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11graph_10_v3 - ENGG1007 FCS ENGG1007 Foundations of...

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1 ENGG1007 FCS ENGG1007 Foundations of Computer Science Click to edit Master subtitle style Graphs Prof. Francis Chin, Dr SM Yiu November 4/5, 2010 (Chapter 9)
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2 ENGG1007 FCS What is a Graph? A simple graph G = ( V , E ) consists of Ø V , the set of vertices (nodes, points) , and Ø E , the set of edges (lines) Undirected graphs are graphs where edges have no direction ( unordered pairs of V ). Example: V = {a, b, c, d, e} E = {{a,b}, {a, c}, {c, d}, {c, e}, {d, e}} Directed graphs are graphs where edges have directions, i.e., E consists of ordered pairs of V Example: V = {u, x, y, z} E = {(x, y), (x, z), (y, u), (z, x), (z, u)} a b c d e z x y u
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3 ENGG1007 FCS Simple Graphs and Multigraphs Simple graphs - at most one edge between any pair of vertices. Multi-graphs - having multiple edges between the same pair of vertices, or even with self-loops. Examples: Ø Multiple roads between same pair of cities If x , y are two vertices, and e = { x , y } is in E , then Ø x and y are adjacent Ø e is incident with x and y Ø x and y are the endpoints of e Degree ( v ) = number of edges incident with vertex v Ø deg ( y ) = 2 , deg ( u ) = 2 , deg ( x ) = 3 , deg ( z ) = 5 z x y u
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4 ENGG1007 FCS Graph properties: Degrees By pigeon hole principle, there exist two vertices with the same degree. For an undirected graph G = ( V , E ): e.g. sum of degrees = 3+2+5+2 = 12, | E | = 6 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 ): | | ) ( deg ) ( deg E v v V v V v = = + - | | 2 ) deg( E v V v = z x y u
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5 ENGG1007 FCS Other Applications – 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 Edge = move
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6 ENGG1007 FCS Game tree - Tic tac toe Vertices = state of the game represented by (position, whose move) position = (s1, s2, s3, s4, s5, s6, s7, s8, s9 ), where s i g { x, 0, - } Example: initial state, v0 = ( ( -, -, -, -, -, -, -, -, - ), 0 ) v5 = ( ( x, -, 0, 0, 0, -, x, -, - ), x ) What are valid vertices? Assuming 0 is the first move, say vi = ( ( x, -, x, 0, 0, -, x, -, - ), x ) ? If ‘0’ is the first moves, number(x)=number(0)-1 or number(0) say vj = ( ( x, -, x, 0, 0, -, x, 0, - ), x ) ? When number(x)=number(0), whose move = 0 else = x X X s 1 s 2 s 4 s 5 s 9 s 6 s 3 s 8 s 7
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7 ENGG1007 FCS Game tree - Tic tac toe Example: initial state, v0 = ( ( -, -, -, -, -, -, -, -, - ), 0 ) v5 = ( ( x, -, 0, 0, 0, -, x, -, - ), x ) Edges = move (reachability denoted by directed edges) Example: states (positions) reached by v5 (v5, v6 ), where v6 = ( ( x, x, 0, 0, 0, -, x, -, - ), 0 ) (v5, v6‘), where
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