Cuts and Connectivity Cuts and Connectivity CSE, IIT KGP Vertex - - PowerPoint PPT Presentation

CSE, IIT KGP

Cuts and Connectivity Cuts and Connectivity

CSE, IIT KGP

Vertex Cut and Connectivity Vertex Cut and Connectivity

• A

A separating set separating set or

• r vertex cut

vertex cut of a graph G

• f a graph G

is a set is a set S S ⊆ ⊆ V(G) V(G) such that G such that G− −S has more S has more than one component. than one component.

– – A graph G is A graph G is k k-

• connected

connected if every vertex cut has if every vertex cut has at least at least k k vertices. vertices. – – The The connectivity connectivity of G, written as

• f G, written as κ

κ(G), is the (G), is the minimum size of a vertex cut. minimum size of a vertex cut.

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Edge Edge-

• connectivity

connectivity

• A

A disconnecting set disconnecting set of edges is a set

• f edges is a set F

F⊆ ⊆ E(G) E(G) such that G such that G – – F has more than one F has more than one component. component.

– – A graph is A graph is k k-

• edge

edge-

• connected

connected if every if every disconnecting set has at least disconnecting set has at least k k edges. edges. – – The The edge connectivity edge connectivity of G, written as

• f G, written as κ

κ' '(G), is the (G), is the minimum size of a disconnecting set. minimum size of a disconnecting set.

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Edge Cut Edge Cut

• Given

Given S,T S,T⊆ ⊆ V(G) V(G), we write [S,T] for the set , we write [S,T] for the set

• f edges having one endpoint in S and the
• f edges having one endpoint in S and the
• ther in T.
• ther in T.

– – An An edge cut edge cut is an edge set of the form [S,S is an edge set of the form [S,S′ ′], ], where S is a nonempty proper subset of V(G). where S is a nonempty proper subset of V(G).

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Results Results

• κ

κ(G) (G) ≤ ≤ κ′ κ′(G) (G) ≤ ≤ δ δ(G) (G)

• If S is a subset of the vertices of a graph G,

If S is a subset of the vertices of a graph G, then: then: |[S,S |[S,S′ ′]| = [ ]| = [Σ Σv

v∈ ∈S S d(v)]

d(v)] – – 2e(G[S]) 2e(G[S])

• If G is a simple graph and |[S,S

If G is a simple graph and |[S,S′ ′]| < ]| < δ δ(G) for (G) for some nonempty proper subset S of V(G), some nonempty proper subset S of V(G), then |S| > then |S| > δ δ(G). (G).

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More results… More results…

• A graph G having at least three vertices is

A graph G having at least three vertices is 2 2-

• connected if and only if each pair

connected if and only if each pair u,v u,v∈ ∈V(G) V(G) is connected by a pair of internally disjoint is connected by a pair of internally disjoint u,v u,v-

• paths in G.

paths in G.

• If G is a k

If G is a k-

• connected graph, and G

connected graph, and G′ ′ is is

• btained from G by adding a new vertex
• btained from G by adding a new vertex y

y adjacent to at least adjacent to at least k k vertices in G, then G vertices in G, then G′ ′ is is k k-

• connected.

connected.

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And more … And more …

• If n(G)

If n(G) ≥ ≥ 3, then the following conditions are 3, then the following conditions are equivalent (and characterize 2 equivalent (and characterize 2-

• connected graphs)

connected graphs) (A) (A) G is connected and has no cut vertex. G is connected and has no cut vertex. (B) (B) For all For all x,y x,y∈ ∈ V(G) V(G), there are internally disjoint , there are internally disjoint x,y x,y-

• paths

paths (C) (C) For all For all x,y x,y∈ ∈ V(G) V(G), there is a cycle through , there is a cycle through x x and and y y. . (D) (D) δ δ(G) (G) ≥ ≥ 1, and every pair of edges in G lies on a 1, and every pair of edges in G lies on a common cycle common cycle

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x,y x,y-

• separator

separator

• Given

Given x,y x,y ∈ ∈ V(G), V(G), a set a set S S ⊆ ⊆ V(G) V(G) − − {x,y} {x,y} is an is an x,y x,y-

• separator

separator or a

• r a x,y

x,y-

• cut

cut if if G G− −S S has no has no x,y x,y-

• path.

path. – – Let Let κ κ(x,y) (x,y) be the minimum size of an be the minimum size of an x,y x,y-

• cut.

cut.

• Let

Let λ λ(x,y) (x,y) be the minimum size of a set of pair be the minimum size of a set of pair-

• wise

wise internally disjoint internally disjoint x,y x,y-

• paths.

paths. – – Let Let λ λ(G) (G) be the largest be the largest k k such that such that λ λ(x,y) (x,y) ≥ ≥ k k for for all all x,y x,y∈ ∈ V(G). V(G). – – For X,Y For X,Y⊆ ⊆ V(G), an X,Y V(G), an X,Y-

• path

path is a path having is a path having first vertex in first vertex in X, X, last vertex in last vertex in Y, Y, and no other and no other vertex in X vertex in X∪ ∪Y. Y.

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Menger’s Menger’s Theorem Theorem

• If

If x,y x,y are vertices of a graph G and are vertices of a graph G and x,y x,y∉ ∉E(G), then E(G), then the minimum size of an the minimum size of an x,y x,y-

• cut equals the

cut equals the maximum number of pair maximum number of pair-

• wise internally disjoint

wise internally disjoint x,y x,y-

• paths.

paths.

• [Corollary] The connectivity of G equals the

[Corollary] The connectivity of G equals the maximum maximum k k such that such that λ λ(x,y) (x,y) ≥ ≥ k k for all for all x,y x,y∈ ∈V(G). V(G). The edge connectivity of G equals the maximum The edge connectivity of G equals the maximum k k such that such that λ′ λ′(x,y) (x,y) ≥ ≥ k k for all for all x,y x,y∈ ∈V(G). V(G).