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10.4 Spanning Trees

10.4 Spanning Trees. Def. Def: Let G be a simple graph. A spanning tree of G is a subgraph of G that is a tree containing every vertex of G See handout for examples. Thm . 1. Thm 1.: A simple graph is connected iff it has a spanning tree Recall some def: Connected ____

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10.4 Spanning Trees

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  1. 10.4 Spanning Trees

  2. Def • Def: Let G be a simple graph. A spanning tree of G is a subgraph of G that is a tree containing every vertex of G • See handout for examples

  3. Thm. 1 • Thm 1.: A simple graph is connected iff it has a spanning tree • Recall some def: • Connected ____ • Spanning tree______ • Tree_________

  4. Proof of Thm. 1 A simple graph is conn.iff it has a spanning tree: Proof Suppose G has a spanning tree T Because it is spanning, ________ Because it is a tree, 10.1 Thm. 1 says _________ Since T is a subgraph of G, G is ________ • Suppose G is connected If G is NOT a tree it must ___________ Remove an edge. The resulting graph has ___ edge and contains ___vertices of G and is ________ Repeat until _____ This is possible because______________

  5. Algorithms for constructing spanning trees • See handout and use the following methods • Depth first (backtracking) • Start with a root • Form a path by adding vertices as long as possible (without adding a circuit) • When you can’t add any more, go back to previous one and add more… • Breath first • Start with a root • Add all edges incident to this vertex (level 1), arbitrarily order them • For each vertex in level 1, add each edge incident (as long as it doesn’t form a circuit),…

  6. Depth example a d i j c e f h k b g Start at f

  7. Breadth example a b c l d e f g h i j m k start at e

  8. Use backtracking to find a subset, if possible, … • Of the set {27, 24, 19, 14, 11, 8} with the sum of 20

  9. Use backtracking to find a subset, if possible, … • Of the set {27, 24, 19, 14, 11, 8} with the sum of 41

  10. Use backtracking to find a subset, if possible, … • Of the set {27, 24, 19, 14, 11, 8} with the sum of 60

  11. Ex with colors • See if a graph has 3 colors– use a tree

  12. 10.5 Minimum spanning trees • Prim’s Algorithm • Start with smallest weight • Successively add edges that are incident, choosing smallest weights, and not forming a circuit • Stop after n-1 edges selected (with n vertices) • Kruskal’s Algorithm • Start with smallest weight • Successively add edges that are smallest weight (not necessarily incident) and not forming a circuit • Stop after n-1 edges selected (with n vertices) • See handout or book ex

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