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 $$L(v)\subseteq V(H)$$L(v)⊆V(H) for each vertex $$v\in V(G)$$v∈V(G), and an integer k. The task is to decide whether there exists a set $$W \subseteq V(G)$$W⊆V(G) of size at most k such that there is a homomorphism from $$G {\setminus } W$$G\W to H respecting the lists. We show that DL-Hom($${H}$$H), parameterized by k and |H|, is fixed-parameter tractable for any $$(P_6,C_6)$$(P6,C6)-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($${H}$$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.