Towards the Construction of a Fast Algorithm for the Vertex Separation Problem on Cactus Graphs

1. Minko Markov Sofia University, Faculty of Mathematics and Informatics minkom@fmi.uni-sofia.bg Towards the Construction of a Fast Algorithm for the Vertex Separation Problem on Cactus Graphs

2. Structure of the presentation • Background • Vertex Separation of Trees and Unicyclics • Vertex Separation of Cacti • Boudaried Cacti and Stretchability • Decomposition of Boundaried Cacti • Main Theorem for Stretchability on Boundaried Cacti Minko Markov, Faculty of Mathematics and Informatics, Sofia University

3. Vertex Separation (VS) of Layouts and Graphs • An NP-complete problem on undirected ordinary graphs • Do not confuse “Vertex Separation” with “Vertex Separator” • The definition of Vertex Separation is based on the definition of linear layout Minko Markov, Faculty of Mathematics and Informatics, Sofia University

4. VS of Layouts and Graphs (2) vsL(G)=2 G = (V,E) u v w L vs(G) = min {vsL(G) | L is a layout ofG} = 2 u v w y x x 1 2 2 2 0 y πL(u) = {u} πL(v) = {u,v} πL(w) = {v,w} πL(y) = {w,y} πL(x) = ∅ Minko Markov, Faculty of Mathematics and Informatics, Sofia University

5. Node Search Number (SN) monotonous (progressive) search u v w S = u+ v+ w+ u— y+ v— x+ y— x— w— sns(G) = 3 x y (u,v)is clean (u,v), (u,w),(v,w), (v,y), and(w,y)are clean all edges are contaminated all edges are clean (u,v), (u,w) and(v,w)are clean Minko Markov, Faculty of Mathematics and Informatics. This research is supported by Sofia University Science Fund under project "Discrete Structures"

6. VS is equivalent to SN • For every graph G, vs(G) = sn(G) − 1 • Optimal searches define unique optimal layouts, optimal layouts define multitudes of optimal searches u v L = u v w y x, vsL(G) = 2 w S =u+ v+ w+u−y+v−x+y− x− w−,sns(G) = 3 x y Minko Markov, Faculty of Mathematics and Informatics, Sofia University

7. Fast algorithms for VS on restric-ted graphs • O(n) for trees (Ellis, Sudborough, Turner, 1994) • O(n lg n) on unicyclic graphs (Ellis, Markov, 2004), improved to O(n) (Chou, Ko, Ho, Chen, 2006) • O(bc + c2 + n) on block graphs (Chou et al., 2008) • O(n) on 3-Cycle-Disjoint Graphs—a strict subclass of cactus graphs (Yang, Zhang, Cao 2010) Minko Markov, Faculty of Mathematics and Informatics, Sofia University

8. Cactus graphs (cacti) Minko Markov, Faculty of Mathematics and Informatics, Sofia University

9. Rooted Cacti Minko Markov, Faculty of Mathematics and Informatics, Sofia University

10. VS of trees – O(n) algorithm by Ellis, Sudborough, Turner (1994) • Theorem (EST, 1994): If T is a tree and k ≥ 1, then vs(T) ≤ k iff every vertex induces at most two subtrees of vs = k. v vs = k vs = k < k < k < k Minko Markov, Faculty of Mathematics and Informatics, Sofia University

11. k k k-critical subtree • T is a rooted tree, vs(T) = k, and the root induces two subtrees of vs = k. ... < k < k Minko Markov, Faculty of Mathematics and Informatics, Sofia University

12. p p k k Label of a tree lab(T) = (k, p, q), k > p > q T T1 T2 q vs(T2)=q vs(T1)=p vs(T)=k Minko Markov, Faculty of Mathematics and Informatics, Sofia University

13. The EST algorithm lab = (9) lab = ? lab3 = (8,5,2c) lab1 = (5,2) lab2 = (7,6,5) 9 8 7 6 5 4 3 2 1 lab1: _ _ _ ▲ _ _ ● _ lab2: _ ▲ ▲ ● _ _ _ _ lab3: ▲ _ _ ▲ _ _ ▲ _ ● _ _ _ _ _ _ _ ● _ _ _ _ _ _ ● _ _ _ _ _ _ ● _ _ ● _ _ ● _ _ _ ● _ _ _ _ _ _ _ _ lab: Minko Markov, Faculty of Mathematics and Informatics, Sofia University

14. VS < k VS < k The VS backbone of a tree • the easiest kind of rooted tree of VS k 1 VS = k K−1 K−1 K−1 VS < k Minko Markov, Faculty of Mathematics and Informatics, Sofia University

15. VS < k VS = k The VS backbone of a tree • the second best kind (VS = k) 1 VS = k K K−1 K−1 VS < k Minko Markov, Faculty of Mathematics and Informatics, Sofia University

16. VS = k VS = k The VS backbone of a tree • an even harder rooted tree of VS k 1 VS = k K K−1 K VS < k Minko Markov, Faculty of Mathematics and Informatics, Sofia University

17. VS = k VS = k The VS backbone of a tree 1 • the hardest kind VS = k 1 K−1 K−1 1 VS = k-1 VS = k-1 K−1 K K VS < k Minko Markov, Faculty of Mathematics and Informatics, Sofia University

18. The backbone of a non-rooted tree 1 1 1 VS = k K−1 K−1 K−1 K−1 K−1 K−1 K−1 K−1 K−1 K VS < k Minko Markov, Faculty of Mathematics and Informatics, Sofia University

19. Vertex Separation of Cacti • Theorem (M.M., 2007). Let G be a cactus and k ≥ 1. Then vs(G) ≤ k iff: • Every vertex induces at most two cacti of separation k, all others are < k. • In every cycle there exist vertices u and v (not necessarily distinct) such that G⊝[u,v] is k-stretchable. Minko Markov, Faculty of Mathematics and Informatics, Sofia University

20. G⊝[u,v] G⊝[u,v] G u v Minko Markov, Faculty of Mathematics and Informatics, Sofia University

21. Stretchability k w.r.t. u and v K u v Minko Markov, Faculty of Mathematics and Informatics, Sofia University

22. The idea behind the theorem • Definition: a c-path (cactus path) in a cactus is a linear order of vertices and cycles b d p r i k g m n o f h u v w x a s1 s2 s3 c e j l q t s4 C = a s1 f g h s2 m n o s3 u v w x Minko Markov, Faculty of Mathematics and Informatics, Sofia University

23. The backbone of a cactus K f d j b a c i s e h K−1 K−1 K−1 K−1 K−1 K−1 K−1 K−1 Minko Markov, Faculty of Mathematics and Informatics, Sofia University

24. The root and the backbone G K f d j i b c a s e h G1 r lab(G(r)) = ( K, lab(G1(r)) ) Minko Markov, Faculty of Mathematics and Informatics, Sofia University

25. The root and the backbone i c G1 G K f d j i b c a s e h r lab(G(r)) = ( K, lab(G1(r)) ) Minko Markov, Faculty of Mathematics and Informatics, Sofia University

26. k-1 k-1 k k k k The cacti pitfall Minko Markov, Faculty of Mathematics and Informatics, Sofia University

27. k-2 k-2 The cacti pitfall k k k k Minko Markov, Faculty of Mathematics and Informatics, Sofia University

28. The cacti pitfall Minko Markov, Faculty of Mathematics and Informatics, Sofia University

29. The solution for cacti? • Take Stretchability w.r.t. k vertex pairs as the primary problem • Consider bounaried cacti, the boundary being the vertices w.r.t. which we stretch • The original problem reduces to this one – just take an empty boundary Minko Markov, Faculty of Mathematics and Informatics, Sofia University

30. Boundaried cactus • A cactus G in which some cycles s1, …, sn have two boundary vertices each. All boundary vertices are of degree 2. • Let the boundary pair in si be ‹ui, wi›. The search game on G is performed so that n searchers are placed on U = {u1, …, un} initially and at the end, each of W = {w1, …, wn} must have a searcher. • The boundary is ‹U, W›. Minko Markov, Faculty of Mathematics and Informatics, Sofia University

31. The Residual VS of a boundaried cactus • Let G be a boundaried cactus with n vertex pairs in the boundary. Let k be the stretchability of G w.r.t. the boundary. Then rvs(G) = k – n. We proved k – n > 0 always. • From now on we consider RVS of boundaried cacti. VS of cacti is a special case of RVS. Minko Markov, Faculty of Mathematics and Informatics, Sofia University

32. RVS of Boundaried Cacti • Theorem: Let G be a boundaried cactus, boundary ‹U, W›, and m ≥ 1. Then rvs(G) ≤ m iff: • Every nonboundary vertex induces at most two boundaried cacti of rvs m, all others are < m. • In every cycle there are nonboundary vertices x and y (not necessarily distinct) such that G is (k+1)-stretchable w.r.t. ‹U  {x}, W  {y}› or ‹U  {y}, W  {x}›. Minko Markov, Faculty of Mathematics and Informatics, Sofia University

33. How to prove k-stretchability • It is more rigorous to use the VS definition and terminology, not the NSN L is k-stretchable iff the separation of any vertex is (k – the number of intervals it is in) {u,v,w} : left, {x,y} :right ris the rightmost neighbour ofx and y u x y v r w layout L Minko Markov, Faculty of Mathematics and Informatics, Sofia University

34. How to prove k-stretchability • It is easier to modify L into an extended layoutL* and consider its VS L is k-stretchable iffL* has VS ≤ k L is k-stretchable iff the separation of any vertex is (k – the number of intervals it is in) u u x y v v w w layout L layout L* Minko Markov, Faculty of Mathematics and Informatics, Sofia University

35. Proof of the theorem, part I • Consider an optimal extended layout L*. Consider the leftmost and rightmost nonboundary vertices a and z. G z s1 s2 a s3 s4 G1 rvs(G) ≤ 5 → rvs(G1) ≤ 4, i.e. vs(G1) ≤ 4 Minko Markov, Faculty of Mathematics and Informatics, Sofia University

36. THE END Minko Markov, Faculty of Mathematics and Informatics, Sofia University