复旦大学课程

❖ (a)strongly connected ❖ (b)connected directed ❖ (c)weakly connected ❖ strongly connected components: G1,G2,…,Gω
❖ V ={v1,v2,v3,v4,v5,v6,v7, v8} ❖ V1={v1,v7,v8}, V2={v2,v3,v5,v6}, V3={v4}, ❖ strongly connected components :
V1=Km{xm1,nv,xed2re,tnxico3e,txse4s}aa, nVcdo2m=cp{oylen1t,teya2ibn,yisp3a,yra4tl,iylte5}eg,drgaepsh:jVoi1nhinags or vVe'r1t=ic{exs1,xin2,xV3,2y,4,yan5}d, VV'22=h{yas1,yn2,yv3,exr4t}ic, es and
❖ 1)We prove V(G)=V1∪V2, V1∩V2= ❖ Let vV1∩V2, ❖ there is an odd simple circuit in G such that
these edges of the simple circuit p1∪p2 ❖ each edge joins a vertex of V1 to a vertex of V2
not contain any odd simple circuit. ❖ Let C=(v0,v1,…,vm,v0) be an simple circuit of G
❖ (2)G does not contain any odd simple circuit, we prove G is bipartite
❖ Since a graph is bipartite iff each component of it is, we may assume that G is connected.
❖ Pick a vertex uV,and put V1={x|l(u,x) is even simple path} ,and V2={y|l(u,y) is odd simple path}
❖ Definition 15: A graph is called connectivity if there is a path between every pair of distinct vertices of the graph. Otherwise , the graph is disconnected.
❖ A graph that is not connected is the union of two or more connected subgraphs, each pair of which has no vertex in common. These disjoint connected subgraphs are called the connected components of the graph
e1,e2,…,en of G such that e1=(v0=u,v1), e2=(v1,v2), …, en=(vn-1,vn=v), and no edge occurs more than
once in the edge sequence. A path is called simple if no vertex appear more than once. A circuit is a path that begins and ends with the same vertex. A circuit
❖ Example: Let G be a simple graph. If G has n vertices, e edges, and ω connected components , then
n e 1 (n )(n 1)
2
Proof: e≥n-ω Let us apply induction on the number of edges of G. e=0, isolated vertex，has n components ，n=ω, 0=e≥n-ω=0，the result holds Suppose that result holds for e=e0-1 e=e0, Omitting any edge , G', (1)G' has n vertices, ω components and e0-1 edges. (2)G' has n vertices, ω+1 components and e0-1 edges
❖ G(V1),G(V2),G(V3)
❖5.2.4 Bipartite graph
❖ Definition18: A simple graph is called bipartite
if its vertex set V can be partioned into two
disjoint sets V1 and V2 such that every edge in the graph connects a vertex in V1 and a vertex in V2. (so that no edge in G connects either two vertices in V1 or two vertices in V2).The symbol
y2,uj+1,,u2n,y1,y2) from u to y2, ❖ Simple path (u,u1,u2,,uj-1,y2),simple circuit
(y2,uj+1,,u2n,y1,y2) ❖ j is odd number
❖ j is even number
5.3Euler and Hamilton paths
contains all edges joining vertices in V1.
❖ K3,3 , K2,3。
❖ The graph is not bipartite ❖ Theorem 5.5:A graph is bipartite iff it does not
contain any odd simple circuit. ❖ Proof:(1)Let G be bipartite , we prove it does
The graph is an Euler circuit. The result holds 2) Suppose that result holds for em e=m+1， (G)≥2. By the theorem 5.4, there is a simple circuit C in the graph G
❖ (e1,e2,e7,e1,e2,e9)is not a path ❖ (e1,e2,e7,e6,e9)is a path from a to e ❖ (e1,e2,e9)is a path from a to e, is a simple
path. ❖ (a,b,c,e)
❖ Definition 17: 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 connected directed graph if there is a path from a to b or 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 pair vertices in the underlying undirected graph.
❖Proof:(1)Let connected multigraph
G have an Euler circuit, then each of
its vertices has even degree.
❖ (v0,v1,…,vi, …,vk),v0=vk
❖First note that an Euler circuit
begins with a vertex v0 and continues with an edge incident to v0, say {v0,v1}. The edge {v0,v1} contributes one to d(v0).
❖ Thus each of G’s vertices ) we prove that each edge of G joins a vertex of V1 and a vertex V2
❖ If it has a edge joins two vertices y1 and y2 of V2 ❖ odd simple path
❖ (u=u0,u1,u2,,u2n,y1,y2),even path ❖ y2ui(0i2n) ❖ There is uj so that y2=uj. The path (u,u1,u2,,uj-1,
❖ 5.3.1 Euler paths ❖ Definition 19: A path in a graph G is called an
Euler path if it includes every edge exactly once. An Euler circuit is an Euler path that is a circuit ❖ Theorem 5.6: A connected multigraph has an Euler circuit if and only if each of its vertices has even degree.
If G is connected, then the number of edges of G has at least n-1 edges. Tree.
❖5.2.3 Connectivity in directed graphs
❖ Definition 16: Let n be a nonnegative integer and G be a directed graph. A path of length n from u to v in G is a sequence of edges
❖ (2)Suppose that G is a connected multigraph and the degree of every vertex of G is even.
❖ Let us apply induction on the number of edges of G
❖ 1)e=1,loop
2. e 1 (n )(n 1)
2
❖ Let G1,G2,…,Gωbe ω components of G. Gi has n，i vaenrdtices for i=1,2,…, ω, and n1+n2+…+nω=n
ei
1 2
ni (ni
1)
1 (n )(n 1)，
2
The complete graph on n-ω+1 vertices and ω-1 isolated vertices
is simple if the vertices v0,v1,…,vn are all distinct.
(e1,e2,e7,e1,e2,e7)is not a circuit (e1,e2,e7,e6,e12) is a circuit (e1,e2,e7) is a simple circuit. (a,b,c,a)