11graph_10_v3

11graph_10_v3 - ENGG1007 oundations of Computer Science...

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Unformatted text preview: ENGG1007 oundations of Computer Science Foundations of Computer Science Graphs Graphs Prof. Francis Chin, Dr SM Yiu November 4/5, 2010 (Chapter 9) 1 ENGG1007 FCS hat is a Graph? hat is a Graph? What is a Graph? What is a Graph? A simple graph simple graph G = ( V , E ) consists of the set of ertices (nodes points) ertices (nodes points) and ¾ V , the set of vertices (nodes, points) vertices (nodes, points) , and ¾ E , the set of edges (lines edges (lines) Undirected graphs Undirected graphs are graphs where edges have no d direction ( unordered pairs of V ). Example: = {a, b, c, d, e} a c V {a, b, c, d, e} E = {{a,b}, {a, c}, {c, d}, {c, e}, {d, e}} Directed graphs Directed graphs are graphs where edges have directions, b e i.e., E consists of ordered pairs of V Example: = {u x y z} x y 2 V {u, x, y, z} E = {(x, y), (x, z), (y, u), (z, x), (z, u)} z u ENGG1007 FCS imple imple raphs and raphs and ultigraphs ultigraphs Simple Simple Graphs and Graphs and Multigraphs Multigraphs Simple graphs Simple graphs- at most one edge between any pair of ertices vertices. Multi Multi-graphs graphs- having multiple edges between the same pair of vertices, or even with self-loops. Examples: ¾ Multiple roads between same pair of cities x y If x , y are two vertices, and e = { x , y } is in E , then ¾ x and y are adjacent adjacent ¾ e is incident incident with x and y ¾ x and y are the endpoints endpoints of e Degree ( v ) = number of edges incident with vertex v z u 3 g ( ) g ¾ deg ( y ) = 2 , deg ( u ) = 2 , deg ( x ) = 3 , deg ( z ) = 5 ENGG1007 FCS raph properties: Degrees raph properties: Degrees Graph properties: Degrees Graph properties: Degrees By pigeon hole principle, there exist two vertices with the ame degree same degree. For an undirected graph G = ( V , E ): x y e.g. sum of degrees = 3+2+5+2 = 12, | E | = 6 | | 2 ) deg( E v V v = ∑ ∈ z u Proof: every edge is counted twice in counting degrees. Fact: Any undirected graph has an even number of odd- degree vertices (as the sum of degrees is even) For a directed graph G = ( V , E ): 4 | | ) ( deg ) ( deg E v v V v V v = = ∑ ∑ ∈ + ∈ − ENGG1007 FCS ther Applications ther Applications raph Models raph Models Other Applications Other Applications Graph Models Graph Models Acquaintanceship vertices = people edges = two people know each other Competition (tournaments) vertices = teams edges = matches (directed edges to show winners and losers) Game tree vertices = state of the game dge ove 5 Edge = move ENGG1007 FCS ame tree ame tree- ic ic c e Game tree Game tree Tic Tic tac tac toe toe Vertices = state of the game s 1 s 2 s 3 represented by (position, whose move) position = (s 1 , s 2 , s 3 , s 4 , s 5 , s 6 , s 7 , s 8 , s 9 ), where s ∈ { x, 0, - } s 4 s 5 s 9 s 6 s 8 s 7 i Example: initial state, v 0 = ( ( -, -, -, -, -, -, -, -, - ), 0 ) ( ( x 0 0 0 x x ) X v 5 = ( ( x, -, 0, 0, 0, -, x, -, - ), x ) What are valid vertices? Assuming 0 is the first move, X say v i = ( ( x, -, x, 0, 0, -, x, -, - ), x ) ?= ( ( x, -, x, 0, 0, -, x, -, - ), x ) ?...
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This note was uploaded on 02/10/2011 for the course ENGG 1007 taught by Professor Unknown during the Spring '11 term at HKU.

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11graph_10_v3 - ENGG1007 oundations of Computer Science...

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