Cuts and Connectivity Cuts and Connectivity CSE, IIT KGP
Vertex Cut and Connectivity Vertex Cut and Connectivity • A A separating set separating set or or vertex cut vertex cut of a graph G of a graph G • ⊆ V(G) − S has more S ⊆ such that G − is a set S V(G) such that G S has more is a set 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 k k vertices. vertices. at least of G, written as κ κ (G), is the – The The connectivity connectivity of G, written as (G), is the – minimum size of a vertex cut. minimum size of a vertex cut. CSE, IIT KGP
Edge- -connectivity connectivity Edge ⊆ E(G) F ⊆ • A A disconnecting set disconnecting set of edges is a set of edges is a set F E(G) • such that G – – F has more than one F has more than one such that G component. component. – A graph is A graph is k k- -edge edge- -connected connected if every if every – disconnecting set has at least k k edges. edges. disconnecting set has at least of G, written as κ κ ' – The The edge connectivity edge connectivity of G, written as '(G), is the (G), is the – minimum size of a disconnecting set. minimum size of a disconnecting set. CSE, IIT KGP
Edge Cut Edge Cut ⊆ V(G) S,T ⊆ • Given Given S,T V(G) , we write [S,T] for the set , we write [S,T] for the set • of edges having one endpoint in S and the of edges having one endpoint in S and the other in T. other in T. ′ ], is an edge set of the form [S,S ′ – An An edge cut edge cut 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). CSE, IIT KGP
Results Results • κ κ (G) ≤ κ′ κ′ (G) ≤ δ δ (G) (G) ≤ (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: ′ ]| = [ Σ v |[S,S ′ ]| = [ Σ d(v)] – – 2e(G[S]) 2e(G[S]) |[S,S S d(v)] ∈ S v ∈ ′ ]| < δ (G) for If G is a simple graph and |[S,S ′ ]| < δ • If G is a simple graph and |[S,S (G) for • some nonempty proper subset S of V(G), some nonempty proper subset S of V(G), δ (G). then |S| > δ (G). then |S| > CSE, IIT KGP
More results… More results… • A graph G having at least three vertices is A graph G having at least three vertices is • ∈ V(G) u,v ∈ 2- -connected if and only if each pair connected if and only if each pair u,v V(G) 2 is connected by a pair of internally disjoint is connected by a pair of internally disjoint u,v- - paths in G. paths in G. u,v ′ is connected graph, and G ′ • If G is a k If G is a k- -connected graph, and G is • obtained from G by adding a new vertex y y obtained from G by adding a new vertex ′ is vertices in G, then G ′ adjacent to at least k k vertices in G, then G is adjacent to at least k- -connected. connected. k CSE, IIT KGP
And more … And more … ≥ 3, then the following conditions are If n(G) ≥ • 3, then the following conditions are • If n(G) equivalent (and characterize 2- -connected graphs) connected graphs) equivalent (and characterize 2 (A) G is connected and has no cut vertex. G is connected and has no cut vertex. (A) ∈ V(G) x,y ∈ (B) For all For all x,y V(G) , there are internally disjoint , there are internally disjoint (B) x,y - -paths paths x,y ∈ V(G) x,y ∈ (C) For all For all x,y V(G) , there is a cycle through , there is a cycle through x x and and y y . . (C) (D) δ δ (G) ≥ 1, and every pair of edges in G lies on a (G) ≥ 1, and every pair of edges in G lies on a (D) common cycle common cycle CSE, IIT KGP
x,y- - separator separator x,y ∈ V(G), ⊆ V(G) − {x,y} x,y ∈ S ⊆ V(G) − • Given x,y V(G), a set a set S {x,y} is an is an x,y x,y- - • Given − S G − separator or a or a x,y x,y- -cut cut if if G S has no has no x,y x,y- - path. path. separator Let κ κ (x,y) – Let (x,y) be the minimum size of an be the minimum size of an x,y x,y- - cut. cut. – λ (x,y) Let λ • (x,y) be the minimum size of a set of pair be the minimum size of a set of pair- -wise wise • Let internally disjoint x,y x,y- - paths. paths. internally disjoint λ (G) λ (x,y) ≥ k Let λ such that λ (x,y) ≥ – Let (G) be the largest be the largest k k such that k for for – ∈ V(G). x,y ∈ all x,y V(G). all ⊆ V(G), an X,Y For X,Y ⊆ – For X,Y V(G), an X,Y- - path path is a path having is a path having – first vertex in X, X, last vertex in last vertex in Y, Y, and no other and no other first vertex in ∪ Y. vertex in X ∪ Y. vertex in X CSE, IIT KGP
Menger’s Theorem Theorem Menger’s ∉ E(G), then x,y ∉ • If x,y x,y are vertices of a graph G and are vertices of a graph G and x,y E(G), then • If the minimum size of an x,y x,y- - cut equals the cut equals the the minimum size of an maximum number of pair- -wise internally disjoint wise internally disjoint maximum number of pair x,y- - paths. paths. x,y • [Corollary] The connectivity of G equals the • [Corollary] The connectivity of G equals the λ (x,y) ≥ k ∈ V(G). such that λ (x,y) ≥ x,y ∈ maximum k k such that k for all for all x,y V(G). maximum The edge connectivity of G equals the maximum k k The edge connectivity of G equals the maximum λ′ (x,y) ≥ k ∈ V(G). such that λ′ (x,y) ≥ x,y ∈ k for all for all x,y V(G). such that CSE, IIT KGP
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