5. Biconnected Components of A Graph

1. 5. Biconnected Components of A Graph If one city’s airport is closed by bad weather, can you still fly between any other pair of cities? If one computer in a network goes down, can a message be sent between any other pair of computers in the network? If any one vertex (and edges incident with it) is removed from a connected graph, is the remaining subgraph still connected?

2. G I B G B I C B E C E E F F F J J H H D D A A (a) graph. (b) Its biconnected components Def’n: Let G=(V, E) be a connected, undirected graph. A vertex aV is said to be an articulation point if there exist vertex v and w such that (1) v, w and a are distinct (2) Every path between v and w must contain a. Alternatively, a is an articulation point of G if removing a splits G into two or more parts. subgraphs

3. Def’n:A graph G =(V, E) is said to be biconnected if and only if it has no articulation points. G2 G3 G1 which of the above are biconnected? Def’n: Let G' = (V', E') be a biconnected subgraph of a graph G = (V, E). G' is said to be a biconnected component of G if G' is maximal i.e., not contained in any other biconnected subgraph of G. Example is shown in the previous page !!!

4. Observation G = (V, E) Two edge e1and e2 in E is said to be related if e1 = e2 or if there is a cycle containing both e1 and e2. (e, e)  Re  E (e1 , e2)  R and (e2 , e3)  R  (e1 , e3)  R (e1 , e2)  R  (e2 , e1)  Rwhy ? Each subgraph consisting of the edges in an equivalent class and the incident vertex is a biconnected component !!! Can you prove it ?

5. x a y Lemma: For 1 i k , Let Gi =(Vi , Ei) be the biconnected components of a connected undirected graph G = (V, E). Then (1) Gi is biconnected for each 1 i k , (2) For all i j , Vi Vj contains at most one vertex. (3) a is an articulation point of G if and only if a Vi Vj for some ij [ proof ] (1) Trivial why ? (2) suppose that two distinct vertices v and w are in Vi  Vj , i j C1 v Gi Why? w C2 Gj v (3) x () () a y w

6. F A A B B E E C C D D A A F B B C C D D E E A is not an articulation point A is an articulation point Can you now characterize an articulation point when A is the root ?

7. A E B D I F C H G A B C Can you characterize D ? D E F G H I

8. v w One or more bicomponents An articulation point v in a depth-first search tree. Every path from root to w passes through v.

9. v' v w

10. Theorem : Let G = (V, E) be a connected, undirected graph and let S = (V, T) be a depth first spanning tree for G. Vertex a is an articulation point of G if and only if one of the following is true: (1) a is the root and a has two or more sons. (2) a is not the root and for some son s of a, there is no back edge between any descendant of s ( including s itself ) and a proper ancestor of a.

11. [proof] i) The root is an articulation point if and only if it has two or more sons. y x () () a y x x y

12. (ii) () a y x Since a is not the root, either x or y is a proper descendant of a !!! why ? y a x WLOG, let x be a proper descendant of a a a s s This is not possible. Why ? x x y  Only this Two cases: (1) (2) a a no back edge back edge s s x x

13. case (2) back edge y is not a descendant of a. y a x y is a descendant of a. w' w This is not possible. s s’ a y x () Easy, Exercise

14. An algorithm for finding all biconnected components of a graph G = (V, E) z v w biconnected component z

15. How to detect an articulation point ? When to report a biconnected component ? Assumption Depth First Search

16. Depth First Search number B F C A G =(V, E) D E 1 A A B C D E F 123654 2 B 6 D 3 C 4 5 F E Any relation between DFS numbers and articulation point ?

17. A 1 Assume that (a,b) a b Tree edge : (a,b) a < b Back edge : (a,b) a > b 2 B 6 D 3 C 4 5 F E w’ If there is a back edge from x to a proper ancestor of v, then v is reachable from x. v w x

18. How far back in the DFS tree from each vertex ? BACK[v] v can back to BACK[v] by following (1) tree edges (2) back edges Initially, BACK[v] = v, where v is a DFS number why ? BACK[v]:= min{BACK(v), min{w| (x,w) is a back edge, x is a descendant of v, and w is an ancestor of v.} w v x

19. What if v BACK(w), when w v (back up to v) ? assuming that BACK(w) = min{BACK(x) | xT(w)} v is an articulation point !!! why ? Now, how to update BACK(v) ? (i) vw (back edge) (ii) vw (backing up to v) case(i) BACK[v]:= min{BACK[v], w} : back edge why ? case(ii) BACK[v]:= min{BACK[v], BACK[w]} : back up to v why ? v x w w v w

20. How to detect and when ? v when ?  ? or  ? why ? w How ? v BACK(w) !!! How to report ? Well, ……. A set of edges.

21. r z y v b a w B W A when v is reached, pop edges until (v, w) is reached W (v, w) (y, v) A (y, a) (z, y) (r, z) Edge Stack

22. 1 A 1/1 2 D A 10 F 2/2 J D 3 9 H 4 E 4/4 E F 5 7 B 3/1 B 5/5 G C C (b) Proceed forward ; initialize values of back ; Detect back edge FA ; update back[F] ; 6/4 8 I 6 (c) Continue forward ; Detect back edge CE ; update back[C] (a)The complete depth- first search tree. 5/4 7/5 4/4 E 4/4 5/4 7/7 G B G B B B 5/4 5/4 I 8/5 I 8/5 C C 6/4 (e) Forward to G and I ; detect back edge IB ; update back[I] ; back[I]<dfsNumber[G] (f) Back up to G updating back[G]; back[G]=dfsNumber[B] ; remove bicomponent ; (g) Back up to B ; back[B]=dfsNumber[E] ; remove bicomonent ; (d) back[C]<dfsNumber[B] so back up to B updating back[B]. The action of the bicomponent algorithm on the graph in Fig. 4.25 (detecting the first two bicomponents)

23. procedure SEARCHB(v); begin mark v "old"; DFSNUMBER[v]  COUNT; COUNT  COUNT + 1; BACK[v]  DFSNUMBER[v]; for each vertex w on L[v] do Stack edges, here if w is marked "new" then begin add(v, w) to T; FATHER[w] v SEARCHB(w) if BACK[w]  DFSNUMBER[v] then a biconnected component has been found ; Pop, here BACK[v]  MIN(BACK[v], BACK[w]) end else if w is not FATHER[v] then BACK[v]  MIN(BACK[v], DFSNUMBER[w]) end O(|Ei|) Report O(|Ei|) why ?

24. Algorithm : Bicomp. T := Ø ; for all v in V do mark v as "new" end for all "new" vertex v in V do count := 1 ; SEARCHB(v) |V| + (|E1| + |E2| + …) = |V| + |E| end Theorem : Algorithm Bicomp correctly finds the biconnected components of G = (V, E) in O(|E| + |V|) time. [proof ] See the next page. O(|V|)

25. (i) Time complexity O(|V| + |E|) (ii) Correctness Root ?  Even if the root is not an articulation point, it can be treated as an articulation point !!! why ? The biconnected component containing the root is emitted from the root.

26. (By induction on the # of biconnected components in G ) Reporting a biconnected component (i) w v (SEARCHB(w) is completed) (ii) v  BACK(w) v w The edges above (v, w) on STACK is exactly those edges in the biconnected component containing (v, w). (b = 1) Trivial why ? Depth First Search !!! In this case, v must be the root !!! why ? (b = i) Assume the induction hypothesis. (b = i+1) (v,w)  b-1=i

27. K-connectivity k = 1 connectivity k = 2 bi-connectivity k = 3 tri-connectivity k = 4 no known efficient algorithm

28. 6. Strongly Connected Components of a Digraph Def'n : Let G = (V, E) be a directed graph. We can partition V into equivalent classes Vi, 1i r, such that vertices v and w are equivalent if and only if there is a path from v to w and w to v. Let Ei, 1i r, be the set of edges connecting the pairs of vertices in Vi. The graph Gi = (Vi, Ei) are called the strongly connected components of G. A graph is said to be strongly connected if it has only one strongly connected component. G1 1 2 3 G3 G=(V, E) 8 9 7 4 5 6 G2

29. A A B H B H E D E D F G C F G C J I J I K K (a) A digraph (b) Its strong components. The condensation of the digraph

30. G = (V, E) A Weakly Connected Component

31. How to find the strongly connected components of a graph ? Assumption : Depth First Search tree edge descendant edge cross edge back edge

32. v1 v9 v12 v5 G = (V, E) v2 v4 v13 v11 v10 v3 v6 v16 v14 v15 v7 v8 v17 v18 v1 13 v13 1 T = (V, ET) v2 v14 2 14 15 v15 v3 3 16 v16 4 v4 9 v10 18 v6 17 v17 v18 6 5 v5 10 v12 v7 12 7 v11 11 v9 8 v8 Spanning Forest

33. Lemma : Let Gi= (Vi, Ei) be a strongly connected component of a directed graph G = (V, E). Let T = (V, ET) be a depth-first search spanning forest of G. Then the vertices of Gi together with the edges which are common to both Ei and ET form a subtree of the spanning tree T. [ proof ] v,w  Vi v w Depth first number Gi=(Vi, Ei) WLOG, let v < w. In Gi, P v w why ? x = min{ y| y is in P } Note : x may be v itself. In T, x Once P reaches a descendant of x in T, it cannot leave the subtree of x !!! why? p

34.  w is a descendant of x !!! why ? Now, all vertices between x and w are also descendants of x. why ? DFS !!! ( by the way of T is connected ) Since x v < w, v is also a descendant of x.  Any pair of vertices in Gihave a common ancestor in T. This is also in Gi. why ?  r min{ y| y is a common ancestor of the vertices in Gi} u Gi why?

35. (r1, r2,····, rk) Observation i < j Either riis to the left of rj ora descendant of rj. why ?

36. v1 v6 v9 v3 G = (V, E) v2 v7 v8 v4 v5 v1 1 v2 2 v3 3 6 v6 4 v4 v9 v7 9 7 v5 5 v8 strongly connected components 8 (r1, r2, r3, ····, rk)  the sequence of roots in the order in which 4 6 1 the DFS of these vertices are terminated.

37. Lemma : For each i, 1 i k, Gi consists of those vertices which are descendant of ri but are in none of G1, G2,····, Gi-1. [proof] The root rj for j > i cannot be a descendant of ri, since the call of SEARCH(rj) terminates after SEARCH(ri).

38. r w w v or LOWLINK[v] = min { {v}  { w| there is a cross or back edge from a descendant of v to w, and the root of the strongly connected component containing w is an ancestor of v} } LOWLINK[v] v Why ?

39. Lemma : Let G = (V, E) be a directed graph. A vertex v is the root of a strongly connected component if and only if LOWLINK[v] = v. [proof] (  ) Suppose that v is the root of a strongly connected component of G. By definition of LOWLINK, LOWLINK[v] v for all vV. why ? Suppose that LOWLINK[v] < v. Then, there are vertices w and r such that [1] w is reached by a cross or back edge from a descendant of v. [2] r is the root of the strongly connected component containing w. [3] r is an ancestor of v. [4] w < v (2)  r is an ancestor of wr w (2),(4)  r w < v r < v (3)  r is a proper ancestor of v. r and v must be in the same strongly connected component !!! why ? v is not the root of a strongly connected component. #  LOWLINK[v] = v. r w w v r v

40. (  ) Now, suppose that LOWLINK[v] = v. Assume that v is not the root of the strongly connected component containing v. Then, there is some proper ancestor r of v, which is the root, i.e., r Then , there is a path P from v to r . Why? Consider the first edge of P from a descendant of v to a vertex w that is not a descendant of v. w<v Why? w v P P w The path P goes from v to w to r. Why? There also exists a path from r to v. Why? r and w are in the same strongly connected component. LOWLINK[v] w < v. #

41. How to compute LOWLINK[v] v LOWLINK[v] := v LOWLINK[v] = min{ w, LOWLINK[v] } LOWLINK[v] = min{ LOWLINK[w], LOWLINK[v] } w v v w w v w

42. How to report Gi x y v w LOWLINK[w] = w g g w v y x

43. Procedure SEARCHC(v): begin mark v "old"; DFSNUMBER[v] COUNT; COUNTCOUNT + 1; O(|V|) LOWLINK[v]  DFSNUMBER[v]; push v on STACK; for each vertex w on L[v] do O(|E|) if w is marked "new" then begin SEARCHC(w); LOWLINK[v]  MIN(LOWLINK[v], LOWLINK[w]) end back edge or cross edge else {not new} if DFSNUMBER[w] < DFSNUMBER[v]need a table and w is on STACK then LOWLINK[v]  MIN(DFSNUMBER[w], LOWLINK[v]) if LOWLINK[v] = DFSNUMBER[v] then begin repeat begin pop x from top of STACK; print x end until x=v; print "end of strongly connected component" end end

44. Algorithm : (Strongly connected components of a directed graph) procedure StrongCC begin count := 1; for all v in V do mark v "new" O(|V|) end STACK := Ø while there exists a vertex v marked "new" do SEARCHC(v) O(|E|) end. end Optimal !!! O( |V| + |E| )

45. Theorem : "StrongCC" correctly finds the strongly connected components of G in O(|V| + |E|) time. [proof] Time complexity : Already shown correctness : ( By induction on # of calls to SEARCHC) "Whenever SEARCHC terminates, LOWLINK(v)is correctly computed." Exercise.