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Chapter 4

Chapter 4 . Connectivity and Paths. 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.

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Chapter 4

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  1. Chapter 4 Connectivity and Paths Ch. 4. Connectivity and paths

  2. 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. • 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. Ch. 4. Connectivity and paths

  3. Example: Connectivity of Kn4.1.2 • 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. • 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. Ch. 4. Connectivity and paths

  4. Example: Connectivity of Kn4.1.2 Delete one vertex Delete four vertices Delete two vertices Ch. 4. Connectivity and paths

  5. Example: Connectivity of Km,n4.1.2 • Consider a bipartition X,Y of Km,n. • 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}. • The connectivity of K3,3 is 3; the graph is 1-connected, 2-connected, and 3-connected, but not 4-connected. Ch. 4. Connectivity and paths

  6. 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) • 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. Ch. 4. Connectivity and paths

  7. Edge-Connectivity 24.1.7 • 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) Ch. 4. Connectivity and paths

  8. Edge-Connectivity 3 S Disconnecting set Edge cut Ch. 4. Connectivity and paths

  9. Theorem 4.1.9: If G is a simple graph, then (G)  ’(G)  (G)min vertex-cutminedge-cutmin degree 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).  minedge-cut Incident edges is an Edge cut Ch. 4. Connectivity and paths

  10. Graph Theory Theorem 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 Sis adjacent to every vertex of S, then |[S, S ]| = |S ||S|≥n(G)-1≥ (G), and the desired inequality holds. 10 Ch. 4. Connectivity and paths

  11. S S Theorem 4.1.9 2 Proof: Continue  • Otherwise,we choose xS and y Swithxy. • 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). Ch. 4. Connectivity and paths

  12. 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. Ch. 4. Connectivity and paths

  13. 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 components1 Proof: • Let F = [S,S] be an edge cut. • Suppose that G-F has exactly two components, and let F’  F. • 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.  Ch. 4. Connectivity and paths

  14. Proposition4.1.15:If G is a connected graph, then an edge cut F is a bond if and only if G-F has exactly two components2 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= AB, 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 B  S S A Ch. 4. Connectivity and paths

  15. 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. Ch. 4. Connectivity and paths

  16. 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. • 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. Ch. 4. Connectivity and paths

  17. Proposition4.1.19: 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.  Ch. 4. Connectivity and paths

  18. Proposition4.1.19: Two blocks in a graph share at most one vertex. Proof: Continue • When delete one vertex v from Bi, what remains is connected. • 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. Ch. 4. Connectivity and paths

  19. Depth First Search • Depth first search: Explore always from the most recently discovered vertex that has unexplored edges. • Use stack • Breadth first search: Explores from the oldest vertex. • Use queue Ch. 4. Connectivity and paths

  20. Graph Theory Example of Depth First Search 4.1.21 Depth-first search from u u, a, b, c, d, e, f, g • From u go to a, b, c, d • From d, try to get to b but b is visited • From d, no other unvisited edge to go, so go back to c • Similarly, no other unvisited edge to go from c , go back to b • From b go to e • No other unvisited edge to go from e, go back to b and then a • From a, go to f and then g • From g go to u but u is visited • From g go back to f, a, and then u f g e d b c a u 20 Ch. 4. Connectivity and paths

  21. Example of Depth First Search 4.1.21 • Other Depth-first search paths from u • u, a, f, g, b, e, d, c • u, g, f, a, b, c, d, e f g f e d g e d b c a u b c a u Ch. 4. Connectivity and paths

  22. Lemma4.1.22: If T is a spanning tree of a connected graph G grown by DFS from u, then every edge of G not in T consists of two vertices v,w such that v lies on the u,w-path in T. Proof: • Let v w be an edge of G, with v encountered before w in the depth-first search. • Because v w is an edge, we cannot finish v before w is added to T. • Hence w appears somewhere in the subtree formed before finishing v, and the path from w to u contains v. • See example in the next page Ch. 4. Connectivity and paths

  23. Lemma4.1.22: If T is a spanning tree of a connected graph G grown by DFS from u, then every edge of G not in T consists of two vertices v,w such that v lies on the u,w-path in T. g f w e u c a v Ch. 4. Connectivity and paths

  24. Algorithm: 4.1.23Computing the blocks of a graph 1/2 • Input: A connected graph G. • Idea: Build a depth-first search tree T of G, discarding portions of T as blocks are identified. Maintain one vertex called ACTIVE. • Initialization: Pick a root x∈V(H); make xACTIVE; set T={x}. Ch. 4. Connectivity and paths

  25. Algorithm4.1.23 : Computing the blocks of a graph2/2 • Iteration: Let v denote the current active vertex. 1) If v has an unexplored incident edge vw, then 1A) If wV(T), then add vw to T, mark vw explored, make wACTIVE. 1B) If w∈V(T), then w is an ancestor of v; mark vw explored. 2) If v has no more unexplored incident edges, then 2A) If v ≠ x, and w is the parent of v, make wACTIVE. If no vertex in the current subtree T’ rooted at v has an explored edge to an ancestor above w, then V(T’)∪{w} is the vertex set of a block; record this information and delete V(T’) from T. ( see example in the next page) 2B) If v = x, terminate. Ch. 4. Connectivity and paths

  26. Algorithm4.1.23 : Computing the blocks of a graph2/2 Example of Step 2A: f g v w - Consider the subgraph in blue color which is the current subtree T’ rooted at v . - Since no vertex in T’ has an explored edge to an ancestor above w, then V(T’)∪{w} is the vertex set of a block Ch. 4. Connectivity and paths

  27. Theorem 4.2.2 :A graph G having at least three vertices is 2-connected if and only if for each pair u,v∈V(G) there exist internally disjoint u,v-paths in G. (Whitney [1932a]) • Sufficiency: • When G has internally disjoint u,v-paths, deletion of one vertex cannot separate u from v. • Since this condition is given for every pair u,v, deletion of one vertex cannot make any vertex unreachable from any other. • We conclude that G is 2-connected. • Necessity: by induction method Ch. 4. Connectivity and paths

  28. Lemma: (Expansion Lemma) If G is a k-connected graph, and G’ is obtained from G by adding a new vertex y with at least k neighbors in G, then G’ is k-connected.4.2.3 Proof: We prove that a separating set S of G’ must have size at least k. • If yS, then S-{y} separates G, so |S|k+1. • If yS and N(y)S, then |S |  k. • Otherwise, y and N(y)-S lie in a single component of G’-S. • Thus again S must separate G and |S|  k. Ch. 4. Connectivity and paths

  29. Theorem 4.2.4. For a graph G with at least three vertices, the following conditions are equivalent. A) G is connected and has no cut-vertex. B) For all x, yV(G), there are internally disjoint x, y-paths C) For all x, y V(G), there is a cycle through x and y. D) (G)  1, and every pair of edges in G lies on a common cycle. Ch. 4. Connectivity and paths

  30. Corollary 4.2.6: If G is 2-connected, then the graph G’ obtained by subdividing an edge of G is 2-connected. Proof: • Let G’ be formed from G by adding vertex w to subdivide uv. To show that G’ is 2-connected, it suffice to find a cycle through arbitrary edges e,f of G’(by Theorem 4.2.4D). • Since G is 2-connected, any two edges of G lie on a common cycle (Theorem 4.2.4D). • When our given edges e, f of G’ lie in G, a cycle through them in G is also in G’, unless it uses uv, in which case we modify the cycle. Here “modify the cycle” means “replace the edge uv with the u, v-path of length 2 through w”. • When e ∈ E(G) and f ∈ {uw, wv}, we modify a cycle passing through e and uv in G. When {e, f}={uw, wv}, we modify a cycle through uv. Ch. 4. Connectivity and paths

  31. Network Flow Problems 4.3 • A network is a digraph with a nonegative capacity c(e) on each edge e and a distinguished source vertex s and sink vertex t. • Vertices are also called nodes. • A flow f assigns a value f(e) to each edge e. • We write f+(v) for the total flow on edges leaving v and f –(v) for the total flow on edges entering v. • A flow is feasible if it satisfies the capacity constraints 0≤f(e)≤c(e) for each edge and the conservation constraints f+ (v) = f –(v) for each node v{s,t}. Ch. 4. Connectivity and paths

  32. Maximum Network Flow • The value val(f) of a flow f is the net flow f –(t)-f +(t) into the sink. A maximum flow is a feasible flow of maximum value. Ch. 4. Connectivity and paths

  33. Example of Max Flow • The zero flowassigns flow 0 to each edge • It is feasible. v u (0)1 (0)2 (0)2 (0)1 s t f (0)2 (0)2 (0)1 x y Ch. 4. Connectivity and paths

  34. v u (1)1 (1)2 (0)2 (1)1 s t f (0)2 (1)2 (1)1 x y Example of Max Flow • In the network below we illustrate a non-zero feasible flow. • Capacities are shown in bold, flow values in parentheses. • Our flow f assigns f(sx) = f(vt) = 0, and f(e) = 1 for every other edge e. This is a feasible flow of value 1. Ch. 4. Connectivity and paths

  35. u v (1)1 (1)2 (1)2 v (0)1 u (1)1 s t f (1)2 (0)2 (1)2 (1)1 (1)2 s t f (1)1 x y (0)2 (1)2 (1)1 x y Example of Max Flow • A path from the source to the sink with excess capacity would allow us to increase flow. • In this example, no path remains with excess capacity, but the flow f’ with f’(vx) = 0 and f’(e) = 1 for e ≠ vx has value 2. Ch. 4. Connectivity and paths

  36. f-Augmenting Path 4.3.4 • When f is a feasible flow in a network N, an f-augmenting path is a source-to-sink path P in the underlying graph G such that for each e ∈ E(P), a) if P follows e in the forward direction, then f(e) < c(e). b) if P follows e in the backward direction, then f(e)>0. • Let ε(e)=c(e) - f(e) when e is forward on P, and let ε(e)=f(e) when e is backward on P. The tolerance of P is mine∈E(P)ε(e). Ch. 4. Connectivity and paths

  37. + + - + - - - + New Flow after Augmenting • The edges of P incident to an internal vertex v of P occur in one of the four ways shown below. • In each case, the change to the flow out of v is the same as the change to the flow into v, so f ⁻(v) = f ⁺(v). Ch. 4. Connectivity and paths

  38. Lemma. If P is an f-augmenting path with tolerance z, then changing flow by +z on edges followed forward by P and by –z on edges followed backward by P produces a feasible flow f’ with val(f’) = val(f)+z. Proof: • The definition of tolerance ensures that 0 ≤ f’(e) ≤ c(e) for every edge e, so the capacity constraints hold. • We need only check vertices of P, since flow elsewhere has not changed. • For every vertex v, f+(v) = f–(v) • Finally, the net flow into the sink t increases by z. - - + + + + - Ch. 4. Connectivity and paths

  39. Source/sink cut • In a network, a source/sink cut [S,T] consists of the edges from a source set S to a sink set T, where S and T partition the set of nodes, with s∈S and t∈ T. • The capacity of the cut [S, T],written cap(S, T), is the total of the capacities on the edges of [S, T]. • Keep in mind that in a digraph [S, T] denotes the set of edges with tail in S and head in T. Thus the capacity of a cut [S, T] is completely unaffected by edges from T to S. Ch. 4. Connectivity and paths

  40. Ford-Fulkerson Labeling Alg. For Max- flow14.3.9 • Input: A feasible flow f in a network. • Output: An f-augmenting path or a cut with capacity val(f). • Idea: Find the nodes reachable from s by paths with positive tolerance. Reaching t completes an f-augmenting path. During the search, R is the set of nodes labeled Reached, and S is the subset of R labeled Searched. Ch. 4. Connectivity and paths

  41. Ford-Fulkerson Labeling Alg. For Max- flow2 Initialization: R = {s}, S = . Iteration: Choose v ∈ R-S. • For each exiting edge vw with f(vw) < c(vw) and w ∉R, add w to R. • For each entering edge uv with f(uv>0) and u ∉ R, add u to R. • Label each vertex added to R as “reached”, and record v as the vertex reaching it. After exploring all edges at v, add v to S. • If the sink t has been reached (put in R), then trace the path reaching t to report an f-augmenting path and terminate. If R = S, then return the cut [S, Ŝ] and terminate. Otherwise, iterate. Ch. 4. Connectivity and paths

  42. Theorem 4.3.11 In every network, the maximum value of a feasible flow equals the minimum capacity of a source/sink cut Ch. 4. Connectivity and paths

  43. Ch. 4. Connectivity and paths

  44. Ch. 4. Connectivity and paths

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