2. 1 IM.CJCU Hsin-Hung Chou The Node-Searching Problem on Special Graphs 周信宏 長榮大學 資訊管理學系 chouhh@mail.cjcu.edu.tw

3. 2 IM.CJCU Hsin-Hung Chou Outline Introduction Properties of the problem Previous results Result on unicyclic graphs Conclusions

4. 3 IM.CJCU Hsin-Hung Chou Outline Introduction graph searching problem problem definition Properties of the problem Previous results Result on unicyclic graphs Conclusions

5. 4 Graph-searching problem was first proposed by Parsons, 1976. Goal: To find a search strategy using the least number of searchers to capture the fugitive. Graph-Searching Problem IM.CJCU Hsin-Hung Chou

6. 5 Problem Definition Variations: Operations: Place a searcher Remove a searcher Move along an edge Clearing rules: Move along an edge Guard two endpoints of an edge RulesOperations Edge- Node- Mixed- Place Remove Move Place Remove Place Remove Move Guard Move Guard IM.CJCU Hsin-Hung Chou

7. 6 Node-Searching Problem a b c d e # of searchers = 3 The node-searching problem was first proposed by Kirousis and Papadimitriou, 1986. IM.CJCU Hsin-Hung Chou

8. 7 Outline Introduction Properties of the problem progressive strategy related problems Previous results Result on unicyclic graphs Conclusions

9. 8 Recontamination IM.CJCU Hsin-Hung Chou a b c d e

10. 9 Kirousis and Papadimitriou showed that there exists an optimal search strategy without recontamination for any graph. Progressive Strategy IM.CJCU Hsin-Hung Chou We can only consider search strategies without recontamination. There exists an optimal search strategy in which no vertex is visited twice by a searcher, and in which every searcher is deleted immediately after all the edges incident on it have been cleared.

11. 10 Interval Model IM.CJCU Hsin-Hung Chou e b a c d 12876543109 abcdeabcde

12. 11 Guard Sets IM.CJCU Hsin-Hung Chou 12876543109 abcdeabcde a bc c cd c de ea X1X1 ab X2X2 X3X3 X4X4 X5X5 X6X6 X7X7 X8X8 X9X9

13. 12 Path-decomposition IM.CJCU Hsin-Hung Chou a bc c cd c de ea X1X1 ab X2X2 X3X3 X4X4 X5X5 X6X6 X7X7 X8X8 X9X9 1. 2. For every edge (u,v)  E(G), there exists an X i containing both u and v. 3. if i < j < k.

14. 13 Path-width Problem IM.CJCU Hsin-Hung Chou The width of a path-decomposition is max { |X i | | 1  i  r} – 1. The path-width of G is the minimum width over all path-decompositions of G. The path-width of G is equal to the node-search number of G minus one.

15. 14 Guard Sequence IM.CJCU Hsin-Hung Chou 12876543109 abcdeabcde c d e a b

16. 15 Linear Layout IM.CJCU Hsin-Hung Chou c d e a b A linear layout of a graph G is a one-to-one mapping L : V(G)  {1,2, …, |V(G)|}. 1 2 3 4 5

17. 16 Cut Numbers IM.CJCU Hsin-Hung Chou The i-th cut number of L, denoted by cut L (i), is the number of vertices which are mapped to integers less than or equal to i and adjacent to a vertex mapped to an integer larger than i. c d e a b 1 2 3 4 5 cut L (i) : 1 2 1 2 0

18. 17 Vertex Separation IM.CJCU Hsin-Hung Chou The vertex separation with respect to G and L: vs L (G) = max {cut L (i) | 1  i  |V(G)|}. The vertex separation of G: vs(G) = min {vs L (G) | L is a linear layout of G}. The vertex separation of G is equal to the node-search number of G minus one.

19. 18 Related Problems path-width problem vertex separation problem interval thickness problem gate matrix layout problem narrowness problem The node-searching problem is equivalent to IM.CJCU Hsin-Hung Chou

20. 19 IM.CJCU Hsin-Hung Chou Outline Introduction Properties of the problem Previous results Result on unicyclic graphs Conclusions

21. 20 Previous Results IM.CJCU Hsin-Hung Chou Classes of graphsComplexity general with bounded degree 3NP-complete planar with bounded degree 3NP-complete chordalNP-complete chordal bipartiteNP-complete bipartiteNP-complete bipartite distance-hereditaryNP-complete co-comparabilityNP-complete starlikeNP-complete circleNP-complete latticeNP-complete

22. 21 Previous Results (cont.) IM.CJCU Hsin-Hung Chou Classes of graphsComplexity k-starlike (for a fixed k)O(mn k ) splitO(mn) co-chordal O(n  +1 ),  =2.37… intervalO(m+n) permutation O(n  pw) partial k-tree (for a fixed k)  (n 4k+3 ) cograph, treeO(n) circular-arcO(n 2 ) block unicyclic O(bc+c 2 +n) O(n)

23. 22 IM.CJCU Hsin-Hung Chou Outline Motivation Avenue system on trees Previous results Result on unicyclic graphs motivation avenue system on trees linear-time algorithm Conclusions

24. 23 6-tree IM.CJCU Hsin-Hung Chou k-trees Recursive definition of k-trees: A k-clique is a k-tree. If T = (V,E) is a k-tree and C is a k-clique of T and x  V(T), then T’ = (V  {x},E  {cx | c  C}) is a k-tree. 5-tree

25. 24 5-tree IM.CJCU Hsin-Hung Chou Partial k-trees A graph is a partial k-trees if it is a spanning subgraph of a k-tree. partial 5-tree

26. 25 IM.CJCU Hsin-Hung Chou Unicyclic Graphs A unicyclic graph is a graph composed of a tree with one extra edge. A unicyclic graph is a partial 2-tree.

27. 26 IM.CJCU Hsin-Hung Chou Results on Unicyclic Graphs Hans L. Bodlaender and Ton Kloks, “Efficient and constructive algorithms for the pathwidth and treewidth of graphs”, Journal of Algorithms, 21(no.2):pp. 358–402, 1996. Time complexity:  (n 4k+3 ) for a partial k-tree with fixed k. J. A. Ellis and M. Markov, “Computing the vertex separation of unicyclic graphs”, Information and Computation, 192:pp. 123–161, 2004. Time complexity: O(n log n). Our result: O(n).

28. 27 IM.CJCU Hsin-Hung Chou Outline Motivation Avenue system on trees Previous results Result on unicyclic graphs motivation avenue system on trees linear-time algorithm Conclusions

29. 28 IM.CJCU Hsin-Hung Chou Results on Trees J. A. Ellis, I. H. Sudborough, and J. S. Turner, “The vertex separation and search number of a graph”, Information and Computation, 113(no. 1):pp. 50–79, 1994. Search number: O(n); Optimal search strategy: O(n log n). S. L. Peng, C. W. Ho, T. S. Hsu, M. T. Ko, and C. Y. Tang, “A linear-time algorithm for constructing an optimal node-search strategy of a tree”, LNCS 1449:pp. 279–288, 1998. K. Skodinis, “Construction of linear tree-layouts which are optimal with respect to vertex separation in linear time”, Journal of Algorithms, 47:pp. 40–59, 2003. Optimal search strategy: O(n).

30. 29 IM.CJCU Hsin-Hung Chou kk kk kk  k+1 [PAR76] For any tree T and an integer k  2, ns(T)  k+1 if and only if there exists a vertex v with at least three branches having search numbers at least k. Parsons’ Lemma branch

31. 30 IM.CJCU Hsin-Hung Chou Example ns(T) = 3

32. 31 IM.CJCU Hsin-Hung Chou Hub ns(T 1 ) = 2 ns(T) = 3 ns(T 2 ) = 2 ns(T 3 ) = 2

33. 32 IM.CJCU Hsin-Hung Chou Example ns(T) = 3

34. 33 IM.CJCU Hsin-Hung Chou ns(T 2 ) = 3 ns(T 1 ) = 3 ns(T) = 3 Critical Vertices

35. 34 IM.CJCU Hsin-Hung Chou Outlet Vertices ns(T’) = 3 ns(T) = 3

36. 35 IM.CJCU Hsin-Hung Chou Avenue on Trees [MEG88] For any tree T, a path P = [v 1, v 2,..., v r ] is an avenue of T, if the following conditions hold: If r = 1, then v 1 is a hub. If r > 1, then each of v 1 and v r is an outlet vertex, and for every j, 2  j  r-1, v j is a critical vertex. [MEG88] For any tree T, T has an avenue. If the length of the avenue is at least two, then the avenue is unique.

37. 36 IM.CJCU Hsin-Hung Chou Dynamic Programming Every tree under construction has a specified vertex called the root. The algorithm computes the search number based on the tree decomposition using dynamic programming.

38. 37 IM.CJCU Hsin-Hung Chou Types of Rooted Trees

39. 38 IM.CJCU Hsin-Hung Chou Label of Trees Label: u p = u u1u1 Vertices: Types: M,M, …, M, H(E,I) a1a1 u2u2 a2a2 u3u3 a3a3

40. 39 k k IM.CJCU Hsin-Hung Chou Merge Rules – Case 1 = (k+1’) k If there exist at least three labels containing k which is the maximum element in all labels, then …

41. 40 k k IM.CJCU Hsin-Hung Chou Merge Rules – Case 2 = (k+1’) < k If there exist exactly two labels containing k and one of them contains a k-critical vertex, then … < k k

42. 41 k k IM.CJCU Hsin-Hung Chou Merge Rules – Case 3 = (k) < k If there exist exactly two labels containing k and neither of them contains a k-critical vertex, then … < k

43. 42 < k k IM.CJCU Hsin-Hung Chou Merge Rules – Case 4 = (k+1’) < k If there exist exactly one label containing k and it contains a k-critical vertex x, and ns(T u [x]) = k, then … < k k x

44. 43 < k k IM.CJCU Hsin-Hung Chou Merge Rules – Case 5 = (k)& (T u [x]) < k If there exist exactly one label containing k and it contains a k-critical vertex x, and ns(T u [x]) < k, then … < k k x

45. 44 < k k IM.CJCU Hsin-Hung Chou Merge Rules – Case 6 = (k’) < k If there exists exactly one label containing k and it contains no k-critical vertex, then … < k

46. 45 IM.CJCU Hsin-Hung Chou Outline Motivation Avenue system on trees Previous results Result on unicyclic graphs motivation avenue system on trees linear-time algorithm Conclusions

47. 46 IM.CJCU Hsin-Hung Chou Oriented Search Strategy A search strategy in which u is the start vertex and v is the end vertex is called an oriented search strategy for G from u to v. c v u a b start end # of searchers = 4 os(G,u,v) = os(G,v,u). os(G,u) = min { os(G,u,v) | v  V(G) }. ns(G) = min { os(G,u,v) | u,v  V(G) }.

48. 47 IM.CJCU Hsin-Hung Chou Main Algorithm (Phase 1) Compute the labels of rooted constituent trees.

49. 48 IM.CJCU Hsin-Hung Chou Main Algorithm (Phase 1) Compute the label of rooted U-e. ns(U-e)  ns(U)  ns(U-e) + 1.

50. 49 IM.CJCU Hsin-Hung Chou Main Algorithm (Phase 1) Construct the label array of U. Example:  ={ 1 =(9,8,6,4’), 2 =(8’), 3 =(7,6)}. ]1[  A ],,6[ 3 2 v I ],,4[ 1 4 v H ],,6[ 1 3 v M ]9[  A ]8[  A ]7[  A ]6[  A ]5[  A ]4[  A ]3[  A ]2[  A ],,7[ 3 1 v M ],,8[ 2 1 v E],,8[ 1 2 v M ],,9[ 1 1 v M i ALL n i H n i E n i I n i M n i ptr 1 1 1 2 2 8 7 6 4 0 1 11 1 11 1 000 00 0 0 0 0 0 0 0 0 H E I

51. 50 IM.CJCU Hsin-Hung Chou Main Algorithm (Phase 2) Decide ns(U) = k or k+1, where k = ns(U-e), based on the labels of U-e and constituent trees.

52. 51 IM.CJCU Hsin-Hung Chou Phase 2 (Case 1) One k-critical constituent tree (containing k-critical vertex). k k T’ u u c If ns(T’) = k, then ns(U) = k+1; else ns(U) = k. U’ If ns(U’-e) < k-1, then ns(U) = k; else ns(U) = ns(U’)+1. e

53. 52 IM.CJCU Hsin-Hung Chou Phase 2 (Case 2) Three or more k-non-critical constituent trees. k ns(U) = k+1. k k

54. 53 IM.CJCU Hsin-Hung Chou Phase 2 (Case 3) Exactly two k-non-critical constituent trees. k k u v If os(U’,u,v)  k, then ns(U) = k; else ns(U) = k+1. U’

55. 54 IM.CJCU Hsin-Hung Chou Phase 2 (Case 4) Exactly one k-non-critical constituent tree. If os(U’,u,v)  k, then ns(U) = k; else ns(U) = k+1. k k-1 u v U’

56. 55 IM.CJCU Hsin-Hung Chou Phase 2 (Case 5) No k-non-critical constituent tree. If os(U’,u,v)  k, then ns(U) = k; else ns(U) = k+1. k-1 u v U’

57. 56 IM.CJCU Hsin-Hung Chou Conclusion unicyclic  cactus  partial k-tree (O(n 4k+3 )). Unicyclic graph: O(n). Cactus graph:

58. 57 IM.CJCU Hsin-Hung Chou Thank you for your attention!

59. 58 IM.CJCU Hsin-Hung Chou Q & A