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The complexity of counting homomorphisms to cactus graphs modulo 2

Andreas Göbel, Leslie Ann Goldberg, David Richerby
2014 ACM Transactions on Computation Theory  
In this paper, we study the complexity of counting homomorphisms modulo 2.  ...  We show that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time.  ...  The complexity of counting graph homomorphisms modulo 2 The first results on the complexity of counting graph homomorphisms modulo 2 were obtained by Faben and Jerrum [9, 10] who made some important  ... 
doi:10.1145/2635825 fatcat:2p355za4hfeaxklgat57527qlm

Counting Homomorphisms to K4-minor-free Graphs, modulo 2 [chapter]

Jacob Focke, Leslie Ann Goldberg, Marc Roth, Stanislav Živný
2021 Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)  
As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most 3, as well as a full classification for the problem of counting list homomorphisms, modulo 2  ...  We study the problem of computing the parity of the number of homomorphisms from an input graph G to a fixed graph H.  ...  Furthermore we thank Holger Dell for pointing out the tight conditional lower bound for counting independent sets modulo 2 in [5] .  ... 
doi:10.1137/1.9781611976465.137 fatcat:mdxd2toiofd4pmgy6dhibm7jwq

Counting Homomorphisms to Trees Modulo a Prime

Andreas Göbel, J. A. Gregor Lagodzinski, Karen Seidel, Michael Wagner
2018 International Symposium on Mathematical Foundations of Computer Science  
In this article we study the complexity of # p HomsToH, the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p.  ...  This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph H when counting modulo 2.  ...  Acknowledgements The first author would like to thank Leslie Ann Goldberg and David Richerby for fruitful discussions during the early stages of this work.  ... 
doi:10.4230/lipics.mfcs.2018.49 dblp:conf/mfcs/0001L018 fatcat:p2vafh2ianasbghm2led4rsc4m

Counting Homomorphisms to Square-Free Graphs, Modulo 2 [article]

Andreas Göbel, Leslie Ann Goldberg, David Richerby
2015 arXiv   pre-print
We study the problem HomsToH of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph H.  ...  This confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of treewidth-2 graphs called cactus graphs.  ...  Counting modulo 2 Although counting modulo 2 produces a one-bit answer, the complexity of such problems has a rather different flavour from the complexity of decision problems.  ... 
arXiv:1501.07539v4 fatcat:nredn4kh4zdqhemqx7povzau4a

The complexity of parity graph homomorphism: an initial investigation [article]

John Faben, Mark Jerrum
2013 arXiv   pre-print
Given a graph G, we investigate the question of determining the parity of the number of homomorphisms from G to some other fixed graph H.  ...  , and the problem of counting homomorphisms for other moduli.  ...  As mentioned above, the complexity of exactly counting the number of homomorphisms to a given graph H was characterised by Dyer and Greenhill. They proved the following theorem.  ... 
arXiv:1309.4033v1 fatcat:wturhivjbrd2lexdtfmumefqzy

Counting Homomorphisms Modulo a Prime Number

Amirhossein Kazeminia, Andrei A. Bulatov, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science  
One of the most well studied problems in this area is #GraphHom(H) -the problem of finding the number of homomorphisms from a given graph G to the graph H.  ...  Counting problems in general and counting graph homomorphisms in particular have numerous applications in combinatorics, computer science, statistical physics, and elsewhere.  ...  It is the problem of counting homomorphisms from a given partially H-labelled graph G to a fixed graph H modulo prime p.  ... 
doi:10.4230/lipics.mfcs.2019.59 dblp:conf/mfcs/KazeminiaB19 fatcat:ei2sfugtdvco7iwb3sclfsvgna

Computational Counting (Dagstuhl Seminar 18341)

Ivona Bezáková, Leslie Ann Goldberg, Mark R. Jerrum, Marc Herbstritt
2018 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 17341 "Computational Counting".  ...  A total of 43 researchers from all over the world, with interests and expertise in different aspects of computational counting, actively participated in the meeting.  ...  Modular counting provides a rich setting in which to study the structure of homomorphism problems. In this talk we will discuss the complexity of counting graph homomorphisms modulo 2.  ... 
doi:10.4230/dagrep.7.8.23 dblp:journals/dagstuhl-reports/BezakovaGJ17 fatcat:yp3oqvgo4fal5lbio5yeqt4uje

On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order [article]

J. A. Gregor Lagodzinski, Andreas Göbel, Katrin Casel, Tobias Friedrich
2022 arXiv   pre-print
First, we introduce the study of quantum graphs to the study of modular counting homomorphisms.  ...  We study the problem of counting the number of homomorphisms from an input graph G to a fixed (quantum) graph H̅ in any finite field of prime order ℤ_p.  ...  We are going to apply the dot product for the construction of the partially H-labelled bip-graphs, where the properties of the dot product are given by Corollary 3.22.  ... 
arXiv:2011.04827v3 fatcat:uoac3dmpkrdvtoeknpb66dpqiy

Complexity classification of counting graph homomorphisms modulo a prime number [article]

Andrei A.Bulatov, Amirhossein Kazeminia
2021 arXiv   pre-print
While the complexity of exact counting of graph homomorphisms (Dyer and Greenhill, 2000) and the counting CSP (Bulatov, 2013, and Dyer and Richerby, 2013) is well understood, counting modulo some natural  ...  As a part of this investigation we develop techniques that widen the spectrum of reductions available for modular counting and apply to the general CSP rather than being limited to graph homomorphisms.  ...  Automorphism group For a relational structure H, an automorphism is an injective homomorphism into itself. The automorphisms of H form a group with respect to composition denoted Aut(H).  ... 
arXiv:2106.04086v1 fatcat:2idkouvvpzgtxmde5ea42qzb7q

Counting Homomorphisms to K_4-minor-free Graphs, modulo 2 [article]

Jacob Focke, Leslie Ann Goldberg, Marc Roth, Stanislav Živný
2021 arXiv   pre-print
As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most 3, as well as a full classification for the problem of counting list homomorphisms, modulo 2  ...  We study the problem of computing the parity of the number of homomorphisms from an input graph G to a fixed graph H.  ...  Furthermore we thank Holger Dell for pointing out the tight conditional lower bound for counting independent sets modulo 2 in [5] .  ... 
arXiv:2006.16632v4 fatcat:sbdgfnihbbfg7k5brm25ijapru

Large induced subgraphs via triangulations and CMSO [article]

Fedor Fomin, Ioan Todinca, Yngve Villanger
2013 arXiv   pre-print
task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo m. 2) "Minimum Γ-deletion", where for a fixed finite set of graphs Γcontaining a planar  ...  For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure  ...  Thilikos for fruitful discussions and useful suggestions on the topic of the paper.  ... 
arXiv:1309.1559v1 fatcat:f6ljhfu3dbdtdf32yr7ljjf4pe

Large Induced Subgraphs via Triangulations and CMSO

Fedor V. Fomin, Ioan Todinca, Yngve Villanger
2015 SIAM journal on computing (Print)  
For a given graph G = (V, E), the task is to maximize |X| subject to the following: there is a set F ⊆ V such that X ⊆ F , the subgraph G[F ] induced by F is of treewidth at most t, and structure (G[F  ...  Special cases of this optimization problem are the following generic examples.  ...  Thilikos for fruitful discussions and useful suggestions on the topic of the paper.  ... 
doi:10.1137/140964801 fatcat:zmhblcqudrb5ffkuwmtybcdfye

Electrical networks and the Grove algebra [article]

Yibo Gao, Thomas Lam, Zixuan Xu
2022 arXiv   pre-print
We develop the combinatorics of double groves to study the grove algebra, and find a quadratic Gr\"obner basis for the grove ideal.  ...  We study the ring of regular functions on the space of planar electrical networks, which we coin the grove algebra.  ...  We call the underlying unweighted graph of a cactus network a cactus graph. Groves. A grove F on a cactus network Γ is a spanning forest, such that each component is connected to the boundary.  ... 
arXiv:2208.12798v1 fatcat:miqpt6m3efdprcwcri7lu4mlzq

Orbits of Braid Groups on Cacti

G. Jones, A. Zvonkin
2002 Moscow Mathematical Journal  
Motivated by the problem of the topological classification of polynomials, a problem that goes back to 19th century researchers, we discuss several techniques for investigating orbits of braid groups on  ...  By exceptional here we mean primitive and not equal to Sn or An, where n is the degree.  ...  J. is grateful to LaBRI (Université Bordeaux I) for its hospitality. A. Z. is grateful to Nikolai Adrianov, Askold Khovanskii and Smilka Zdravkovska for a number of extremely valuable discussions.  ... 
doi:10.17323/1609-4514-2002-2-1-127-160 fatcat:z7onpbxhmnad3cnvhmsw7kbtw4

Counting Homomorphisms to Cactus Graphs Modulo 2 *

Andreas Göbel, Ann Leslie, David Goldberg, Richerby
unpublished
In this paper, we study the complexity of counting homomorphisms modulo 2.  ...  We show that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time.  ...  Since H 1 is an induced subgraph of H, it is a cactus graph. By Theorems 10 and 13, ⊕HomsToH 1 is ⊕P-hard. By Lemma 2, ⊕HomsToH is ⊕P-hard.  ... 
fatcat:arevgojgsbddfpgjd4o73bf4sq
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