwise
  edge-disjoint w, z-paths.