Disjoint Paths and Connected Subgraphs for H-Free Graphs

The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct pairs. We determine, with an exception of two cases, the complexity of the Disjoint Paths problem for H-free graphs. If k is fixed, we obtain the k-Disjoint Paths problem, which is known to be polynomial-time solvable on the class of all graphs for every k ≥ 1. The latter does no longer hold if we need to connect vertices from terminal sets instead of terminal pairs. We completely classify the complexity of k-Disjoint Connected Subgraphs for H-free graphs, and give the same almost-complete classification for Disjoint Connected Subgraphs for H-free graphs as for Disjoint Paths.

[1]  Bruce A. Reed,et al.  The disjoint paths problem in quadratic time , 2012, J. Comb. Theory, Ser. B.

[2]  Richard Bellman,et al.  Dynamic Programming Treatment of the Travelling Salesman Problem , 1962, JACM.

[3]  Daniël Paulusma,et al.  Contracting to a Longest Path in H-Free Graphs , 2018, ISAAC.

[4]  Daniël Paulusma,et al.  On partitioning a graph into two connected subgraphs , 2009, Theor. Comput. Sci..

[5]  Michal Pilipczuk,et al.  Solving the 2-Disjoint Connected Subgraphs Problem Faster than 2n , 2012, Algorithmica.

[6]  Russell Merris,et al.  Split graphs , 2003, Eur. J. Comb..

[7]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[8]  M. Held,et al.  A dynamic programming approach to sequencing problems , 1962, ACM National Meeting.

[9]  Neil Robertson,et al.  Graph Minors .XIII. The Disjoint Paths Problem , 1995, J. Comb. Theory B.

[10]  Egon Wanke,et al.  Vertex disjoint paths on clique-width bounded graphs , 2004, Theor. Comput. Sci..

[11]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[12]  Michal Pilipczuk,et al.  Solving the 2-Disjoint Connected Subgraphs Problem Faster than 2n , 2014, Algorithmica.

[13]  Gerhard J. Woeginger,et al.  Partitioning graphs into connected parts , 2009, Theor. Comput. Sci..

[14]  F. Fomin,et al.  Path Contraction Faster Than 2 , 2019 .

[15]  Fedor V. Fomin,et al.  Path Contraction Faster Than 2n , 2019, ICALP.

[16]  Egon Wanke,et al.  Vertex Disjoint Paths on Clique-Width Bounded Graphs , 2004, LATIN.

[17]  Fedor V. Fomin,et al.  Path Contraction Faster than 2n , 2020, SIAM J. Discret. Math..

[18]  Richard M. Karp,et al.  On the Computational Complexity of Combinatorial Problems , 1975, Networks.

[19]  W. Marsden.  I and J , 2012 .

[20]  Erik Jan van Leeuwen,et al.  Finding Disjoint Paths in Split Graphs , 2014, Theory of Computing Systems.

[21]  Derek G. Corneil,et al.  Complement reducible graphs , 1981, Discret. Appl. Math..

[22]  Jan Arne Telle,et al.  Connecting Terminals and 2-Disjoint Connected Subgraphs , 2013, WG.

[23]  Alan P. Sprague,et al.  Disjoint Paths in Circular Arc Graphs , 1996, Nord. J. Comput..

[24]  Alon Itai,et al.  On the Complexity of Timetable and Multicommodity Flow Problems , 1976, SIAM J. Comput..

[25]  van Leeuwen,et al.  Optimization and approximation on systems of geometric objects , 2009 .

[26]  Daniël Paulusma,et al.  On partitioning a graph into two connected subgraphs , 2011, Theor. Comput. Sci..

[27]  Yossi Shiloach,et al.  A Polynomial Solution to the Undirected Two Paths Problem , 1980, JACM.

[28]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

[29]  Stephan Olariu,et al.  Paw-Fee Graphs , 1988, Inf. Process. Lett..

[30]  Maarten Löffler,et al.  Removing local extrema from imprecise terrains , 2010, Comput. Geom..

[31]  M. Held,et al.  A dynamic programming approach to sequencing problems , 1962, ACM National Meeting.