, Volume 78, Issue 1, pp 110–146 | Cite as

List H-Coloring a Graph by Removing Few Vertices



In the deletion version of the list homomorphism problem, we are given graphs G and H, a list \(L(v)\subseteq V(H)\) for each vertex \(v\in V(G)\), and an integer k. The task is to decide whether there exists a set \(W \subseteq V(G)\) of size at most k such that there is a homomorphism from \(G {\setminus } W\) to H respecting the lists. We show that DL-Hom(\({H}\)), parameterized by k and |H|, is fixed-parameter tractable for any \((P_6,C_6)\)-free bipartite graph H; already for this restricted class of graphs, the problem generalizes Vertex Cover, Odd Cycle Transversal, and Vertex Multiway Cut parameterized by the size of the cutset and the number of terminals. We conjecture that DL-Hom(\({H}\)) is fixed-parameter tractable for the class of graphs H for which the list homomorphism problem (without deletions) is polynomial-time solvable; by a result of Feder et al. (Combinatorica 19(4):487–505, 1999), a graph H belongs to this class precisely if it is a bipartite graph whose complement is a circular arc graph. We show that this conjecture is equivalent to the fixed-parameter tractability of a single fairly natural satisfiability problem, Clause Deletion Chain-SAT.


Fixed-parameter tractability Graph homomorphism Treewidth reduction Iterative compression Shadow removal 



We thank the anonymous referees for their comments that helped improve the presentation, and for pointing out a problem that required some technical work to fix.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Weizmann Institute of ScienceRehovotIsrael
  2. 2.Simon Fraser UniversityBurnabyCanada
  3. 3.Institute for Computer Science and ControlHungarian Academy of Sciences (MTA SZTAKI)BudapestHungary

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