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

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 7 October 2014Minko 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 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University

4. vs(G) = min {vs L (G) | L is a layout of G} = 2 VS of Layouts and Graphs (2) 7 October 2014 vs L (G)=2 0 x y v u w uvwyx 1222 L G = (V,E) π L (u) = {u}π L (v) = {u,v}π L (w) = {v,w}π L (y) = {w,y} π L (x) = ∅ Minko Markov, Faculty of Mathematics and Informatics, Sofia University

5. (u,v), (u,w), (v,w), (v,y), and (w,y) are clean Node Search Number (SN) 7 October 2014 Minko Markov, Faculty of Mathematics and Informatics. This research is supported by Sofia University Science Fund under project "Discrete Structures" w+w+ all edges are contaminated (u,v), (u,w) and (v,w) are clean x y vu w u+u+ v+v+ x+x+ y+y+ v—v— u—u— y—y— w—w— x—x— sn s (G) = 3 S = (u,v) is cleanall edges are clean monotonous (progressive) search

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 7 October 2014 x y vu w L = u v w y x, vs L (G) = 2 S = u + v + w + u − y + v − x + y − x − w −, sn s (G) = 3 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 + c 2 + 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) 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University

8. Cactus graphs (cacti) 7 October 2014 Minko Markov, Faculty of Mathematics and Informatics, Sofia University

9. Rooted Cacti 7 October 2014Minko 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. 7 October 2014 v vs = k < k Minko Markov, Faculty of Mathematics and Informatics, Sofia University

11. k-critical subtree  T is a rooted tree, vs(T) = k, and the root induces two subtrees of vs = k. 7 October 2014 kk < k... Minko Markov, Faculty of Mathematics and Informatics, Sofia University

12. Label of a tree 7 October 2014 T2T2 TT1T1 lab(T) = (k, p, q), k > p > q pp kk q vs(T)=kvs(T 1 )=pvs(T 2 )=q Minko Markov, Faculty of Mathematics and Informatics, Sofia University

13. The EST algorithm 7 October 2014 ▲ _ _ ▲ _ _ ▲ _ lab: lab = ? lab 1 = (5,2) lab 1 : 9 8 7 6 5 4 3 2 1 lab 2 = (7,6,5) lab 3 = (8,5,2 c ) lab 2 : lab 3 : _ _ _ ▲ _ _ ● _ _ ▲ ▲ ● _ _ _ _ ● _ _ _ _ _ _ _ _ ● _ _ _ _ _ _ _ ● _ _ _ _ _ _ ● _ _ _ _ _ _ ● _ _ ● _ _ _ lab = (9) Minko Markov, Faculty of Mathematics and Informatics, Sofia University

14. The VS backbone of a tree  the easiest kind of rooted tree of VS k 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University VS < k VS = k 1 K−1

15. The VS backbone of a tree  the second best kind (VS = k) 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University VS = k VS < k VS = k 1 K K−1

16. The VS backbone of a tree  an even harder rooted tree of VS k 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University VS = k VS < k VS = k 1 K K−1 K

17. The VS backbone of a tree  the hardest kind 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University VS = k VS < k VS = k 1 K K−1 K VS = k-1 1 1 K−1

18. The backbone of a non-rooted tree 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University VS < k VS = k 1 K−1 1 1

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. 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University

20. G ⊝ [u,v] 7 October 2014 uv G G ⊝ [u,v] Minko Markov, Faculty of Mathematics and Informatics, Sofia University

21. Stretchability k w.r.t. u and v 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University u v K

22. The idea behind the theorem  Definition: a c-path (cactus path) in a cactus is a linear order of vertices and cycles 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University a b e g c ki d tq rp onm lj hfvux s3s3 s2s2 s1s1 w s4s4 C = a s 1 f g h s 2 m n o s 3 u v w x

23. The backbone of a cactus 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University b f d he cai s j K−1 K

24. The root and the backbone 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University r b f d he c a i s j K lab(G( r )) = ( K, G lab(G 1 (r)) ) G1G1

25. The root and the backbone 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University r f d he s lab(G( r )) = ( K, i j bc a K G lab(G 1 (r)) ) G1G1 i c

26. The cacti pitfall 7 October 2014 k-1 k k k k Minko Markov, Faculty of Mathematics and Informatics, Sofia University

27. The cacti pitfall 7 October 2014 kk k-2 kk Minko Markov, Faculty of Mathematics and Informatics, Sofia University

28. The cacti pitfall 7 October 2014Minko 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 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University

30. Boundaried cactus  A cactus G in which some cycles s 1, …, s n have two boundary vertices each. All boundary vertices are of degree 2.  Let the boundary pair in s i be ‹u i, w i ›. The search game on G is performed so that n searchers are placed on U = {u 1, …, u n } initially and at the end, each of W = {w 1, …, w n } must have a searcher.  The boundary is ‹U, W›. 7 October 2014Minko 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. 7 October 2014Minko 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}›. 7 October 2014Minko 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 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University L is k-stretchable iff the separation of any vertex is (k – the number of intervals it is in) urx {u,v,w} : left, {x,y} : right vwy r is the rightmost neighbour of x and y layout L

34. How to prove k-stretchability  It is easier to modify L into an extended layout L* and consider its VS 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University x uvw y layout L layout L* uvw L is k-stretchable iff L* has VS ≤ k

35. Proof of the theorem, part I  Consider an optimal extended layout L*. Consider the leftmost and rightmost nonboundary vertices a and z. 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University a z s3s3 s2s2 s1s1 s4s4 rvs(G) ≤ 5 → rvs(G 1 ) ≤ 4, i.e. vs(G 1 ) ≤ 4 G1G1 G

36. THE END 7 October 2014Minko Markov, Faculty of Mathematics and Informatics, Sofia University