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, G 1 ∪ G 2 = ( V 1 ∪ V 2 , E 1 ∪ E 2 ) , where G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) 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 v 1 , v 2 , . . . , v n represents vertices and e 1 , e 2 , . . . , e m represnts the edges of a graph An incidence matrix is the n × m matrix M = [ m ij ] where the value 1 is when edge e j is incident with v i otherwise, 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 G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) are isomorphic , if there exists a one-to-one and onto (‘bijective’) function from V 1 to V 2 that preserves the adjacency property (i.e., ( u , v ) ∈ E 1 → ( f ( u ) , f ( v )) ∈ E 2 ) 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 v 1 , v 2 , . . . , v n 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 v i to v j , where r is a positive integer, equals the ( i , j ) th entry of A r . ✍ Lecturer: Jiho Noh CS275 - Discrete Mathematics Fall 2019 14 / 14
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