Abstract
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.
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Notes
The notation \(x_1\rightarrow x_2\rightarrow \dots \rightarrow x_\ell \) is a shorthand for \((x_1 \rightarrow x_2) \wedge (x_2 \rightarrow x_3) \wedge \cdots \wedge (x_{\ell -1} \rightarrow x_\ell )\).
For background on MSO, we refer the reader to, e.g., the textbook of Flum and Grohe [15]; very briefly, MSO is a logical language that allows quantification over the elements and subsets of the universe of a relational structure.
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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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Supported by ERC Starting Grant PARAMTIGHT (No. 280152) and OTKA Grant NK105645.
R. Chitnis: Most of this work was done when the author was a student at University of Maryland, College Park. Supported in part by NSF CAREER award 1053605, NSF Grant CCF-1161626, ONR YIP award N000141110662, DARPA/AFOSR Grant FA9550-12-1-0423, a University of Maryland Research and Scholarship Award (RASA) and a Summer International Research Fellowship from University of Maryland.
L. Egri: Supported by NSERC and FQRNT.
A preliminary version of the paper appeared in the proceedings of ESA 2013.
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Chitnis, R., Egri, L. & Marx, D. List H-Coloring a Graph by Removing Few Vertices. Algorithmica 78, 110–146 (2017). https://doi.org/10.1007/s00453-016-0139-6
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DOI: https://doi.org/10.1007/s00453-016-0139-6