List Homomorphism: Beyond the Known Boundaries

Abstract

Given two graphs G and H, and a list $L(u)\subseteq V(H)$ associated with each $u\in V(G)$, a list homomorphism from G to H is a mapping $f:V(G)\rightarrow V(H)$ such that (i) for all $u\in V(G)$, $f(u) \in L(u)$, and (ii) for all $u,v\in V(G)$, if $uv\in E(G)$ then $f(u)f(v)\in E(H)$. 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 $O(n^{k^2-3k+4})$ 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 $n^{o(k)}$, unless the Exponential Time Hypothesis (ETH) fails. We complement this result with a matching upper bound and another positive result.

• 1. There is a $O(n^{8k+3})$ 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.