23 Graphs and Trees

23 Graphs and Trees - Handout #23 CS103 April 21, 2010...

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Unformatted text preview: Handout #23 CS103 April 21, 2010 Robert Plummer Graphs and Trees More on Graphs We discussed graphs when we were studying relations. Here are some slightly different and extended definitions: Definition An undirected graph is an ordered pair (V, E) where (i) V is a non-empty set of vertices, and (ii) E is an edge set consisting of unordered pairs of distinct vertices. Unless we say otherwise, we will be talking about undirected graphs. (The reference for these definitions is: Introduction to Algorithms, 2 nd Ed. , Thomas Cormen, et al. This is a classic CS text that you will probably encounter in CS161.) Definitions If (u, v) is an edge, we say (u, v) is incident on vertices u and v, that u is adjacent to v, and that v is adjacent to u. The degree of a vertex is the number of edges incident on it. A path of length k from a vertex u to a vertex u' in a graph G = (V, E) is a sequence (v , v 1 , v 2 , ..., v k ) of vertices from V such that u = v , u' = v k , and (v i-1 , v i ) ∈ E for i = 1, 2, ..., k. A path is simple if all its vertices are distinct. A path (v , v 1 , v 2 , ..., v k ) forms a (simple) cycle if k ≥ 3, v = v k , and v 1 , v 2 , ..., v k are distinct. A graph with no cycles is acyclic . A graph is connected if there is a path between every pair of vertices. 1 2 3 4 5 6 7 – 2 – Trees Definition A tree is an acyclic, connected, graph in which one vertex has been designated as the root (node 1 in the example below): We normally draw the root of a tree at the top, and we refer to the vertices as nodes . Here are some properties of trees: Let G = (V, E) be an undirected graph. The following statements are equivalent: 1. G is a tree. 2. Any two vertices of G are connected by a unique simple path. 3. G is connected, but if any edge in E is removed, the result is not connected. 4. G is connected, and |E| = |V| - 1. 5. G is acyclic, and |E| = |V| - 1. 6. G is acyclic, but if an edge is added to E, the resulting graph has a cycle. Definitions Consider a node x in a tree with root r. Any node on the unique path from r to x is an ancestor of x. If y is an ancestor of x, x is a descendant of y. If the last node on the path from r to x is (y, x), then y is the parent of x and x is the child of y. The number of children of a node is called the degree of the node. Nodes with the same parent are siblings . A node with no children is a leaf , and a node that is not a leaf is an interior nod e. The length of the path from the root to a node is the depth (or level ) of the node, and the length of the longest path from the root to a leaf is the height of the tree. 1 2 3 4 5 6 7 – 3 – Definitions The subtree rooted at x is the tree consisting of x, its descendants, and the edges connecting them....
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23 Graphs and Trees - Handout #23 CS103 April 21, 2010...

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