CS275 - Discrete Mathematics Chapter 10. Graphs Lecturer: Jiho Noh - - PowerPoint PPT Presentation

CS275 - Discrete Mathematics

Chapter 10. Graphs Lecturer: Jiho Noh Fall 2019

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 1 / 14

10.2. Graph Terminology and Special Types of Graphs Modifying Graphs

Subgraphs

When edges and vertices are removed from a graph, without removing endpoints of any remaining edges, a smaller graph is obtained, which is called a subgraph of the original graph. A subgraph of G = (V , E) is a graph H = (W , F), where W ⊆ V and F ⊆ E A subgraph H of G is a proper subgraph of G if H = G subgrpah induced by a subset W of the vertex set V ✍

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 2 / 14

10.2. Graph Terminology and Special Types of Graphs Modifying Graphs

Removing/Adding Edges

Given a graph G = (V , E) and an edge e ∈ E, we can produce a new graph as such by removing or adding the edge to G: G − e = (V , E − {e}) G + e = (V , E ∪ {e}) Union of Graphs, G1 ∪ G2 = (V1 ∪ V2, E1 ∪ E2), where G1 = (V1, E1) and G2 = (V2, E2) Edge contraction removes an edge e with endpoints u and v and merges u and v into a new single vertex w.

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 3 / 14

10.3. Representing Graphs and Graph Isomorphism Representing Graphs

Representing Graphs

Adjacency Lists list of adjacent vertices to each vertex of a graph. ✍ Adjacency Matrices A matrix representation for a simple graph n × n zero-one matrix where the value indicates the adjacency between vertices of a graph with n vertices ✍ (Note, for multigraphs, the values represent the number of edges between vertices)

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 4 / 14

10.3. Representing Graphs and Graph Isomorphism Representing Graphs

Representing Graphs

Incidence Matrices Suppose that v1, v2, . . . , vn represents vertices and e1, e2, . . . , em represnts the edges of a graph An incidence matrix is the n × m matrix M = [mij] where the value 1 is when edge ej is incident with vi

• therwise, 0

✍

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 5 / 14

10.3. Representing Graphs and Graph Isomorphism Isomorphism of Graphs

Isomorphism

Do graphs have the same structure when we ignore the identities of their vertices? Isomorphism: iso → same/equal, morph → shape G1 = (V1, E1) and G2 = (V2, E2) are isomorphic, if there exists a

• ne-to-one and onto (‘bijective’) function from V1 to V2 that preserves the

adjacency property (i.e., (u, v) ∈ E1 → (f (u), f (v)) ∈ E2) This function is called an isomorphism. ✍

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 6 / 14

10.3. Representing Graphs and Graph Isomorphism Isomorphism of Graphs

How to determine if the graphs are isomorphic?

answer: There’s no efficient algorithm for this! There are some techniques that shows if the graphs are NOT isomorphic, by inspecting the Graph Invariants.

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 7 / 14

10.3. Representing Graphs and Graph Isomorphism Isomorphism of Graphs

Graph Invariants

Isomorphic graphs MUST have: the same number of vertices the same number of edges the same degree from the correspnding vertices between the isomorphic graphs These are called graph invariants. ✍

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 8 / 14

10.4. Connectivity Path

Path

Path is a sequence of edges that begins of a vertex of a graph and travels from vertex to vertex along edges of the graph. Terminology: length, circuit/cycle, simple path, pass through/traverse ✍

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 9 / 14

10.4. Connectivity Connectedness in Undirected Graphs

Connectedness in Undirected Graphs

An undirected graph is connected if there is a path between every pair of distinct vertices of the graph. (↔ disconnected) A connected component of a graph G is connected subgraph of G that is not a proper subgraph of another connected subgraph of G. That is, a connected component of a graph is a maximal connected subgraph of G. ✍

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 10 / 14

10.4. Connectivity Connectedness in Undirected Graphs

Cut Vertices, Cut Edges

The removal from a graph of a vertex and all incident edges may produce a subgraph with more connected components. Such vertices are called cut vertices (or articulation points). Analogously, we call such edges cut edges (or bridge). ✍

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 11 / 14

10.4. Connectivity Connectedness in Undirected Graphs

Connectedness in Directed Graphs

A directed graph is strongly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph A directed graph is weakly connected if there is a path between every two vertices in underlying undirected graph strongly connected components — strongly connected subgraphs in directed graphs

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 12 / 14

10.4. Connectivity Connectedness in Undirected Graphs

simple circuit of length k as a graph invariant

The existence of a “simple circuit of length k”, where k is a positive integer greater than 2, is an invariant under graph isomorphism. This can help in proving given two graphs are not isomorphic. ✍

{(1, 3), (1, 4), (1, 5), (2, 3), (2, 5), (3, 4)} vs. {(1, 2), (1, 3), (1, 5), (2, 3), (3, 4), (4, 5)}

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 13 / 14

10.4. Connectivity Counting Paths Between Vertices

Counting Paths Between Vertices

Thm 10.2.2 — Counting Paths Let G be a graph with adjacency matrix A with respect to the ordering v1, v2, . . . , vn of the vertices of the graph (with directed or undirected edges, with multiple edges and loops allowed). The number of different paths of length r from vi to vj, where r is a positive integer, equals the (i, j)th entry of Ar. ✍

Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 14 / 14