Graphs and Trees

1. Graphs and Trees • This handout: • Trees • Minimum Spanning Tree Problem • www.ohio.edu/people/melkonia/math3050/.../trees.ppt

2. Terminology of Graphs: Cycles, Connectivity and Trees • A path that begins and ends at the same node is called a cycle. Example: • Two nodes are connected if there is a path between them. • A graph is connected if every pair of its nodes is connected. • A graph is acyclic if it doesn’t have any cycle. • A graph is called a tree if it is connected and acyclic. Example:

3. Examples/Applications of Trees • Family tree • Evolutionary tree • Possibility tree • Prime number factorization tree • Spanning tree of a telecom network (considered later in this handout) • Hydrocarbon molecules

4. Properties of Trees • Definition: Let T be a tree. If T has at least 3 nodes, then • A node of degree 1 in T is called a leaf (or a terminal node). • A node of degree greater than 1 is called an internal node. • Lemma: Every tree with more than one node has at least one leaf. • Theorem: For any positive integer n, any tree with n nodes has n-1 edges. Proof by induction (blackboard).

5. Properties of Trees • Lemma: If G is any connected graph, C is a cycle in G, and one of the edges of C is removed from G, then the graph that remains is still connected. • Theorem: For any positive integer n, if G is a connected graph with n vertices and n-1 edges, then G is a tree. • Theorem: For any positive integer n, if G is an acyclic graph with n vertices and n-1 edges, then G is a tree. Proofs on blackboard.

6. Rooted Trees r • Definition: A rooted tree is a tree in which one vertex is distinguished from the others and is called the root. • The level of a vertex is the number of edges along the unique path between it and the root. • The height of a rooted tree is the maximum level to any vertex of the tree. • The children of a vertex v are those vertices that are adjacent to v and one level farther away from the root than v. • When every vertex in a rooted tree has at most two children, the tree is called a binary tree. a b c d

7. 3 3 b b d d 2 2 8 8 a a 3 3 5 5 7 7 4 4 e e c c 4 4 Minimum Spanning Tree Problem • Given: Graph G=(V, E), |V|=n Cost function c: E  R . • Goal: Find a minimum-cost spanning tree for V i.e., find a subset of arcs E*  E which connects any two nodes of V with minimum possible cost. • Example: Min. span. tree: G*=(V,E*) G=(V,E) Red bold arcs are in E*

8. Algorithm for solving the Minimum Spanning Tree Problem • Initialization: Select any node arbitrarily, connect to its nearest node. • Repeat • Identify the unconnected node which is closest to a connected node • Connect these two nodes Until all nodes are connected Note: Ties for the closest node are broken arbitrarily.

9. 3 3 b b d d 2 2 8 8 a a 3 3 5 5 7 7 4 4 e e c c 4 4 The algorithm applied to our example • Initialization: Select node a to start. Its closest node is nodeb. Connect nodesa and b. • Iteration 1: There are two unconnected node closest to a connected node: nodes c and d (both are 3 units far from node b). Break the tie arbitrarily by connecting node c to node b. Red bold arcs are in E*; thin arcs representpotential links.

10. 3 3 b b d d 2 2 8 8 a a 3 3 5 5 7 7 4 4 e e c c 4 4 The algorithm applied to our example • Iteration 2: The unconnected node closest to a connected node is noded (3far from node b). Connect nodes b and d. • Iteration 3:The only unconnected node left is node e. Its closest connected node is node c (distance between c and e is 4). Connect node e to node c. • All nodes are connected. The bold arcs give a min. spanning tree.