Elementary Graph Algorithms

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

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.

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

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.

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

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

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

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

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.

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

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

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

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

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

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.

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

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)

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

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)

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.

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