Graphs: Definitions

Graphs: Definitions • A graph G is a pair (V, E) where • V is a set of vertices • E is a set of edges

Graphs: Definitions • A graph G is a pair (V, E) where • V is a set of vertices V = {A, B, C, D, E, F} • E is a set of edges E = {(A,B), (A,D), (B,C), (C,D), (C,E), (D,E)} B C A F D E

Graphs: Definitions • Each edge is a pair (v, w), where v, w V • V = {A, B, C, D, E, F} • E = {(A,B), (A,D), (B,C), (C,D), (C,E), (D,E)} B C A F D E

Graphs: Definitions • If edge pairs are ordered, the graph is directed, otherwise undirected. B C A F D E

Graphs: Definitions • If edge pairs are ordered, the graph is directed, otherwise undirected. • We draw edges in undirected graphs with lines with no arrow heads. B C A F This is an undirected graph. D E

Graphs: Definitions • If edge pairs are ordered, the graph is directed, otherwise undirected. • We draw edges in undirected graphs with lines with no arrow heads. (B, C) and (C, B) mean the same edge B C A F This is an undirected graph. D E

Graphs: Definitions • If edge pairs are ordered, the graph is directed, otherwise undirected. • We draw edges in directed graphs with lines with arrow heads. This edge is (B, C). (C, B) wouldmean a directed edge from C to B B C A F This is a directed graph. D E

Graphs: Definitions • Vertex w is adjacent to v if and only if (v, w)  E. B C A F D E

Graphs: Definitions • Vertex w is adjacent to v if and only if (v, w)  E. • In a directed graph the order matters: • B is adjacent to A in this graph, but A is not adjacent to B. B C A F D E

Graphs: Definitions • Vertex w is adjacent to v if and only if (v, w)  E. • In an undirected graph the order does not matter: • we say B is adjacent to A and that A is adjacent to B. B C A F D E

Graphs: Definitions • In some cases each edge has a weight (or cost) associated with it. 4 3 B C A 4.5 1.2 7.5 F – 2.3 D E

Graphs: Definitions • In some cases each edge has a weight (or cost) associated with it. • The costs might be determined by a cost function, c: E  R, where Ris the real numbers (or integers). 4 3 B C A 4.5 1.2 7.5 F – 2.3 D E

Graphs: Definitions • In some cases each edge has a weight (or cost) associated with it. • The costs might be determined by a cost function, c: E  R, where Ris the real numbers (or integers). E.g., c(A, B) = 3, c(D,E) = – 2.3, etc. 4 3 B C A 4.5 1.2 7.5 F – 2.3 D E

Graphs: Definitions • In some cases each edge has a weight (or cost) associated with it. • When no edge exists between two vertices, we say the cost is infinite. E.g., c(C,F) =  4 3 B C A 4.5 1.2 7.5 F – 2.3 D E

Graphs: Definitions • Let G = (V, E) be a graph. • A subgraph of G is a graph H = (V*, E*) such that V*  V and E*  E. (Since H is a graph, it must be the case that E*  V*  V*.) B C A F D E

Graphs: Definitions • Let G = (V, E) be a graph. • A subgraph of G is a graph H = (V*, E*) such that V*  V and E*  E. (Since H is a graph, it must be the case that E*  V*  V*.) E.g., V* = {A, C, D}, E* = {(C, D)}. B C A F D E

Graphs: Definitions • Let G = (V, E) be a graph. • A subgraph of G is a graph H = (V*, E*) such that V*  V and E*  E. (Since H is a graph, it must be the case that E*  V*  V*.) E.g., V* = {A, C, D}, E* = {(C, D)}. C A D

Graphs: Definitions • Let G = (V, E) be a graph. • A path in the graph is a sequence of vertices • w , w , . . . , w such that (w , w )  E for1 i  N–1. 1 2 N i i+1 E.g., A, B, C, E is a path in this graph B C A F D E

Graphs: Definitions • Let w , w , . . . , w be a path. • The length of the path is the number of edges, N–1, one less than the number of vertices in the path. 1 2 N E.g., the length of path A, B, C, E is 3. B C A F D E

Graphs: Definitions • Let w , w , . . . , w be a path in a directed graph. • Since each edge (w , w ) in the path is ordered, • the arrows on the path are always directed along • the path. 1 2 N i i+1 E.g., A, B, C, E is a path in this directed graph, but . . . B C A F D E

Graphs: Definitions • Let w , w , . . . , w be a path in a directed graph. • Since each edge (w , w ) in the path is ordered, • the arrows on the path are always directed along • the path. 1 2 N i i+1 . . . but A, B, C, D is not a path, since (C, D) is not an edge. B C A F D E

Graphs: Definitions A path is simple if all vertices in it are distinct, except that the first and last could be the same. E.g., the path A, B, C, E is simple . . . B C A F D E

Graphs: Definitions A path is simple if all vertices in it are distinct, except that the first and last could be the same. . . . and so is the path A, B, C, E, D, A. B C A F D E

Graphs: Definitions A path is simple if all vertices in it are distinct, except that the first and last could be the same. . . . but the path A, B, C, E, D, C is not simple, since C is repeated. B C A F D E

Graphs: Definitions If G is an undirected graph, we say it is connected if there is a path from every vertex to every other vertex. This undirected graph is not connected. B C A F D E

Graphs: Definitions If G is an undirected graph, we say it is connected if there is a path from every vertex to every other vertex. This one is connected. B C A F D E

Graphs: Definitions If G is an directed graph, we say it is strongly connected if there is a path from every vertex to every other vertex. This directed graph is strongly connected. B C A F D E

Graphs: Definitions If G is an directed graph, we say it is strongly connected if there is a path from every vertex to every other vertex. This directed graph is not strongly connected; e.g., there’s no path from D to A. B C A F D E

Graphs: Definitions If G is directed and not strongly connected, but the underlying graph (without direction to the edges) is connected, we say that G is weakly connected. This directed graph is notstrongly connected, but it is weakly connected, since . . . B C A F D E

Graphs: Definitions If G is directed and not strongly connected, but the underlying graph (without direction to the edges) is connected, we say that G is weakly connected. . . . since the underlying undirected graph is connected. B C A F D E

Graphs: Definitions • A cycle in a graph is a path of length at least 1 such that w = w . 1 N The path A, B, C, E, D, A is a cycle. B C A F D E

Graphs: Definitions • A graph with no cycles is called acyclic. This graph is acyclic. B C A F D E

Graphs: Definitions • A graph with no cycles is called acyclic. This directed graph is not acyclic, . . . B C A F D E

Graphs: Definitions • A graph with no cycles is called acyclic. . . . but this one is. A Directed Acyclic Graph is often called simply a DAG. B C A F D E

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. Incomplete: B C A D E

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. Complete: B C A D E

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. • Suppose G = (V, E) is complete. Can you express |E| as a function of |V|? This graph has |V| = 5 vertices and |E| = 10 edges. Complete: B C A D E

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. • Suppose G = (V, E) is complete. Can you express |E| as a function of |V|? Make a small table: Complete: |V| |E| 1 0 A

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. • Suppose G = (V, E) is complete. Can you express |E| as a function of |V|? Complete: |V| |E| B 1 0 A 2 1

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. • Suppose G = (V, E) is complete. Can you express |E| as a function of |V|? Complete: |V| |E| B C 1 0 A 2 1 3 3

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. • Suppose G = (V, E) is complete. Can you express |E| as a function of |V|? Complete: |V| |E| B C 1 0 A 2 1 3 3 4 6 D

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. • Suppose G = (V, E) is complete. Can you express |E| as a function of |V|? Complete: |V| |E| B C 1 0 A 2 1 3 3 4 6 5 10 D E

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. • Suppose G = (V, E) is complete. Can you express |E| as a function of |V|? |V| |E| B C 1 0 A 2 1 3 3 F 4 6 5 10 D E 6 ?

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. • Suppose G = (V, E) is complete. Can you express |E| as a function of |V|? |V| |E| B C 1 0 A 2 1 3 3 F 4 6 5 10 D E 6 15

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. • Suppose G = (V, E) is complete. Can you express |E| as a function of |V|? The pattern, for getting from N–1 vertices to N, is that we add N–1 new edges . . . |V| |E| B C 1 0 +1 A 2 1 +2 3 3 F +3 4 6 +4 5 10 D +5 E 6 15

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. • Suppose G = (V, E) is complete. Can you express |E| as a function of |V|? . . .so the total number of edges, when |V| = N is . . . |V| |E| B C 1 0 +1 A 2 1 +2 3 3 F +3 4 6 +4 5 10 D +5 E 6 15

Graphs: Definitions • A complete graph is one that has an edge between every pair of vertices. • Suppose G = (V, E) is complete. Can you express |E| as a function of |V|? . . .so the total number of edges, when |V| = N is i = N(N – 1)/2 = |V|(|V| – 1)/2 |V| |E| B C 1 0 +1 A 2 1 +2 N–1 3 3 F +3 4 6 i = 0 +4 5 10 D +5 E 6 15

Graphs: Definitions • A free tree is a connected, acyclic, undirected graph. This is a free tree. B C A F D E

Graphs: Definitions • A free tree is a connected, acyclic, undirected graph. • “Free” refers to the fact that there is no vertex designated as the “root.” B C A F D E

root Graphs: Definitions • A free tree is a connected, acyclic, undirected graph. • If some vertex is designated as the root, we have a rootedtree. B C A F D E

Graphs: Definitions