Graphs Todays announcements: PA3 due 29 Nov 23:59 Final Exam, 10 - - PowerPoint PPT Presentation

Graphs

Today’s announcements:

◮ PA3 due 29 Nov 23:59 ◮ Final Exam, 10 Dec 12:00, OSBO A

Today’s Plan

◮ Graph representation ◮ Graph terminology

Division

• 1. Start at vertex 0 and leading digit.

2.

• 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 Applications

Storing things that are graphs by nature

◮ Road networks ◮ Airline flights ◮ Relationships between people, things ◮ Room connections in Hunt the Wumpus

Compilers

◮ call graph - which functions call which others ◮ control flow graph - which fragments of code can follow which

• thers

◮ dependency graphs - which variables depend on which others

Others

◮ circuits, class hierarchies, meshes, networks of computers, ...

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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

• 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. Vertices in the largest complete subgraph in G:
• 7. How many connected components are in G?
• 8. How many edges in a spanning forest?
• 9. How many simple paths connect 0 and 9?
• 10. Can you draw G with no edge crossings?

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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

Number of edges

How many edges in a simple connected graph on n vertices? Minimum Maximum

• 1. If n = 1 or 2, minimally connected has

edges.

• 2. IH: minimally connected on n − 1 vertices has

edges.

• 3. Minimally connected graphs are acyclic.
• 4. If deg(v) ≥ 2 for all v ∈ V then graph has cycle.
• 5. So minimally connected graph contains v with deg(v) = 1.
• 6. Remove this v (and one incident edge).
• 7. Remaining graph is min. conn. and has

edges (by IH).

• 8. Original n vertex graph has

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

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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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