# 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 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(√(n) 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^2/3^3/2 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.

## Authors

• 11 publications
• 29 publications
• ### Almost all string graphs are intersection graphs of plane convex sets

A string graph is the intersection graph of a family of continuous arcs...
03/18/2018 ∙ by János Pach, et al. ∙ 0

• ### Complexity of the list homomorphism problem in hereditary graph classes

A homomorphism from a graph G to a graph H is an edge-preserving mapping...
10/07/2020 ∙ by Karolina Okrasa, et al. ∙ 0

• ### Optimality Program in Segment and String Graphs

Planar graphs are known to allow subexponential algorithms running in ti...
12/24/2017 ∙ by Édouard Bonnet, et al. ∙ 0

• ### Fine-grained complexity of graph homomorphism problem for bounded-treewidth graphs

For graphs G and H, a homomorphism from G to H is an edge-preserving map...
06/19/2019 ∙ by Karolina Okrasa, et al. ∙ 0

• ### Computing Maximum Independent Set on Outerstring Graphs and Their Relatives

A graph G with n vertices is called an outerstring graph if it has an in...
03/17/2019 ∙ by Prosenjit Bose, et al. ∙ 0

• ### On grounded L-graphs and their relatives

We consider the graph class Grounded-L corresponding to graphs that admi...
08/13/2018 ∙ by Vít Jelínek, et al. ∙ 0