Recognizing Graphs Close to Bipartite Graphs

Authors Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, Daniël Paulusma

Thumbnail PDF


  • Filesize: 0.49 MB
  • 14 pages

Document Identifiers

Author Details

Marthe Bonamy
Konrad K. Dabrowski
Carl Feghali
Matthew Johnson
Daniël Paulusma

Cite AsGet BibTex

Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, and Daniël Paulusma. Recognizing Graphs Close to Bipartite Graphs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree.
  • degenerate graphs
  • near-bipartite graphs
  • reconfiguration graphs


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Bradley Baetz and David R. Wood. Brooks' vertex-colouring theorem in linear time. CoRR, abs/1401.8023, 2014. Google Scholar
  2. Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, and Daniël Paulusma. Independent feedback vertex set for P₅-free graphs. Manuscript, 2017. Google Scholar
  3. Paul S. Bonsma and Luis Cereceda. Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theoretical Computer Science, 410(50):5215-5226, 2009. Google Scholar
  4. Andreas Brandstädt, Synara Brito, Sulamita Klein, Loana Tito Nogueira, and Fábio Protti. Cycle transversals in perfect graphs and cographs. Theoretical Computer Science, 469:15-23, 2013. Google Scholar
  5. Andreas Brandstädt, Peter L. Hammer, Van Bang Le, and Vadim V. Lozin. Bisplit graphs. Discrete Mathematics, 299(1-3):11-32, 2005. Google Scholar
  6. Andreas Brandstädt, Van Bang Le, and Thomas Szymczak. The complexity of some problems related to graph 3-colorability. Discrete Applied Mathematics, 89(1-3):59-73, 1998. Google Scholar
  7. Leizhen Cai and Derek G. Corneil. A generalization of perfect graphs - i-perfect graphs. Journal of Graph Theory, 23(1):87-103, 1996. Google Scholar
  8. Paul A. Catlin. Brooks' graph-coloring theorem and the independence number. Journal of Combinatorial Theory, Series B, 27(1):42-48, 1979. Google Scholar
  9. Paul A. Catlin and Hong-Jian Lai. Vertex arboricity and maximum degree. Discrete Mathematics, 141(1-3):37-46, 1995. Google Scholar
  10. Luis Cereceda. Mixing graph colourings. PhD thesis, London School of Economics, 2007. Google Scholar
  11. Konrad K. Dabrowski, Vadim V. Lozin, and Juraj Stacho. Stable-Π partitions of graphs. Discrete Applied Mathematics, 182:104-114, 2015. Google Scholar
  12. François Dross, Mickaël Montassier, and Alexandre Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree. CoRR, abs/1606.04394, 2016. Google Scholar
  13. Tomás Feder, Pavol Hell, Sulamita Klein, and Rajeev Motwani. List partitions. SIAM Journal on Discrete Mathematics, 16(3):449-478, 2003. Google Scholar
  14. Carl Feghali, Matthew Johnson, and Daniël Paulusma. A reconfigurations analogue of Brooks' Theorem and its consequences. Journal of Graph Theory, 83(4):340-358, 2016. Google Scholar
  15. Michael Randolph Garey, David S. Johnson, and Larry J. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1(3):237-267, 1976. Google Scholar
  16. Martin Grötschel, László Lovász, and Alexander Schrijver. Polynomial algorithms for perfect graphs. Annals of Discrete Mathematics, 21:325-356, 1984. Google Scholar
  17. Chính T. Hoàng and Van Bang Le. On P₄-transversals of perfect graphs. Discrete Mathematics, 216(1-3):195-210, 2000. Google Scholar
  18. Matthew Johnson, Dieter Kratsch, Stefan Kratsch, Viresh Patel, and Daniël Paulusma. Finding shortest paths between graph colourings. Algorithmica, 75(2):295-321, 2016. Google Scholar
  19. Jan Kratochvíl and Ingo Schiermeyer. On the computational complexity of (𝒪,𝒫)-partition problems. Discussiones Mathematicae Graph Theory, 17(2):253-258, 1997. Google Scholar
  20. László Lovász. Coverings and coloring of hypergraphs. Congressus Numerantium, VIII:3-12, 1973. Google Scholar
  21. Vadim V. Lozin. Between 2- and 3-colorability. Information Processing Letters, 94(4):179-182, 2005. Google Scholar
  22. Nadimpalli V. R. Mahadev and Uri N. Peled. Threshold graphs and related topics, volume 56 of Annals of Discrete Mathematics. North-Holland, Amsterdam, 1995. Google Scholar
  23. Martín Matamala. Vertex partitions and maximum degenerate subgraphs. Journal of Graph Theory, 55(3):227-232, 2007. Google Scholar
  24. Colin McDiarmid and Nikola Yolov. Recognition of unipolar and generalised split graphs. Algorithms, 8(1):46-59, 2015. Google Scholar
  25. George B. Mertzios and Paul G. Spirakis. Algorithms and almost tight results for 3-colorability of small diameter graphs. Algorithmica, 74(1):385-414, 2016. Google Scholar
  26. Neeldhara Misra, Geevarghese Philip, Venkatesh Raman, and Saket Saurabh. On parameterized independent feedback vertex set. Theoretical Computer Science, 461:65-75, 2012. Google Scholar
  27. Yuma Tamura, Takehiro Ito, and Xiao Zhou. Algorithms for the independent feedback vertex set problem. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E98-A(6):1179-1188, 2015. Google Scholar
  28. Yongqi Wu, Jinjiang Yuan, and Yongcheng Zhao. Partition a graph into two induced forests. Journal of Mathematical Study, 29:1-6, 1996. Google Scholar
  29. Aifeng Yang and Jinjiang Yuan. Partition the vertices of a graph into one independent set and one acyclic set. Discrete Mathematics, 306(12):1207-1216, 2006. Google Scholar