图论第四章

15
Graph Theory
Example of Blocks
4.1.17
If H is a block of G, then H as a graph has no cut-vertex, but H may contain vertices that are cut-vertices of G.
Hence any path in Bi from every vertex in Bi-{v} to any in V(B1)∩V(B2)-{v} is retained. Since the blocks have at least two common vertices, deleting a single vertex leaves a vertex in the intersection. Paths from all vertices to that vertex are retained, so B1∪B2 cannot be disconnected by deleting one vertex.
2
Graph Theory
Example: Connectivity of Kn
4.1.2
Because a clique has no separating set, we need to adopt a convention for its connectivity.
– This explains the phrase “or has only one vertex” in Definition 4.1.1.
1
Proof: The edges incident to a vertex v of minimum degree form an edge cut; hence ’(G) (G) . It remains to show that (G) ’(G).
min edge-cut Incident edges is
The graph G-F’ contains the two components of G-F plus at least one edge between them, making it connected. Hence F is a minimal disconnecting set and is bond.
Graph Theory
Chapter 4
Connectivity and Paths
1
Graph Theory
Connectivity
4.1.1
A separating set or vertex cut of a graph G is a set S⊆V(G) such that G-S has more than one component.
– A graph is k-edge-connected if every disconnecting set has at least k edges.
– The edge-connectivity of G, written ’(G), is the minimum size of a disconnecting set.
The connectivity of K3,3 is 3; the graph is 1connected, 2-connected, and 3-connected, but not 4connected.
5
Graph Theory
Edge-Connectivity 1
4.1.7
A disconnecting set of edges is a set F⊆E(G) such that G-F has more than one component. (also called a cut)
We obtain (Kn)=n-1, while (G)≤n(G)-2 when G is not a complete graph.
– With this convention, most general results about connectivity remain valid on complete graphs.
The connectivity of G, written (G), is the minimum size of a vertex set S such that G-S is disconnected or has only one vertex. A graph G is k-connected if its connectivity is at least k.
6
Graph Theory
Edge-Connectivity 2
4.1.7
Given S, T⊆V(G), we write [S,T] for the set of edges having one end-point in S and the other in T.
An edge cut is an edge set of the form [S,Ŝ], where S is a nonempty proper subset of V(G) and Ŝ denotes V(G)-S. (also called a cut-set)
S
S
11
Graph Theory
Bond
4.1.14
A bond is a minimal nonempty edge cut.
Here “minimal” means that no proper nonempty subset is also an edge cut.
– We characterize bonds in connected graphs.

13
Graph Theory
Proposition 4.1.15:If G is a connected graph, then an edge
cut F is a bond if and only if G-F has exactly two components 2
Proof: continue Conversely, suppose that G-F has more than two components. Since G-F is the disjoint union of G[S] and G[S], one of these has at least two components, say G[S]. Then S= AB, where no edges join A and B. Now the edge cuts [A,A] and [B,B] are proper subsets of F, so F is not a bond
4.1.2
– Every induced subgraph that has at least one vertex from X and from Y is connected. – Hence every separating set of Km,n contains X or Y. – Since X and Y themselves are separating sets (or leave only one vertex), we have (Km,n) = min{m,n}.
B S A
14
S
Graph Theory
Blocks
A block of a graph G is a maximal connected subgraph of G that has no cut-vertex.
– If G itself is connected and has no cut-vertex, then G is a block.
7
Graph Theory
Edge-Connectivity 3
S
S
Disconnecting set
Edge cut
8
Graph Theory
Theorem 4.1.9 : If G is a simple graph, then (G) ’(G) (G)
min vertex-cut min edge-cut min degree
10
Graph Theory
Theorem 4.1.9 2
Proof: Continue
Otherwise,we choose x S and y S with xy.
Let T consist of all neighbors of x in S and all vertices of S -{x} with neighbors in S. Every x, y-path passes through T, so T is a separating set. Also, picking the edges from x to T S and one edge from each vertex of T S to S (shown bold below) yields |T| distinct edges of [S,S ]. Thus ’(G)= |[ S,S ]|≥ |T| ≥ (G).
Proposition 4.1.19: Two blocks in a graph
share at most one vertex.
Proof:
Use contradiction.
– Suppose that blocks B1, B2 have at least two common vertices. – We show that B1∪B2 is a connected subgraph with no cut-vertex, which contradicts the maximality of B1 and B2.
12
Graph Theory
Proposition 4.1.15: If G is a connected graph, then an edge cut F is a bond if and only if G-F has exactly two components 1
Proof: Let F = [S,S] be an edge cut. Suppose that G-F has exactly two components, and let F’ F.
17
Graph Theory
Proposition 4.1.19: Two blocks in a graph
share at most one vertex.
Proof: Continue
When delete one vertex v from Bi, what remains is connected.
3
Graph Theory
Example: Connectivity of Kn
4.1.2
Delete one vertex
Delete two vertices
Delete four vertices
4
Graph Theory
Example: Connectivity of Km,n
Consider a bipartition X,Y of Km,n.
– For example, the graph drawn below has five blocks; three copies of K2, one of K3, and one subgraph that is neither a cycle nor a complete graph.
Graph Theory
an Edge cut
9
Hale Waihona Puke Graph TheoryTheorem 4.1.9 : If G is a simple graph, then (G) ’(G) (G)
vertex-cut edge-cut minimum degree
1
Proof: continue We have observed that (G) n(G)-1 (see Example 4.1.2). Consider a smallest edge cut [S, S ]. If every vertex of S is adjacent to every vertex of S, then |[S, S ]| = |S ||S |≥n(G)-1≥ (G), and the desired inequality holds.