Graphs Todays announcements: PA3 out, due 29 March 11:59p Final - - PowerPoint PPT Presentation

Graphs

Today’s announcements:

◮ PA3 out, due 29 March 11:59p ◮ Final Exam, 12 April 7:00p, SRC A & B

Today’s Plan

◮ Graph representation ◮ Graph terminology

Division

• 1. Start at vertex 0 and leading digit.
• 2. At digit d, follow d black edges and then one red edge, and

move to next digit. Repeat.

• 3. Divisible by 6 (or 7) iff end at vertex 0.

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1 2 3 4 5 6 1 2 3 4 5

Greek gods

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MYTH

E R O S G A E A N Y X E R E B U S CHAOS P O N T U S U R A N U S H E M E R A A E T H E R T H A N A T O S M O R O S C E R N E M E S I S I A P E T U S O C E A N U S T E T H Y S E U R Y B I A C R E U S R H E A C R O N O S D I O N E P H O E B E C O E U S T H E M I S M N E M N O S Y N E H Y P E R I O N T H E A P R O M E T H E U S P A L L A S E R I S L E T O E O S H E L I O S S E L E N E N E R E U S T H A U M A S D O R I S M E T I S S T Y X I N A C H U S M E L I A P O S E I D O N H E R A Z E U S H E S T I A H A D E S P E R S E P L E I O N E E L E C T R A S T Y X D E M E T E R C H A R O N A R I O N P E R S E P H O N E P E R S E S P A S I P H A E C I R C E C L Y M N E G A L A T E A P R O T O A G A V E A M P H I T R I T E I R I S B I A N I K E I O A P O L L O A R T E M I S D I K E U R A N I A T H A L I A C L I O C A L L I O P E A E T H R A A G E N O R T E L E P H A S S A C Y R E N E A C R I S I U S T H E S T I U S A M B R O S I A E U D O R A P H Y T O E R Y T H E A H E S P E R I A D I O M E D E S H A R M O N I A D A N A E A L C E M E N E O R P H E U S D R Y O P E H E R M E S M I N O S S E M E L E P E R S E U S H E R A C L E S H E L E N M E N E L A U S C A S T O R S Y M A E T H I S P A N A R I A D N E D I O N Y S U S H E R M I O N E A C I S L A T R A M Y S H E B E A R E S A T H E N A A P H R O D I T E A T L A S T R I T O N O E A G R U S M A I A P H O E N I X E U R O P A R O M U L U S R E M U S L E D A T Y N D A R E U S T I T A N S S E A G O D S & N y m p h s D O D E K A T H E O N

• T

W E L V E O L Y M P I A N S H U M A N S & D E M I G O D S M U S E S ZEUS p r i m

• r

d i a l d e i t i e s

LEGEND OF THE MYTH

FAMILY IN THE MYTH MOTHER FATHER CHILDREN COLORS IN THE MYTH PRIMORDIAL DEITIES TITANS SEA GODS AND NYMPHS DODEKATHEON, THE TWELVE OLYMPIANS OTHER GODS GODS OF THE UNDERWORLD MUSES ANIMALS AND HYBRIDS HUMANS AND DEMIGODS Zeus D E M E T E R H E R A M A I A D I O N E S E M E L E L E T O T H E M I S P E R S E P H O N E H E B E A R E S H E R M E S A T H E N A A P H R O D I T E D I O N Y S U S A P O L L O A R T E M I S D I K E E U R O P A D A N A E A L C E M E N E L E D A M N E M O S Y N E M I N O S P E R S E U S H E R A C L E S H E L E N U R A N I A T H A L I A C L I O C A L L I O P E circles in the myth {Google results} 196 M - 8690 K 8690 K - 2740 K 2740 K - 1080 K 1080 K - 2410

• J. KLAWITTER & T

Graph definition

A graph is a pair of sets: G = (V , E).

◮ V is a set of vertices: {v1, v2, . . . , vn}. ◮ E is a set of edges: {e1, e2, . . . , em} where each ei is a pair of

vertices: ei ∈ V × V .

A B C

V = {A, B, C} E = {(A, B), (B, A), (C, B)} If each edge is an ordered pair (i.e. (A, B) = (B, A)) then the graph is directed otherwise undirected.

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Graph vocabulary

2 b e d c a g f 4 3 5 1 8 6 9 7 n k h j m

• l

i p q

Vertices adjacent to v: N(v) = {u|(u, v) ∈ E} Edges incident to v: I(v) = {(u, v)|u ∈ N(v)} Degree of v: deg(v) = |I(v)| Path: Sequence of vertices connected by edges Cycle: Path with same start and end vertex Simple graph: No self-loops or multi-edges

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Graph vocabulary

2 b e d c a g f 4 3 5 1 8 6 9 7 n k h j m

• l

i p q

Subgraph of G = (V , E): (V ′ ⊆ V , E ′ ⊆ E) and if (u, v) ∈ E ′ then u, v ∈ V ′ Complete graph: Maximum number of edges Connected graph: Path between every pair of vertices Connected component: Maximal connected subgraph Acyclic graph: no cycles Spanning tree of G(V , E): Acyclic, connected graph with vertex set V

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Graph Vocabulary: Use the previous graph to answer

• 1. List the edges incident to vertex b:
• 2. What is the degree of vertex d?
• 3. List the vertices adjacent to vertex i:
• 4. Give a path from 0 to 7:
• 5. Give a path from k to h:
• 6. List the vertices in the largest complete subgraph in G:
• 7. How many connected components are in G?
• 8. How many edges in a spanning tree of each component?
• 9. How many simple paths connect 0 and 9?
• 10. Can you draw G with no edge crossings?

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Graph properties

How many edges in a simple connected graph on n vertices? Minimum Maximum In a non-simple, non-connected graph on n vertices? Minimum Maximum

Handshaking Theorem:

If G = (V , E) is an undirected graph, then

• v∈V

deg(v) = 2|E|

Corollary

An undirected graph has an even number of vertices of odd degree.

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Topological Sort

A topological sort is a total order of the vertices of a directed graph G = (V , E) such that if (u, v) is an edge of G then u appears before v in the order.

pants shoes shirt belt watch boxers socks x y means x before y

Topological Sort Algorithm I

• 1. Find each vertex’s in-degree (# of inbound edges)
• 2. While there are vertices remaining

2.1 Pick a vertex with in-degree zero and output it 2.2 Reduce the in-degree of all vertices it has an edge to 2.3 Remove it from the list of vertices

Runtime?

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