15 251 Some Great Theoretical Ideas in Computer Science for Graphs Lecture 17 March 17 2009 What s a tree A tree is a connected graph with no cycles Tree Not Tree Not Tree Tree How Many n Node Trees 1 2 3 4 5 Notation In this lecture n will denote the number of nodes in a graph e will denote the number of edges in a graph Theorem Let G be a graph with n nodes and e edges The following are equivalent 1 G is a tree connected acyclic 2 Every two nodes of G are joined by a unique path 3 G is connected and n e 1 4 G is acyclic and n e 1 5 G is acyclic and if any two nonadjacent nodes are joined by an edge the resulting graph has exactly one cycle To prove this it suffices to show 1 2 3 4 5 1 1 2 1 G is a tree connected acyclic 2 Every two nodes of G are joined by a unique path Proof by contradiction Assume G is a tree that has two nodes connected by two different paths Then there exists a cycle 2 3 2 Every two nodes of G are joined by a unique path 3 G is connected and n e 1 Proof by induction Assume true for every graph with n nodes Let G have n nodes and let x and y be adjacent G1 x y G2 Let n1 e1 be number of nodes and edges in G1 Then n n1 n2 e1 e2 2 e 1 3 43 G is connected and n e 1 4 G is acyclic and n e 1 Proof by contradiction Assume G is connected with n e 1 and G has a cycle containing k nodes k nodes Note that the cycle has k nodes and k edges Start adding nodes and edges until you cover the whole graph Number of edges in the graph will be at least n Corollary Every nontrivial tree has at least two endpoints points of degree 1 Proof by contradiction Assume all but one of the points in the tree have degree at least 2 In any graph sum of the degrees 2e Then the total number of edges in the tree is at least 2n 1 2 n 1 2 n 1 How many labeled trees are there with three nodes 1 2 3 1 3 2 2 1 3 How many labeled trees are there with four nodes a c b d How many labeled trees are there with five nodes 5 labelings 5 x 4 x3 labelings 5 2 labelings 125 labeled trees How many labeled trees are there with n nodes 3 labeled trees with 3 nodes 16 labeled trees with 4 nodes 125 labeled trees with 5 nodes nn 2 labeled trees with n nodes Cayley s Formula The number of labeled trees on n nodes is nn 2 The proof will use the correspondence principle Each labeled tree on n nodes corresponds to A sequence in 1 2 n n 2 that is n 2 numbers each in the range 1 n How to make a sequence from a tree through i from 1 to n 2 Loop Let L be the degree 1 node with the lowest label Define the ith element of the sequence as the label of the node adjacent to L Delete the node L from the tree Example 5 1 8 4 3 2 6 1 3 3 4 4 4 7 How to reconstruct the unique tree from a sequence S Let I 1 2 3 n Loop until S is empty Let i smallest in I but not in S Let s first label in sequence S Add edge i s to the tree Delete i from I 5 Delete s from S 3 1 Add edge a b where I a b 2 8 4 6 1 3 3 4 4 4 7 Spanning Trees A spanning tree of a graph G is a tree that touches every node of G and uses only edges from G Every connected graph has a spanning tree A graph is planar if it can be drawn in the plane without crossing edges Examples of Planar Graphs http www planarity net Faces A planar graph splits the plane into disjoint faces 4 faces Euler s Formula If G is a connected planar graph with n vertices e edges and f faces then n e f 2 Rather than using induction we ll use the important notion of the dual graph Dual put a node in every face and an edge for each edge joining two adjacent faces Let G be the dual graph of G Let T be a spanning tree of G Let T be the graph where there is an edge in dual graph for each edge in G T Then T is a spanning tree for G n eT 1 n f eT eT 2 e 2 f eT 1 Corollary Let G be a simple planar graph with n 2 vertices Then 1 G has a vertex of degree at most 5 2 G has at most 3n 6 edges Proof of 1 In any graph sum of degrees 2e Assume all vertices have degree 6 Then 3n e Furthermore since G is simple 3f 2e So 3n 3f 3e 3 n e f 0 contradict Graph Coloring A coloring of a graph is an assignment of a color to each vertex such that no neighboring vertices have the same color Graph Coloring Arises surprisingly often in CS Register allocation assign temporary variables to registers for scheduling instructions Variables that interfere or are simultaneously active cannot be assigned to the same register Theorem Every planar graph can be 6colored Proof Sketch by induction Assume every planar graph with less than n vertices can be 6 colored Assume G has n vertices Since G is planar it has some node v with degree at most 5 Remove v and color by Induction Hypothesis Not too difficult to give an inductive proof of 5 colorability using same fact that some vertex has degree 5 4 color theorem remains challenging Implementing Graphs Adjacency Matrix Suppose we have a graph G with n vertices The adjacency matrix is the n x n matrix A aij with aij 1 if i j is an edge aij 0 if i j is not an edge Good for dense graphs Example 0 A 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 Counting Paths The number of paths of length k from node i to node j is the entry in position i j in the matrix Ak 0 A2 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 3 2 2 2 2 3 2 2 2 2 3 2 2 2 2 3 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 Adjacency List Suppose we have a graph G with n vertices The adjacency list is the list …
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