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Counting edge-injective homomorphisms and matchings on restricted graph classes [article]

Radu Curticapean, Holger Dell, Marc Roth
2018 arXiv   pre-print
For hereditary classes H of pattern graphs, we complement this result: If the graphs in H have unbounded vertex-cover number even after deleting isolated edges, then counting edge-injective homomorphisms  ...  We consider the #W[1]-hard problem of counting all matchings with exactly k edges in a given input graph G; we prove that it remains #W[1]-hard on graphs G that are line graphs or bipartite graphs with  ...  The authors thank Cornelius Brand and Markus Bläser for interesting discussions, and Johannes Schmitt for pointing out a proof of Lemma 20 and allowing us to use it in this paper.  ... 
arXiv:1702.05447v2 fatcat:sy24us5odjdp5awy4ex4lyjut4

Counting Edge-injective Homomorphisms and Matchings on Restricted Graph Classes

Radu Curticapean, Holger Dell, Marc Roth
2018 Theory of Computing Systems  
We show that #W[1]-hardness persists even when the input graph G comes from restricted graph classes, such as line graphs and bipartite graphs of arbitrary constant girth and maximum degree two on one  ...  For hereditary classes H of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edge-injective homomorphisms, restricted to patterns from H, is #W[1]-hard if no such  ...  The authors thank Cornelius Brand and Markus Bläser for interesting discussions, and Johannes Schmitt for pointing out a proof of Lemma 16.  ... 
doi:10.1007/s00224-018-9893-y fatcat:u2ymwrvajzdehcucdayfhlp6km

Counting Restricted Homomorphisms via Möbius Inversion over Matroid Lattices [article]

Marc Roth
2017 arXiv   pre-print
We use the general theorem to classify the complexity of counting locally injective homomorphisms as well as homomorphisms that are injective in the r-neighborhood for constant r.  ...  Furthermore, we show that the former has "real" FPT cases by considering the subgraph counting problem restricted to trees on both sides.  ...  Acknowledgements The author is very grateful to Holger Dell and Radu Curticapean for fruitful discussions.  ... 
arXiv:1706.08414v1 fatcat:54q2ljvnp5ewboak3d67ij73oq

Parameterized Counting of Partially Injective Homomorphisms

Marc Roth
2021 Algorithmica  
of graph homomorphisms, subgraph counting and, more generally, counting of answers to equi-join queries with inequalities.  ...  The abstract classification theorem is then applied to the problem of counting locally injective graph homomorphisms from small pattern graphs to large target graphs.  ...  Furthermore the author thanks Cornelius Brand for saying "Tutte Polynomial" every once in a while, and Philip Wellnitz for providing valuable feedback on early drafts of the full version of this work.  ... 
doi:10.1007/s00453-021-00805-y fatcat:oetq4yn2nnhy7ovyr47zgch6am

Universality of intervals of line graph order [article]

Jiří Fiala, Jan Hubička, Yangjing Long
2014 arXiv   pre-print
We prove that for every d≥ 3 the homomorphism order of the class of line graphs of finite graphs with maximal degree d is universal.  ...  This means that every finite or countably infinite partially ordered set may be represented by line graphs of graphs with maximal degree d ordered by the existence of a homomorphism.  ...  Acknowledgment We would like to thank to Jaroslav Nešetřil and to the anonymous referees for remarks that improved quality of this paper.  ... 
arXiv:1402.3736v2 fatcat:zdmbkwopnbbs3fxwtu5wreknwq

Color Refinement, Homomorphisms, and Hypergraphs [article]

Jan Böker
2019 arXiv   pre-print
To this end, we show how homomorphisms of hypergraphs and of a colored variant of their incidence graphs are related to each other. This reduces the above statement to one about vertex-colored graphs.  ...  Recent results show that the structural similarity of graphs can be characterized by counting homomorphisms to them: the Tree Theorem states that the well-known color-refinement algorithm does not distinguish  ...  Since, to count homomorphisms from a non-connected graph, one can count homomorphisms from its components instead, we usually restrict ourselves to homomorphism counts from connected graphs.  ... 
arXiv:1903.12432v1 fatcat:gwzzninerzcennxrojtliutcuq

Homomorphisms are a good basis for counting small subgraphs

Radu Curticapean, Holger Dell, Dániel Marx
2017 Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017  
Using the framework of graph motif parameters, we obtain faster algorithms for counting subgraph copies of fixed graphs H in host graphs G: For graphs H on k edges, we show how to count subgraph copies  ...  Finally, we extend graph motif parameters to colored subgraphs and prove a complexity trichotomy: For vertex-colored graphs H and G, where H is from a fixed class H, we want to count color-preserving H-copies  ...  Acknowledgments Thanks a lot to Édouard Bonnet for pointing out [52] and [22] .  ... 
doi:10.1145/3055399.3055502 dblp:conf/stoc/CurticapeanDM17 fatcat:elq7lwyoyffbzbdiwlxycit2jq

Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory [article]

Marc Roth, Philip Wellnitz
2021 arXiv   pre-print
counting homomorphisms from a graph in ℋ to a graph in 𝒢.  ...  Counting homomorphisms from a graph H into another graph G is a fundamental problem of (parameterized) counting complexity theory.  ...  Acknowledgements We thank Karl Bringmann and Holger Dell for fruitful discussions and valuable feedback on early drafts of this work.  ... 
arXiv:1907.03850v2 fatcat:rg3upoulmve7jlsnloszbm22zu

Locally constrained graph homomorphisms—structure, complexity, and applications

Jiří Fiala, Jan Kratochvíl
2008 Computer Science Review  
A graph homomorphism is an edge preserving vertex mapping between two graphs.  ...  Our survey provides an overview of applications, complexity results, related problems, and historical notes on locally constrained graph homomorphisms.  ...  Now we map all edges of one matching onto one of the k edges connecting f (u) and f (x) in H.  ... 
doi:10.1016/j.cosrev.2008.06.001 fatcat:p57dezmpfjhzpf5xahy4f7rtga

Bounded list injective homomorphism for comparative analysis of protein–protein interaction graphs

Isabelle Fagnot, Gaëlle Lelandais, Stéphane Vialette
2008 Journal of Discrete Algorithms  
Motivated by the need for more accurate models, we conclude by giving and discussing three natural extensions of the problem. 1 A k-core is a subgraph of the protein-protein interaction graph in which  ...  In the context of comparative analysis of protein-protein interaction graphs, we use a graph-based formalism to detect the preservation of a given protein complex.  ...  We call #(μ G , μ H )-GRAPH MATCHING WITH ORTHOLOGIES the related counting problem (we refer the reader to [17] for a complete treatment of the #P class).  ... 
doi:10.1016/j.jda.2007.06.002 fatcat:4mtc4rjirbdvzppmddts76l3a4

Counting Answers to Existential Questions [article]

Holger Dell, Marc Roth, Philip Wellnitz
2019 arXiv   pre-print
Our proof also relies on graph minors, and we show a strengthening of the Excluded-Grid-Theorem which might be of independent interest: If the linked matching number is large, then not only can we find  ...  Using ideas stemming from Lov\'asz, we lift complexity results from the class of conjunctive queries to arbitrary existential or universal formulas that might contain inequalities and negations on constraints  ...  Acknowledgements We thank Cornelius Brand, Karl Bringmann, Radu Curticapean, Reinhard Diestel, Joshua Erde, Stephan Kreutzer, Stefan Mengel, Daniel Weißauer, and an anonymous reviewer for discussions and  ... 
arXiv:1902.04960v2 fatcat:sm53qhw6dzanleeutgyztfxqdu

Circulant covers of trivalent circulants

Peter Couperus
2007 Discrete Mathematics  
Given two graphs G 1 and G 2 , one may ask whether or not G 2 is a cover of G 1 .  ...  Feng and Kwak [Typical circulant double coverings of a circulant graph, Discrete Math. 277 (2004) 73 -85] provide a description of typical covers of a circulant graph by another circulant graph, and use  ...  As before, for each k odd, we see there is one isomorphism class over graphs G of this type.  ... 
doi:10.1016/j.disc.2006.08.003 fatcat:iih5ynqpxvet3kdvl2akhrd7aa

Near-Linear Time Homomorphism Counting in Bounded Degeneracy Graphs: The Barrier of Long Induced Cycles [article]

Suman K. Bera, Noujan Pashanasangi, C. Seshadhri
2020 arXiv   pre-print
We focus on the case when the input graph has bounded degeneracy, a commonly studied and practically relevant class for homomorphism counting.  ...  It is known from previous work that for certain classes of H, H-homomorphisms can be counted exactly in near-linear time in bounded degeneracy graphs.  ...  Conclusion In this paper, we study the problem of counting homomorphisms of a fixed pattern H in a graph G with bounded degeneracy.  ... 
arXiv:2010.08083v2 fatcat:oryutbr3ljc3hnyebu5mmjgtha

Packing bipartite graphs with covers of complete bipartite graphs

Jérémie Chalopin, Daniël Paulusma
2014 Discrete Applied Mathematics  
Let K k, be the complete bipartite graph with partition classes of size k and , respectively.  ...  A pseudocovering from a graph G to a graph H is a homomorphism from G to H that becomes a covering to H when restricted to a spanning subgraph of G.  ...  First, a homomorphism from a graph G to a graph H is called locally injective or a partial covering if for every u ∈ V G the restriction of f to the neighborhood of u, i.e., the mapping f u : N G (u) →  ... 
doi:10.1016/j.dam.2012.08.026 fatcat:6ku6d6cqu5gj5cpbq67lie4vj4

Packing Bipartite Graphs with Covers of Complete Bipartite Graphs [chapter]

Jérémie Chalopin, Daniël Paulusma
2010 Lecture Notes in Computer Science  
Let K k, be the complete bipartite graph with partition classes of size k and , respectively.  ...  A pseudocovering from a graph G to a graph H is a homomorphism from G to H that becomes a covering to H when restricted to a spanning subgraph of G.  ...  First, a homomorphism from a graph G to a graph H is called locally injective or a partial covering if for every u ∈ V G the restriction of f to the neighborhood of u, i.e., the mapping f u : N G (u) →  ... 
doi:10.1007/978-3-642-13073-1_25 fatcat:dkf43zipxzevrhggl4wtznv6ke
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