Subexponential algorithms for variants of homomorphism problem in string graphs

We consider the complexity of finding weighted homomorphisms from intersection graphs of curves (string graphs) with $n$ vertices to a fixed graph $H$. We provide a complete dichotomy for the problem: if $H$ has no two vertices sharing two common neighbors, then the problem can be solved in time $2^{O(n^{2/3} \log n)}$, otherwise there is no algorithm working in time $2^{o(n)}$, even in intersection graphs of segments, unless the ETH fails. This generalizes several known results concerning the complexity of computatational problems in geometric intersection graphs. Then we consider two variants of graph homomorphism problem, called locally injective homomorphism and locally bijective homomorphism, where we require the homomorphism to be injective or bijective on the neighborhood of each vertex. We show that for each target graph $H$, both problems can always be solved in time $2^{O(\sqrt{n} \log n)}$ in string graphs. For the locally surjecive homomorphism, defined in an analogous way, the situation seems more complicated. We show the dichotomy theorem for simple connected graphs $H$ with maximum degree 2. If $H$ is isomorphic to $P_3$ or $C_4$, then the existence of a locally surjective homomorphism from a string graph with $n$ vertices to $H$ can be decided in time $2^{O(n^{2/3} \log^{3/2} n)}$, otherwise the problem cannot be solved in time $2^{o(n)}$, unless the ETH fails. As a byproduct, we obtain several results concerning the complexity of variants of homomorphism problem in $P_t$-free graphs. In particular, we obtain the dichotomy theorem for weighted homomorphism, analogous to the one for string graphs.