Midterm Review

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Midterm Review

• 02/26/07

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General Definitions

• G (V, E)
• Vertex (plu vertices)
• Edge (directed, undirected)
• Order, size
• Adjacency, incidency
• Subgraph
• Spanning subgraph
• Induced subgraph

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Navigation

• Walk (closed walk)
• Trail (closed trail circuit)
• Path (closed path cycle)

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Connectivity

• Two vertices u and v are connected
• Graph G is connected
• Connected components
• Equivalence classes of the connectivity relation

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Theorem

• Let G be a graph of order 3 or more. If G
  contains u,v such that G-u and G-v are connected,
  then G itself is connected.

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u-v paths

• Distance between u and v,
• A u-v geodesic,
• Diameter of a graph
• If uv0, v1, , vk is a geodesic,
• distance(u,vi) i for all i

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• Complement of a graph
• Bipartite graphs
• A graph is bipartite iff it contains no odd cycles

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Degrees

• Degree of a vertex
• Min degree, max degree of a graph
• Sum of degrees ?
• Every graph has an even number of odd vertices
• Regular graphs

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Degree sequences

• A non increasing sequence is called graphical if
  
• Theorem
• The sequence s is graphical iff s is graphical

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u vertex with degree d1
d1 vertices of highest degree after u
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Graph Isomorphism

• Definition
• Isomorphism is an equivalence relation over all
  graphs

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Recognizability

• Definition
• The following are recognizable
• Order
• Size
• Degree sequence

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Bridges

• Bridge is an edge such that
• An edge is a bridge if and only if it lies on no
  cycle

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Trees

• Connected, acyclic graphs
• Forest acyclic but not necessarily connected

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Properties On a tree,

• There is a unique path between every pair of
  vertices
• There are at least two leaves (end vertices)
• The order m n 1
• The size of any connected graph (with n vertices
  is) at least n-1

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Minimum Spanning Trees

• Kruskals algorithm add the smallest weight edge
  that does not create a cycle
• Prims algorithm grows a single tree
• They are both instances of a generic MST
  algorithm

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(Vertex) Connectivity

• Cut vertex
• If v is a cut vertex, u and v are not connected
  in G-v, then v lies on every u-v path on G
• G non-separable means G has no cut vertex

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Theorem

• Let G be a graph of order 3 or more.
• G is non-separable iff every two vertices lie on
  a common cycle

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Blocks

• Block maximal non-separable subgraph
• Let R be a relation on edges such that eRf iff
  ((ef) or (e and f lie on a common cycle)
• This is an equivalence relation
• Blocks are equivalence classes of this relation
• Blocks are edge disjoint, share at most one vertex

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Vertex Connectivity

• Vertex cut
• ?(G) minimum vertex cut
• A graph is k-connected if ?(G)? k

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Edge Connectivity

• ?(G)
• ?(G) ? ?(G) ? ?(G)

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Mengers Theorem

• Let u and v be non-adjacent vertices in G
• Then the maximum number of internally disjoint
  u-v paths is equal to the cardinality of the
  minimum u-v separating set.

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Theorem

• A nontrivial graph G is k-connected iff for each
  pair of distinct vertices u,v there are at least
  k internally disjoint u-v paths.

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Corollary

• Let G be a k-connected graph.
• Let S be any subset of k vertices of G.
• Graph H is obtained by adding a vertex to G and
  joining that vertex to the vertices in S.
• Then H is also k-connected.

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Eulerian Graphs

• Idea visit all edges, no edge repetitions
• Recognition Eulerian iff all degrees are even
• Eularian Trail iff all but two degrees are even.
• An Eulerian trail is an open trail.

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Hamiltonian Graphs

• Visit all vertices, no vertex repetitions
• No efficient characterization
• We have seen some necessary and some sufficient
  conditions.

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A necessary condition

• If G is Hamiltonian, for any subset S of
  vertices of components in G-S ? S

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A sufficient condition

• G be a graph of order n ? 3
• If deg(u)deg(v) ? n for each nonadjacent u,v,
  then G is Hamiltonian.

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Another sufficient condition

• Let u,v be two nonadjacent vertices such that
  deg(u)deg(v) ? n.
• G is Hamiltonian iff Ge is Hamiltonian
• We used this to define closure operation.

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A few remarks about the exam

• 90mins, 5 questions
• Unless specified
• All graphs are connected, undirected
• m denotes the size, n denotes the order