4'2 Trees

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4.2 Trees

• Remove some vertices from a connected graph.
• Spanning tree
• Minimal spanning tree (may involve weights)
• Subgraph

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End Lesson 17a
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4.2 Spanning Tree Theorem

• Thm 4.2.1 Every connected graph has a spanning
  tree.
• SPANNING TREE ALGORITHM
• Start your tree with any vertex.
• When your tree contains all the vertices, stop.
• Find a vertex not in the tree but connected by an
  edge to a vertex in the tree. Add vertex and
  edge.
• Try to add another vertex.
• Proof of the Spanning Tree Theorem.

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Lemma A graph G with n vertices has a
spanning tree T with n vertices and n-1 edges.
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Thm 4.2.2
A graph with n vertices and n or more edges
must have at least one cycle. NOTICE A cycle
need not be an Euler cycle.
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Thm 4.2.2 A graph with n vertices and n or
more edges must have at least one cycle.
PROOF for the case of one component If G is a
connected graph with n vertices, construct a
spanning tree T (with n vertices and n-1 edges).
If G has at least n edges also, there is an edge
e of G which is not in T. Combining that edge
with T produces a cycle in G.
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Thm 4.2.2 A graph with n vertices and n or
more edges must have at least one cycle.
PROOF for the case of two components Let G be a
graph with two connected components G1 and G2.
If G has no cycle, then neither G1 nor G2 has a
cycle. Then each component has fewer vertices
than edges, and so does G. The contrapositive of
this is that if G has at least as many vertices
as edges, then G must have a cycle.
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Thm 4.2.2 A graph with n vertices and n or
more edges must have at least one cycle.
PROOF for the case of two components
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Thm 4.2.3 "Daisy Chain" TheoremFor a graph T
with n vertices, the following statements are
equivalent.

• T is a tree.
• T has no cycles and has n-1 edges.
• T is connected and has n-1 edges.
• T is connected but removing any edge will
  disconnect it.
• T has exactly one path between any two vertices.
• T has no cycles, but adding an edge between any
  two vertices will create a cycle.

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MATHEMATICAL SYMBOLS

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