Download presentation

Presentation is loading. Please wait.

Published byTatiana Roser Modified over 4 years ago

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

Similar presentations

Presentation is loading. Please wait....

OK

Discrete Structures Trees (Ch. 11)

Discrete Structures Trees (Ch. 11)

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google