The k-in-a-path problem for claw-free graphs

Transcription

1 The k-in-a-path problem for claw-free graphs Jiří Fiala Univerzita Karlova v Praze Bernard Lidický Univerzita Karlova v Praze Marcin Kamiński Université Libre de Bruxelles Daniël Paulusma University of Durham STACS, Nancy, March 2010

2 Graph Minors N. Robertson, P. D. Seymour, Graph Minors. I. Excluding a Forest, JCTB, 35, 39-61, 1983 N. Robertson, P. D. Seymour, Graph Minors. II. Algorithmic Aspects of Tree-Width, J. of Alg., 7, , 1986 N. Robertson, P. D. Seymour, Graph Minors. III. Planar Tree-Width, JCTB, 36, 49-64, 1984 N. Robertson, P. D. Seymour, Graph Minors. IV. Tree-Width and Well-Quasi-Ordering, JCTB, 48, , 1990 N. Robertson, P. D. Seymour, Graph Minors. V. Excluding a Planar Graph, JCTB, 41, , 1986 N. Robertson, P. D. Seymour, Graph Minors. VI. Disjoint Paths across a Disc, JCTB, 41, , 1986 N. Robertson, P. D. Seymour, Graph Minors. VII. Disjoint Paths on a Surface, JCTB, 45, , 1988 N. Robertson, P. D. Seymour, Graph Minors. VIII. A Kuratowski Theorem for General Surfaces, JCTB, 48, , 1990 N. Robertson, P. D. Seymour, Graph Minors. IX. Disjoint Crossed Paths, JCTB, 49, 40-77, 1990 N. Robertson, P. D. Seymour, Graph Minors. X. Obstructions to Tree- Decomposition, JCTB, 52, , 1991 N. Robertson, P. D. Seymour, Graph Minors. XI. Circuits on a Surface, JCTB, 60, , 1994 N. Robertson, P. D. Seymour, Graph Minors. XII. Distance on a Surface, JCTB, 64, , 1995 N. Robertson, P. D. Seymour, Graph Minors. XIII. The Disjoint Paths Problem, JCTB, 63, , 1995 N. Robertson, P. D. Seymour, Graph Minors. XIV. Extending an Embedding, JCTB, 65, 23-50, 1995 N. Robertson, P. D. Seymour, Graph Minors. XV. Giant Steps, JCTB, 68, , 1996 N. Robertson, P. D. Seymour, Graph Minors. XVI. Excluding a Non-planar Graph, JCTB, 89, 43-76, 2003 N. Robertson, P. D. Seymour, Graph Minors. XVII. Taming a Vortex, JCTB, 77, , 1999 N. Robertson, P. D. Seymour, Graph Minors. XVIII. Tree-decompositions and well-quasi-ordering, JCTB, 89, , 2003 N. Robertson, P. D. Seymour, Graph Minors. XIX. Well-quasi-ordering on a Surface, JCTB, 90, , 2004 N. Robertson, P. D. Seymour, Graph Minors. XX. Wagner s Conjecture, JCTB, 92, , 2004 N. Robertson, P. D. Seymour, Graph Minors. XXI. Graphs with Unique Linkages, JCTB, 99, , 2009 N. Robertson, P. D. Seymour, Graph Minors. XXII. Irrelevant Vertices in Linkage Problems, JCTB, to appear N. Robertson, P. D. Seymour, Graph Minors. XXIII. Nash-Williams Immersion Conjecture, JCTB, to appear 2

3 Graph Minors * Wagner s Conjecture * Structure of graphs with a forbidden minor * Polynomial-time recognition 3

4 Graph Minors * Wagner s Conjecture * Structure of graphs with a forbidden minor * Polynomial-time recognition 3

5 Graph Minors * Wagner s Conjecture * Structure of graphs with a forbidden minor * Polynomial-time recognition 3

6 k-linkages k-linkages Input: graph G and k pairs of vertices of G: (s 1,t 1 ),..., (s k,t k ) Problem: does there exist k vertex-disjoint paths s 1 t 1,... s k t k? k-linkages = k-disjoint Paths 4

7 k-linkages k-linkages Input: graph G and k pairs of vertices of G: (s 1,t 1 ),..., (s k,t k ) Problem: does there exist k vertex-disjoint paths s 1 t 1,... s k t k? k-linkages = k-disjoint Paths 4

8 k-linkages Theorem (Robertson and Seymour) For every k, there exists a polynomial-time algorithm for k-linkages. 5

9 Mutually induced Paths in a graph are mutually induced if * they are vertex disjoint, and * no vertex of one path is adjacent to a vertex of another. 6

10 Mutually induced Paths in a graph are mutually induced if * they are vertex disjoint, and * no vertex of one path is adjacent to a vertex of another. 6

11 Mutually induced Paths in a graph are mutually induced if * they are vertex disjoint, and * no vertex of one path is adjacent to a vertex of another. 6

12 k-induced-linkages k-induced-linkages Input: graph G and k pairs of vertices of G: (s 1,t 1 ),..., (s k,t k ) Problem: does there exist mutually-induced paths s 1 t 1,... s k t k? k-induced-linkages = k-induced-disjoint-paths 7

13 k-induced-linkages k-induced-linkages Input: graph G and k pairs of vertices of G: (s 1,t 1 ),..., (s k,t k ) Problem: does there exist mutually-induced paths s 1 t 1,... s k t k? k-induced-linkages = k-induced-disjoint-paths 7

14 NP-completness D. Bienstock, On the Complexity of Testing for Odd Holes and Induced Odd Paths, Discrete Mathematics, 90, 85-92, 1991 Theorem (Bienstock) 2-Induced-Linkages is NP-complete. 8

15 NP-completness D. Bienstock, On the Complexity of Testing for Odd Holes and Induced Odd Paths, Discrete Mathematics, 90, 85-92, 1991 Theorem (Bienstock) 2-Induced-Linkages is NP-complete. 8

16 Graphs of bounded genus Theorem (Kobayashi, Kawarabayashi) For every k and g, there exists a linear-time algorithm for k-induced-linkages in graphs of genus at most g. Y. Kobayashi, K. Kawarabayashi, Algorithms for finding an induced cycle in planar graphs and bounded genus graphs, ACM-SIAM Symposium on Discrete Algorithms, SODA

17 k-in-a-path k-in-a-path Input: graph G and k vertices of G: t 1,...,t k Problem: does there exist an induced path containing t 1,...,t k (in this order)? k-in-a-path and k-induced-linkages are equivalent (sort of). 10

18 k-in-a-path k-in-a-path Input: graph G and k vertices of G: t 1,...,t k Problem: does there exist an induced path containing t 1,...,t k (in this order)? k-in-a-path and k-induced-linkages are equivalent (sort of). 10

19 Claws = a claw = = K 1,3 11

20 Claw-free graphs M. Chudnovsky, P. D. Seymour, Claw-free Graphs I. Orientable prismatic graphs, JCTB, 97, , 2007 M. Chudnovsky, P. D. Seymour, Claw-free Graphs II. Non-orientable prismatic graphs, JCTB, 98, , 2008 M. Chudnovsky, P. D. Seymour, Claw-free Graphs III. Circular Interval Graphs, JCTB, 98, , 2008 M. Chudnovsky, P. D. Seymour, Claw-free Graphs IV. Decomposition theorem, JCTB, 98, , 2008 M. Chudnovsky, P. D. Seymour, Claw-free Graphs V. Global structure, JCTB, 98, , 2008 M. Chudnovsky, P. D. Seymour, Claw-free Graphs VI. Coloring claw-free graphs, submitted M. Chudnovsky, P. D. Seymour, Claw-free Graphs VII. Quasi-line graphs, submitted 12

21 3-In-a-Path in claw-free graphs Theorem (Lévêque, Lin, Maffray, Trotignon) For every k, there exists a polynomial-time algorithm for 3-In-a-Path (2-Induced-Linkages) in claw-free graphs. Benjamin Lévêque, David Y. Lin, Frédéric Maffray, Nicolas Trotignon, Detecting induced subgraphs, DAM, 157, ,

22 Our main result Theorem For every k, there exists a polynomial-time algorithm for k-in-a-path (k-induced-linkages) in claw-free graphs. 14

23 Sketch of Proof 1. Cleaning the graph. 2. Reduction to quasi-line graphs. 3. Contracting homogeneous pairs. 4. Reduction to (circular) interval graphs. 5. Solving the problem on (circular) interval graphs. 15

24 Cleaning the graph Remove vertices of G that are not on an induced path from t 1 to t k. After cleaning, the graph has no odd antihole on 7 vertices as an induced subgraph. odd antihole = complement of an odd cycle on 5 vertices (induced) 16

25 Cleaning the graph Remove vertices of G that are not on an induced path from t 1 to t k. After cleaning, the graph has no odd antihole on 7 vertices as an induced subgraph. odd antihole = complement of an odd cycle on 5 vertices (induced) 16

26 Cleaning the graph Remove vertices of G that are not on an induced path from t 1 to t k. After cleaning, the graph has no odd antihole on 7 vertices as an induced subgraph. odd antihole = complement of an odd cycle on 5 vertices (induced) 16

27 Sketch of Proof 1. Cleaning the graph. 2. Reduction to quasi-line graphs. 3. Contracting homogeneous pairs. 4. Reduction to (circular) interval graphs. 5. Solving the problem on (circular) interval graphs. 17

28 Quasi-line graphs A graph is quasi-line if the neighborhood of every vertex can be partitioned into two cliques. Fact A claw-free graph is quasi-line iff no vertex has an odd antihole in its neighborhood. 18

29 Quasi-line graphs A graph is quasi-line if the neighborhood of every vertex can be partitioned into two cliques. Fact A claw-free graph is quasi-line iff no vertex has an odd antihole in its neighborhood. 18

30 C 5 in the neighborhood Lemma Let v be a vertex with an induced C 5 in its neighborhood. Removing v does not change the solution of the problem. 19

31 Sketch of Proof 1. Cleaning the graph. 2. Reduction to quasi-line graphs. 3. Contracting homogeneous pairs. 4. Reduction to (circular) interval graphs. 5. Solving the problem on (circular) interval graphs. 20

32 Homogeneous pairs of cliques Two (non-trivial) cliques Q 1 and Q 2 are a homogeneous pair, if every vertex from V \ (V (Q 1 ) V (Q 2 )) is adjacent either to none or to all vertices in Q 1 (Q 2 ). 21

33 Finding homogeneous pairs Theorem (King, Reed) There exists a polynomial-time algorithm for finding a homogeneous pair of cliques in a graph (if one exists). A. King, B. Reed, Bounding χ in terms of ω and δ for quasi-line graphs, Journal of Graph Theory, 59, ,

34 Homogeneous pair contraction Lemma Contracting each of the cliques in a homogeneous pair to a single vertex does not change the solution of the problem (if we do it carefully). 23

35 Homogeneous pair contraction Lemma Contracting each of the cliques in a homogeneous pair to a single vertex does not change the solution of the problem (if we do it carefully). 23

36 Sketch of Proof 1. Cleaning the graph. 2. Reduction to quasi-line graphs. 3. Contracting homogeneous pairs. 4. Reduction to (circular) interval graphs. 5. Solving the problem on (circular) interval graphs. 24

37 Quasi-line graphs Theorem A quasi-line graph with no homogeneous pair of cliques is either a circular interval graph or a composition of interval graphs. M. Chudnovsky, P. D. Seymour, The Structure of Claw-free Graphs, In Surveys in combinatorics 2005, Cambridge, ,

38 Sketch of Proof 1. Cleaning the graph. 2. Reduction to quasi-line graphs. 3. Contracting homogeneous pairs. 4. Reduction to (circular) interval graphs. 5. Solving the problem on (circular) interval graphs. 26

39 Hardness result Theorem When k is a part of the input, k-in-a-path (k-induced-linkages) is NP-complete in claw-free graphs. Why? An induced path in G corresponds to (not necessarily induced) path in the graph H with L(H) = G. Hence, we use Theorem (Karp 1975) When k is part of the input, k-linkages is NP-complete. 27

40 Hardness result Theorem When k is a part of the input, k-in-a-path (k-induced-linkages) is NP-complete in claw-free graphs. Why? An induced path in G corresponds to (not necessarily induced) path in the graph H with L(H) = G. Hence, we use Theorem (Karp 1975) When k is part of the input, k-linkages is NP-complete. 27

41 Hardness result Theorem When k is a part of the input, k-in-a-path (k-induced-linkages) is NP-complete in claw-free graphs. Why? An induced path in G corresponds to (not necessarily induced) path in the graph H with L(H) = G. Hence, we use Theorem (Karp 1975) When k is part of the input, k-linkages is NP-complete. 27

42 Hardness result Theorem When k is a part of the input, k-in-a-path (k-induced-linkages) is NP-complete in claw-free graphs. Why? An induced path in G corresponds to (not necessarily induced) path in the graph H with L(H) = G. Hence, we use Theorem (Karp 1975) When k is part of the input, k-linkages is NP-complete. 27

43 Relaxing k-in-a-path k-in-a-tree Input: graph G and k vertices of G: t 1,...,t k Problem: does there exist an induced tree containing t 1,...,t k? 28

44 3-In-a-Tree Theorem (Chudnovsky, Seymour) There exists a polynomial-time algorithm for 3-In-a-Tree. M. Chudnovsky, P. D. Seymour, The three-in-a-tree problem, Combinatorica, to appear What about k-in-a-tree for k 4? 29

45 3-In-a-Tree Theorem (Chudnovsky, Seymour) There exists a polynomial-time algorithm for 3-In-a-Tree. M. Chudnovsky, P. D. Seymour, The three-in-a-tree problem, Combinatorica, to appear What about k-in-a-tree for k 4? 29

46 k-in-a-tree for claw-free graphs Every induced tree in a claw-free graph is a path. k-in-a-tree and k-in-a-path are the same for claw-free graphs. 30

47 k-in-a-tree for claw-free graphs Every induced tree in a claw-free graph is a path. k-in-a-tree and k-in-a-path are the same for claw-free graphs. 30

48 Open problem What is the computational complexity of finding two mutually-induced cycles in a graph? 31

49 thank you! 32