1. Elementary Graph Algorithms

2. Paths and Cycles
A path is a sequence of vertices P = (v0, v1, …, vk) such that, for 1 ≤ i ≤ k, edge (vi – 1, vi) ∈ E.
Path P is simple if no vertex appears more than once in P.
A cycle is a sequence of vertices C = (v0, v1, …, vk – 1) such that, for 0 ≤ i < k, edge (vi, v(i + 1) mod k) ∈ E.
Cycle C is simple if the path (v0, v1, vk – 1) is simple. a d h b g j e f c i

3. Paths and Cycles
A path is a sequence of vertices P = (v0, v1, …, vk) such that, for 1 ≤ i ≤ k, edge (vi – 1, vi) ∈ E.
Path P is simple if no vertex appears more than once in P.
A cycle is a sequence of vertices C = (v0, v1, …, vk – 1) such that, for 0 ≤ i < k, edge (vi, v(i + 1) mod k) ∈ E.
Cycle C is simple if the path (v0, v1, vk – 1) is simple.
P1 = (g, d, e, b, d, a, h) is not simple. a d h b g j e f c i
P2 = (f, i, c, j) is simple.

4. Paths and Cycles
A path is a sequence of vertices P = (v0, v1, …, vk) such that, for 1 ≤ i ≤ k, edge (vi – 1, vi) ∈ E.
Path P is simple if no vertex appears more than once in P.
A cycle is a sequence of vertices C = (v0, v1, …, vk – 1) such that, for 0 ≤ i < k, edge (vi, v(i + 1) mod k) ∈ E.
Cycle C is simple if the path (v0, v1, vk – 1) is simple.
C1 = (a, h, j, c, i, f, g, d) is simple. a d h b g j e f c i

5. Paths and Cycles
A path is a sequence of vertices P = (v0, v1, …, vk) such that, for 1 ≤ i ≤ k, edge (vi – 1, vi) ∈ E.
Path P is simple if no vertex appears more than once in P.
A cycle is a sequence of vertices C = (v0, v1, …, vk – 1) such that, for 0 ≤ i < k, edge (vi, v(i + 1) mod k) ∈ E.
Cycle C is simple if the path (v0, v1, vk – 1) is simple.
C1 = (a, h, j, c, i, f, g, d) is simple. a d h b g j e f c i
C2 = (g, d, b, h, j, c, i, e, b) is not simple.

6. Subgraphs
A graph H = (W, F) is a subgraph of a graph G = (V, E) if W ⊆ V and F ⊆ E.

7. Subgraphs
A graph H = (W, F) is a subgraph of a graph G = (V, E) if W ⊆ V and F ⊆ E.

8. Spanning Graphs
A spanning graph of G is a subgraph of G that contains all vertices of G.

9. Spanning Graphs
A spanning graph of G is a subgraph of G that contains all vertices of G.

10. Connectivity
A graph G is connected if there is a path between any two vertices in G.
The connected components of a graph are its maximal connected subgraphs.

11. Trees
A tree is a graph that is connected and contains no cycles.

12. Spanning Trees
A spanning tree of a graph is a spanning graph that is a tree.

13. Adjacency List Representation of a Graph
List of vertices
Pointer to head of adjacency list
List of edges
Pointers to adjacency list entries
Adjacency list entry
Pointer to edge
Pointers to endpoints y d w u e c x a v b z w y u x z v d a b c e

14. Graph Exploration Goal: Visit all the vertices in the graph of G
Count them
Number them …
Identify the connected components of G
Compute a spanning tree of G

15. Graph Exploration ExploreGraph(G)
1 Label every vertex and edge of G as unexplored
2 for every vertex v of G
3 do if v is unexplored
4 then mark v as visited
5 S ← Adj(v)
6 while S is not empty
7 do remove an edge (u, w) from S
8 if (u, w) is unexplored
9 then if w is unexplored
10 then mark edge (u, w) as tree edge
11 mark vertex w as visited
12 S ← S ∪ Adj(w)
13 else mark edge (u, w) as a non-tree edge

16. Properties of ExploreGraph
Theorem: Procedure ExploreGraph visits every vertex of G.
Theorem: Every iteration of the for-loop that does not immediately terminate completely explores a connected component of G.
Theorem: The graph defined by the tree edges is a spanning forest of G.

17. Connected Components ConnectedComponents(G)
1 Label every vertex and edge of G as unexplored
2 c ← 0
3 for every vertex v of G
4 do if v is unexplored
5 then c ← c + 1
6 assign component label c to v
7 S ← Adj(v)
8 while S is not empty
9 do remove an edge (u, w) from S
10 if (u, w) is unexplored
11 then if w is unexplored
12 then mark edge (u, w) as tree edge
13 assign component label c to w
14 S ← S ∪ Adj(w)
15 else mark edge (u, w) as a non-tree edge

18. Breadth-First Search Running time: O(n + m) BFS(G)
1 Label every vertex and edge of G as unexplored
2 for every vertex v of G
3 do if v is unexplored
4 then mark v as visited
5 S ← Adj(v)
6 while S is not empty
7 do (u, w) ← Dequeue(S)
8 if (u, w) is unexplored
9 then if w is unexplored
10 then mark edge (u, w) as tree edge
11 mark vertex w as visited
12 Enqueue(S, Adj(w))
13 else mark edge (u, w) as a non-tree edge
Running time: O(n + m)

19. Properties of Breadth-First Search
Theorem: Breadth-first search visits the vertices of G by increasing distance from the root.
Theorem: For every edge (v, w) in G such that v ∈ Li and w ∈ Lj, |i – j| ≤ 1. L0 L4 L3 L1 L2

20. Depth-First Search Running time: O(n + m) DFS(G)
1 Label every vertex and edge of G as unexplored
2 for every vertex v of G
3 do if v is unexplored
4 then mark v as visited
5 S ← Adj(v)
6 while S is not empty
7 do (u, w) ← Pop(S)
8 if (u, w) is unexplored
9 then if w is unexplored
10 then mark edge (u, w) as tree edge
11 mark vertex w as visited
12 Push(S, Adj(w))
13 else mark edge (u, w) as a non-tree edge
Running time: O(n + m)

21. An Important Property of Depth-First Search
Theorem: For every non-tree edge (v, w) in a depth-first spanning tree of an undirected graph, v is an ancestor of w or vice versa.

22. A Recursive Depth-First Search Algorithm
DFS(G)
1 Label every vertex and every edge of G as unexplored
2 for every vertex v of G
3 do if v is unexplored
4 then DFS(G, v)
DFS(G, v)
1 mark vertex v as visited
2 for every edge (v, w) in Adj(v)
3 do if (v, w) is unexplored
4 then if w is unexplored
5 then mark edge (v, w) as tree edge
6 DFS(G, w)
7 else mark edge (v, w) as a non-tree edge