Graphs Today’s announcements:
I PA3 out, due 29 March 11:59p
I Final Exam, 12 April 7:00p, SRC A & B
Today’s Plan 0 0 6 1 I Graph representation 5 1
I Graph terminology 5 2 Division 4 2 1. Start at vertex 0 and leading digit. 3 4 3 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.
1 / 8 Greek gods MYTH Graph of greek mythological figures
CHAOS
s itie l de rdia EBUS imo ER pr E ROS NYX GAEA
PO NTUS
IS NEMES CER MOROS THANATOS ERA AETHER URANUS HEM
S AN TIT
THEA IAPE TUS ERION HYP YNE EMNOS OCEANU MN S TET EMIS HYS EUS TH EU EBE CO NE RYBIA CREU IONE PHO SELE S RHEA CRONOS D S HELIO PR NERE OMET EOS US HEUS
THAUM AS RSE D PE ORIS LETO METIS PALLAS ERIS
STYX INACHUS ME ADES LIA POSEIDON HESTIA H HERA ZEUS SEA PLEION GO E ELEC DS TRA ST & N YX CE ym p CIR hs AE DEMETER SIPH S PA PERSE
P IANS E OLYM TWELV THEON - ES DOD EKA US M CHARON C LYMN E ARION PERSEPHONE G ALATEA LIOPE PR CAL OTO DITE CLIO AGAV HEBE ARES ATHENA APHRO HALIA E AMP ANIA T HITRITE E UR IRIS IS DIK BIA LLO ARTEM NIKE APO AETHR IO ATL A RUS AS ESTIUS OEAG TH IUS ACRIS AGENOR TELEPHASSA CYRENE
S DAREU AMB TRI TYN ROSIA TON EUD LEDA S ORA MAIA RPHEU PHYT O O ERY THEA PHOENIX EUROPA RO HESPE MULUS REMUS DS RIA GO EMI MENE & D ALCE ANS DANAE HUM MONIA DIOMEDES HAR
OR CAST NELAUS ME HELEN ACLES EUS HER PERS DRYOPE HERMES MINOS SEMELE
RMIONE HE
US SYMAETHIS PAN ARIADNE DIONYS
ACIS LATRAMYS
LEGEND OF THE MYTH
FAMILY IN THE MYTH FATHER MOTHER ZEUS Zeus
CHILDREN DEMETER THEMIS HERA MAIA DIONE SEMELE LETO COLORS IN THE MYTH PRIMORDIAL DEITIES TITANS SEA GODS AND NYMPHS DODEKATHEON, THE TWELVE OLYMPIANS OTHER GODS PERSE PHONE HEBE TEMIS DIKE MUSES ARES HERMES ATHENA APHRODITE DIONYSUS APOLLO AR
GODS OF THE UNDERWORLD EUROPA DANAE ALCEMENE LEDA MNEMOSYNE ANIMALS AND HYBRIDS HUMANS AND DEMIGODS
circles in the myth 196 M - 8690 K {Google results} M 8690 K - 2740 K INOS CALLIOPE PERSEUS HERACLES HELEN URANIA THALIA CLIO 2740 K - 1080 K 1080 K - 2410 2 / 8 J. KLAWITTER & T. MCHEDLIDZE Graph definition A graph is a pair of sets: G = (V , E).
I V is a set of vertices: {v1, v2,..., vn}.
I E is a set of edges: {e1, e2,..., em} where each ei is a pair of vertices: ei ∈ V × V . A C V = {A, B, C} B E = {(A, B), (B, A), (C, B)} If each edge is an ordered pair (i.e. (A, B) 6= (B, A)) then the graph is directed otherwise undirected.
3 / 8 Graph vocabulary Vertices adjacent to v: N(v) = {u|(u, v) ∈ E} b a c 8 9 Edges incident to v: I (v) = {(u, v)|u ∈ N(v)} e d
g f 6 7 o Degree of v: deg(v) = |I (v)|
n m 4 5 l Path: Sequence of vertices connected by edges k j 2 3 i h 0 1 Cycle: Path with same start and end vertex p q
Simple graph: No self-loops or multi-edges
4 / 8 Graph vocabulary Subgraph of G = (V , E): (V 0 ⊆ V , E 0 ⊆ E) b and if (u, v) ∈ E 0 then u, v ∈ V 0 a c 8 9 e d Complete graph: Maximum number of edges
g f 6 7 o Connected graph: Path between every pair of n m 4 5 vertices l k j 2 3 Connected component: Maximal connected i subgraph h 0 1 p q Acyclic graph: no cycles
Spanning tree of G(V , E): Acyclic, connected graph with vertex set V 5 / 8 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?
6 / 8 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 X deg(v) = 2|E| v∈V
Corollary An undirected graph has an even number of vertices of odd degree.
7 / 8 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.
belt x y watch means x before y socks
shoes
pants
shirt boxers
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? 8 / 8