# Counting list homomorphisms from graphs of bounded treewidth: tight complexity bounds

## Abstract

*G, H*, and lists

*L*(

*v*) ⊆

*V*(

*H*) for every

*v*∊

*V*(

*G*), a

*list homomorphism*is a function

*f*:

*V(G*) →

*V(H*) that preserves the edges (i.e.,

*uv*∊

*E(G*) implies

*f(u)f(v*) ∊

*E(H*)) and respects the lists (i.e.,

*f(v*) ∊

*L(v*)). Standard techniques show that if

*G*is given with a tree decomposition of width

*t*, then the number of list homomorphisms can be counted in time . Our main result is determining, for every fixed graph

*H*, how much the base

*|V*(

*H*)

*|*in the running time can be improved. For a connected graph

*H*we define irr(

*H*) in the following way: if

*H*has a loop or is nonbipartite, then irr(

*H*) is the maximum size of a set

*S*⊆

*V*(

*H*) where any two vertices have different neighborhoods; if

*H*is bipartite, then irr(

*H*) is the maximum size of such a set that is fully in one of the bipartition classes. For disconnected

*H*, we define irr(

*H*) as the maximum of irr(

*C*) over every connected component

*C*of

*H*. It follows from earlier results that if irr(

*H*) = 1, then the problem of counting list homomorphisms to

*H*is polynomial-time solvable, and otherwise it is #P-hard. We show that, for every fixed graph

*H*, the number of list homomorphisms from (

*G, L*) to

*H*

*G*having width at most

*t*is given in the input, and

*H*) ≥ 2, cannot be counted in time for any

*∊*> 0, even if a tree decomposition of

*G*having width at most

*t*is given in the input, unless the Counting Strong Exponential-Time Hypothesis (#SETH) fails.

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**ISBN (Online)**: 978-1-61197-707-3

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#### History

**Published online**: 5 January 2022

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