# The complexity of counting graph homomorphisms

@article{Dyer2000TheCO, title={The complexity of counting graph homomorphisms}, author={M. Dyer and Catherine S. Greenhill}, journal={Random Struct. Algorithms}, year={2000}, volume={17}, pages={260-289} }

The problem of counting homomorphisms from a general graph G to a fixed graph H is a natural generalisation of graph colouring, with important applications in statistical physics. The problem of deciding whether any homomorphism exists was considered by Hell and Nešetřil. They showed that decision is NPcomplete unless H has a loop or is bipartite; otherwise it is in P. We consider the problem of exactly counting such homomorphisms, and give a similarly complete characterisation. We show that… Expand

#### 178 Citations

The complexity of counting graph homomorphisms (extended abstract)

- Computer Science, Mathematics
- SODA '00
- 2000

The theorems provide the first proof of #P-completeness of the partition function of certain models from statistical physics, such as the Widom-Rowlinson model, even in graphs of maximum degree 3. Expand

Counting, modular counting and graph homomorphisms

- Mathematics, Computer Science
- 2016

This thesis studies the complexity of various problems related to graph homomorphisms, and gives an explicit characterisation of the dichotomy theorem — counting list M -partitions is tractable (in FP) if the matrix M has a structure called a derectangularising sequence. Expand

Counting Homomorphisms Modulo a Prime Number

- Mathematics, Computer Science
- MFCS
- 2019

It is extended to show that the #_p GraphHom(H) problem is#_p P-hard whenever the derived graph associated with H is square-free and is not a star, which completely classifies the complexity of #_ p GraphHom (H) forsquare-free graphs H. Expand

The complexity of counting homomorphisms to cactus graphs modulo 2

- Computer Science, Mathematics
- TOCT
- 2014

It is shown that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time and a dichotomy is given for the case in which H is a tree, which builds upon the work of Faben and Jerrum. Expand

Counting Homomorphisms to Cactus Graphs Modulo 2

- Mathematics, Computer Science
- STACS
- 2014

It is shown that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time and the result builds upon the work of Faben and Jerrum, who gave a dichotomy for the case in which H is a tree. Expand

Counting Homomorphisms to Cactus Graphs Modulo 2ú

- 2016

Abstract A homomorphism from a graph G to a graph H is a function from V (G) to V (H) that preserves edges. Many combinatorial structures that arise in mathematics and computer science can be… Expand

Counting Homomorphisms to Square-Free Graphs, Modulo 2

- Computer Science, Mathematics
- TOCT
- 2016

The following dichotomy is shown: for any H that contains no 4-cycles, ⊕HomsToH is either in polynomial time or is⊕P-complete, which partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of tree-width-2 graphs called cactus graphs. Expand

THE COMPLEXITY OF COUNTING SURJECTIVE

- 2019

4 A homomorphism from a graph G to a graph H is a function from the vertices of G to the 5 vertices of H that preserves edges. A homomorphism is surjective if it uses all of the vertices 6 of H and… Expand

The Complexity of Counting Surjective Homomorphisms and Compactions

- Mathematics, Computer Science
- SIAM J. Discret. Math.
- 2019

A complete characterisation of the complexity of counting surjective homomorphisms from an input graph G to a fixed graph H is given and a dichotomy is pointed out for the complexityof the respective approximate counting problems (in the connected case). Expand

Counting Partitions of Graphs

- Mathematics, Computer Science
- ISAAC
- 2012

It turns out that, among matrices not acccounted for by the existing results on counting homomorphisms, all matrices which do not contain the matrices for independent sets or cliques yield tractable counting problems. Expand

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