1 / 16

Chap 7 Graph

Chap 7 Graph. Def 1: Simple graph G=(V,E) V : nonempty set of vertices E : set of unordered pairs of distinct elements of V called edges Def 2: Multigraph G=(V,E) loops are not allowed in simple or multi graphs

jlancaster
Download Presentation

Chap 7 Graph

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Chap 7 Graph Def 1: Simple graph G=(V,E) V : nonempty set of vertices E : set of unordered pairs of distinct elements of V called edges Def 2: Multigraph G=(V,E) loops are not allowed in simple or multi graphs Def 3: pseudograph (loops allowed )

  2. Chap 7 Graph Def 4: directed graph E : ordered pairs of element of V Def 5:directed multigraph Table 1. Graph Terminology Examples

  3. Chap 7 Graph Graph Terminology • Undirected graph Def 1: adjacent : u and v are adjacent if {u,v} is an edge ; this edge is incident with u and v; u and v are endpoints. Def 2:degree of v : number of edges incident with it ,except that a loop contributes twice to the degree. Example 1 Isolated vertex : degree 0

  4. Chap 7 Graph Theorem 1: The handshaking theorem G=(V,E): an undirected graph with e edges 2 e =  deg (v) Theorem 2: An undirected graph has an even numberof odd-degree vertices vV sd

  5. Chap 7 Graph • Directed graph (u,v): u:initial vertex , v:end vertex Def 3:in-degree , deg-(v) : number of edges with v as end vertex out-degree , deg+(v) Theorem 3:G=(V,E)  deg+ (v) =  deg- (v) = |E| vV vV

  6. Chap 7 Graph Complete graph : Kn , Example 4 Cycle : Cn ,n>= 3 , Example 5 Wheel : Wn ,n>= 3 , Example 6 n-cube :Qn , Example 7 Def 4: bipartite: its vertex set V canbe partitioned into two disjoint nonempty sets V1 and V2 such that every edge connects a vertex in V1 and a vertex V2 .

  7. Chap 7 Graph Example 8 C6 : bipartite Example 9 K3 : not bipartite Example 11 : complete bipartite Def 5:subgraph H=(W,F) of a graph G =(V,E) , WV and F  E . Example 14

  8. Chap 7 Graph • Isomorphism of Graphs Def 1:G1 =(V1,E1) , G2 =(V2 , E2) are isomorphic if there is a one-to-one and onto function f from V1 to V2 such that a , b adjacent in G1 ,iff f(a) and f(b) are adjacent in G2 ,,for all a and b in V1 . Example 8 u1 u2 v1 v2 u3 u4 v3 v4 f(u1)= v1 , f(u2)= v4, f(u3)= v3, f(u4)= v2

  9. Chap 7 Graph Example 9G and H are not isomorphic number of vertices , number of edges ,degrees of the vertices are invariants under isomorphism Example 10Are G and H isomorphic?

  10. Chap 7 Graph • Connectivity Def 1: Path of length n from u to v ; circuit ; simple path / circuit Example 1 Def 2:An undirected graph is connected if there is a path between every pair of distinct vertices of the graph.

  11. Chap 7 Graph Theorem 1 There is a simple path between every pair of distinct vertices of a connected undirected graph. • A graph that is not connected is the union of two or more connected subgraphs , each pair of which has no vertex in common. - connected components • Cut vertices • Cut edge Example 4

  12. Chap 7 Graph Example 6 Are G and H isomorphic? H has a simple circuit of length 3 while G dosen’t - another invariant

  13. Chap 7 Graph Example 7 Are G and H isomorphic? - satisfy all for invariants - f (u1)=v3 , f(u4)=v2, f(u3)=v1,f(u2)=v5.and f(u5)=v4

  14. Chap 7 Graph • Euler and Hamilton Paths - Is there a simplecircuit in Figure 2 that contains every edge? Def 1 : An Euler circuit and an Euler path Example 1,2 Theorem 1 A connected multigraph has an Euler circuit if and only if each of its vertices has even degree .

  15. Chap 7 Graph Theorem 2 A connected multigraph has an Euler paths but not an Euler circuit if and only if it has exactly two vertices of odd degree . Def .Hamilton path : A path X0,X1,…. X n-1, Xn in G=(V,E) is a Hamilton path if V={X0,X1,….X n-1,Xn } and Xi =Xj , for 0 <= i <j <=n. A circuit X0,X1,….X n-1,Xn,X0 is a Hamilton circuit if X0,X1,….X n-1,Xn is a Hamilton path . Figure 8,9 and Example 5

  16. Chap 7 Graph • No known simple necessary and sufficient criteria for the existence of Hamilton circuits . - sufficient conditions - properties used to show that a graph has no H.C. Example 6 • Shortest path problem Example 1 Algorithm 1: Dijkstra’s algorithm Example 2

More Related