Given two graphs G and H, and a list L(u) ⊆ V(H) associated with each u∈ V(G), a list homomorphism from G to H is a mapping f: V(G) → V(H) such that (i) for all u∈ V(G), f(u) ∈ L(u), and (ii) for all u, v∈ V(G), if uv∈ E(G) then f(u) f(v) ∈ E(H). The List Homomorphism problem asks whether there exists a list homomorphism from G to H. Enright, Stewart and Tardos [SIAM J. Discret. Math., 2014] showed that the List Homomorphism problem can be solved in O(nk2-3k+4) time on graphs where every connected induced subgraph of G admits “a multichain ordering” (see the introduction for the definition of multichain ordering of a graph), that includes permutation graphs, biconvex graphs, and interval graphs, where n= | V(G) | and k= | V(H) |. We prove that List Homomorphism parameterized by k even when G is a bipartite permutation graph is W[1]-hard. In fact, our reduction implies that it is not solvable in time no(k), unless the Exponential Time Hypothesis (ETH) fails. We complement this result with a matching upper bound and another positive result. 1.There is a O(n8k+3) time algorithm for List Homomorphism on bipartite graphs that admit a multichain ordering that includes the class of bipartite permutation graphs and biconvex graphs.2.For bipartite graph G that admits a multichain ordering, List Homomorphism is fixed parameter tractable when parameterized by k and the number of layers in the multichain ordering of G. In addition, we study a variant of List Homomorphism called List Locally Surjective Homomorphism. We prove that List Locally Surjective Homomorphism parameterized by the number of vertices in H is W[1]-hard, even when G is a chordal graph and H is a split graph. © 2022, Springer Nature Switzerland AG.