Connected Vertex Cover on AT-Free Graphs

Authors Joydeep Mukherjee, Tamojit Saha

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Joydeep Mukherjee
  • Ramakrishna Mission Vivekananda Educational and Research Institute, Belur, India
Tamojit Saha
  • Ramakrishna Mission Vivekananda Educational and Research Institute, Belur, India
  • Institute of Advancing Intelligence, TCG CREST, Kolkata, India

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Joydeep Mukherjee and Tamojit Saha. Connected Vertex Cover on AT-Free Graphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 54:1-54:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Asteroidal Triple (AT) in a graph is an independent set of three vertices such that every pair of them has a path between them avoiding the neighbourhood of the third. A graph is called AT-free if it does not contain any asteroidal triple. A connected vertex cover of a graph is a subset of its vertices which contains at least one endpoint of each edge and induces a connected subgraph. Settling the complexity of computing a minimum connected vertex cover in an AT-free graph was mentioned as an open problem in Escoffier et al. [Escoffier et al., 2010]. In this paper we answer the question by presenting an exact polynomial time algorithm for computing a minimum connected vertex cover problem on AT-free graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Graph Algorithm
  • AT-free graphs
  • Connected Vertex Cover
  • Optimization


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  1. Esther M Arkin, Magnús M Halldórsson, and Rafael Hassin. Approximating the tree and tour covers of a graph. Information Processing Letters, 47(6):275-282, 1993. Google Scholar
  2. Hari Balakrishnan, Anand Rajaraman, and C Pandu Rangan. Connected domination and steiner set on asteroidal triple-free graphs. In Algorithms and Data Structures: Third Workshop, WADS'93 Montréal, Canada, August 11-13, 1993 Proceedings 3, pages 131-141. Springer, 1993. Google Scholar
  3. Hajo Broersma, Ton Kloks, Dieter Kratsch, and Haiko Müller. Independent sets in asteroidal triple-free graphs. SIAM Journal on Discrete Mathematics, 12(2):276-287, 1999. Google Scholar
  4. Derek G Corneil, Stephan Olariu, and Lorna Stewart. Computing a dominating pair in an asteroidal triple-free graph in linear time. In Workshop on Algorithms and Data Structures, pages 358-368. Springer, 1995. Google Scholar
  5. Derek G Corneil, Stephan Olariu, and Lorna Stewart. Asteroidal triple-free graphs. SIAM Journal on Discrete Mathematics, 10(3):399-430, 1997. Google Scholar
  6. Bruno Escoffier, Laurent Gourvès, and Jérôme Monnot. Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs. Journal of Discrete Algorithms, 8(1):36-49, 2010. Google Scholar
  7. Henning Fernau and David F Manlove. Vertex and edge covers with clustering properties: Complexity and algorithms. Journal of Discrete Algorithms, 7(2):149-167, 2009. Google Scholar
  8. Michael R Garey and David S. Johnson. The rectilinear steiner tree problem is np-complete. SIAM Journal on Applied Mathematics, 32(4):826-834, 1977. Google Scholar
  9. Petr A Golovach, Daniël Paulusma, and Erik Jan van Leeuwen. Induced disjoint paths in at-free graphs. In Algorithm Theory-SWAT 2012: 13th Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 2012. Proceedings 13, pages 153-164. Springer, 2012. Google Scholar
  10. Jiong Guo, Rolf Niedermeier, and Sebastian Wernicke. Parameterized complexity of generalized vertex cover problems. In WADS, pages 36-48. Springer, 2005. Google Scholar
  11. Matthew Johnson, Giacomo Paesani, and Daniël Paulusma. Connected vertex cover for (sp₁+p₅)-free graphs. In Algorithmica82, pages 20-40, 2020. Google Scholar
  12. Dieter Kratsch. Domination and total domination on asteroidal triple-free graphs. Discrete Applied Mathematics, 99(1-3):111-123, 2000. Google Scholar
  13. Dieter Kratsch, Haiko Müller, and Ioan Todinca. Feedback vertex set on at-free graphs. Discrete Applied Mathematics, 156(10):1936-1947, 2008. Google Scholar
  14. Daniel Mölle, Stefan Richter, and Peter Rossmanith. Enumerate and expand: New runtime bounds for vertex cover variants. In Computing and Combinatorics: 12th Annual International Conference, COCOON 2006, Taipei, Taiwan, August 15-18, 2006. Proceedings 12, pages 265-273. Springer, 2006. Google Scholar
  15. Hannes Moser. Exact algorithms for generalizations of vertex cover. Institut für Informatik, Friedrich-Schiller-Universität Jena, 12, 2005. Google Scholar
  16. Andrea Munaro. Boundary classes for graph problems involving non-local properties. Theoretical Computer Science, 692:46-71, 2017. Google Scholar
  17. PK Priyadarsini and T Hemalatha. Connected vertex cover in 2-connected planar graph with maximum degree 4 is np-complete. International Journal of Mathematical, Physical and Engineering Sciences, 2(1):51-54, 2008. Google Scholar
  18. Carla Savage. Depth-first search and the vertex cover problem. Information processing letters, 14(5):233-235, 1982. Google Scholar
  19. Shuichi Ueno, Yoji Kajitani, and Shin'ya Gotoh. On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three. Discrete Mathematics, 72(1-3):355-360, 1988. Google Scholar
  20. Toshimasa Watanabe, Satoshi Kajita, and Kenji Onaga. Vertex covers and connected vertex covers in 3-connected graphs. In 1991 IEEE International Symposium on Circuits and Systems (ISCAS), pages 1017-1020. IEEE, 1991. Google Scholar
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