What is a graph ?

123

45

What is a graph ?

G=(V,E)

V = a set of verticesE = a set of edges

edge = unordered pair of vertices

123

45

What is a graph ?

G=(V,E)

V = {1,2,3,4,5}E = {{1,5}, {3,5}, {2,3}, {2,4}, {3,4}}

123

45

What is a graph ?

G=(V,E)

V = {1,2,3,4,5}E = {{1,5}, {2,3}, {2,4}, {3,4}}

123

45

Connectedness

1 23

45

1 23

45

connected

not connected

How can we check if a graph is connected?

Representing a graph

|V| * |V| symmetric matrix A

with Ai,j = 1 if {i,j} E Ai,j = 0 otherwise

adjacency matrix

Representing a graph

adjacency matrix1 2

3

450 0 1 0 1

0 0 1 1 0

1 1 0 1 0

0 1 1 0 0

1 0 0 0 0

space = (V2)

Representing a graph

adjacency matrix1 2

3

450 0 1 0 1

0 0 1 1 0

1 1 0 1 0

0 1 1 0 0

1 0 0 0 0 space = (V2)

is {u,v} an edge ? (?)list neighbors of v (?)

Representing a graph

adjacency matrix1 2

3

450 0 1 0 1

0 0 1 1 0

1 1 0 1 0

0 1 1 0 0

1 0 0 0 0 space = (V2)

is {u,v} an edge ? (1)list neighbors of v (n)

Representing a graph

adjacency lists

for each vertex v Vlinked list of neighbors of v

Representing a graph

adjacency lists1 2

3

45

space = (E)

1: 3,52: 3,43: 1,2,44: 2,35: 1

Representing a graph

adjacency lists1 2

3

45

space = (E)

1: 3,52: 3,43: 1,2,44: 2,35: 1

is {u,v} an edge ? (?)list neighbors of v (?)

Representing a graph

adjacency lists1 2

3

45

space = (E)

1: 3,52: 3,43: 1,2,44: 2,35: 1

is {u,v} an edge ? (min(dv,du))list neighbors of v (dv)

Representing a graph

adjacency listsspace = (E)1: 3,5

2: 3,43: 1,2,44: 2,35: 1

0 0 1 0 1

0 0 1 1 0

1 1 0 1 0

0 1 1 0 0

1 0 0 0 0

space = (V2)adjacency matrix

is {u,v} in E ? (min{du,dv})neigbors of v ? (dv)

is {u,v} in E ? (1)neigbors of v ? (n)

Counting connected components

How can we check if a graph is connected?

INPUT: graph G given by adjacency list

OUTPUT: number of components of G

BFS (G,v)

seen[v] true enqueue(Q,v) while Q not empty do w dequeue(Q) for each neighbor u of w if not seen[u] then seen[u] true enqueue(Q,u)

G – undirected graph, V={1,...,n}seen[v] = false for all v VQ=queue (FIFO)

Counting connected components

C 0 for all v V do seen[v] false for all v V do if not seen[v] then C++ BFS(G,v)

output G has C connected components

DFS

explore(G,v) visited[v] true for each neighbor u of v if not visited(u) then explore(G,u)

G – undirected graph, V={1,...,n}visited[v] = false for all v V

DFS

explore(G,v) visited[v] true pre[v] clock; clock++ for each neighbor u of v if not visited(u) then explore(G,u) post[v] clock; clock++

G – undirected graph, V={1,...,n}visited[v] = false for all v V

DFS explore(G,v) visited[v] true pre[v] clock; clock++ for each neighbor u of v if not visited(u) then explore(G,u) post[v] clock; clock++

vertex Iv := [pre[v],post[v]]

“interval property” for u,v V either * Iv and Iu are disjoint, or * one is contained in the other

DFS explore(G,v) visited[v] true pre[v] clock; clock++ for each neighbor u of v if not visited(u) then explore(G,u) post[v] clock; clock++A

B

C D

DFS explore(G,v) visited[v] true pre[v] clock; clock++ for each neighbor u of v if not visited(u) then explore(G,u) post[v] clock; clock++A

B

C D

tree edges

Digraphs (directed graphs)

G=(V,E)

V = a set of verticesE = a set of edges

edge = ordered pair of vertices

uv (u,v)

Digraphs (directed graphs)

adjacency listsfor each vertex v Vlinked list of out-neighbors of v

|V| * |V| matrix A

with Ai,j = 1 if (i,j) E Ai,j = 0 otherwise

adjacency matrix

a path = sequence of vertices v1,v2,...,vk such that (v1,v2) E, ... , (vk-1,vk) E

Digraphs (directed graphs)

DAGs (acyclic digraphs)

a cycle = sequence of vertices v1,v2,...,vk such that (v1,v2) E, ... , (vk-1,vk),(vk,v1) E

DAG = digraph with no cycle

Topological sort (linearization)

INPUT: DAG G given by adjacency list

OUTPUT: ordering of vertices such that edges go forward

DFS on digraphs

explore(G,v) visited[v] true pre[v] clock; clock++ for each out-neighbor u of v if not visited(u) then explore(G,u) post[v] clock; clock++

G = digraph, V={1,...,n}visited[v] = false for all v V

DFS on digraphs

A

B

C D

DFS on digraphs

A

B

C D

root

descendant, ancestorchild, parent

DFS on digraphs

A

B

C D

tree edge

DFS on digraphs

A

B

C D

tree edge

DFS on digraphs

A

B

C D

tree edgeback edge

DFS on digraphs

A

B

C D

tree edgeback edge

cross edge

DFS on digraphs

A

B

C D

tree edgeback edge

cross edge

forward edge

Relationships between the intervals?

A

B

C D

tree edgeback edge

cross edge

forward edge

Topological sort using DFS

Lemma: digraph is a DAG if and only if DFS has a back edge.

Topological sort using DFSLemma: digraph is a DAG if and only if DFS has a back edge.

explore(G,v) visited[v] true pre[v] clock; clock++ for each neighbor u of v if not visited(u) then explore(G,u) post[v] clock; clock++

Lemma: in a DAG every edge goes to a vertex with lower post

(strong) connectedness

a digraph G is strongly connectedif for every u,v V there exists a path from u to v in G

(strong) connectedness

How to check if a digraph isstrongly connected?

(strong) connectedness

How to check if a digraph isstrongly connected?

for every uV do DFS(G,u) check if every vV was visited

(strong) connectedness

How to check if a digraph isstrongly connected?

pick some uV DFS(G,u)check if every vV was visitedDFS(reverse(G),u)check if every vV was visited

Strongly connected components

DAG of strongly connected components

Strongly connected components

Lemma: G and reverse(G) havethe same strongly connectedcomponents.

Strongly connected components

DAG of strongly connected components

Strongly connected components

for all v V do color[v] white

for all v V do if color[v]=white then DFS(reverse(G),v) DFS(G,u) (vertices in order post[])