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

2018
*
arXiv
*
pre-print

For hereditary

arXiv:1702.05447v2
fatcat:sy24us5odjdp5awy4ex4lyjut4
*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. ...##
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Counting Edge-injective Homomorphisms and Matchings on Restricted Graph Classes

2018
*
Theory of Computing Systems
*

We show that #W[1]-hardness persists even when the input

doi:10.1007/s00224-018-9893-y
fatcat:u2ymwrvajzdehcucdayfhlp6km
*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. ...##
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Counting Restricted Homomorphisms via Möbius Inversion over Matroid Lattices
[article]

2017
*
arXiv
*
pre-print

We use the general theorem to classify the complexity of

arXiv:1706.08414v1
fatcat:54q2ljvnp5ewboak3d67ij73oq
*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. ...##
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Parameterized Counting of Partially Injective Homomorphisms

2021
*
Algorithmica
*

of

doi:10.1007/s00453-021-00805-y
fatcat:oetq4yn2nnhy7ovyr47zgch6am
*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. ...##
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Universality of intervals of line graph order
[article]

2014
*
arXiv
*
pre-print

We prove that for every d≥ 3 the

arXiv:1402.3736v2
fatcat:zdmbkwopnbbs3fxwtu5wreknwq
*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. ...##
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Color Refinement, Homomorphisms, and Hypergraphs
[article]

2019
*
arXiv
*
pre-print

To this end, we show how

arXiv:1903.12432v1
fatcat:gwzzninerzcennxrojtliutcuq
*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*. ...##
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Homomorphisms are a good basis for counting small subgraphs

2017
*
Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017
*

Using the framework of

doi:10.1145/3055399.3055502
dblp:conf/stoc/CurticapeanDM17
fatcat:elq7lwyoyffbzbdiwlxycit2jq
*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] . ...##
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Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory
[article]

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. ...

##
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Locally constrained graph homomorphisms—structure, complexity, and applications

2008
*
Computer Science Review
*

A

doi:10.1016/j.cosrev.2008.06.001
fatcat:p57dezmpfjhzpf5xahy4f7rtga
*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. ...##
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Bounded list injective homomorphism for comparative analysis of protein–protein interaction graphs

2008
*
Journal of Discrete Algorithms
*

Motivated by the need for more accurate models, we conclude by giving

doi:10.1016/j.jda.2007.06.002
fatcat:4mtc4rjirbdvzppmddts76l3a4
*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*). ...##
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Counting Answers to Existential Questions
[article]

2019
*
arXiv
*
pre-print

Our proof also relies

arXiv:1902.04960v2
fatcat:sm53qhw6dzanleeutgyztfxqdu
*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*...##
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Circulant covers of trivalent circulants

2007
*
Discrete Mathematics
*

Given two

doi:10.1016/j.disc.2006.08.003
fatcat:iih5ynqpxvet3kdvl2akhrd7aa
*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. ...##
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Near-Linear Time Homomorphism Counting in Bounded Degeneracy Graphs: The Barrier of Long Induced Cycles
[article]

2020
*
arXiv
*
pre-print

We focus

arXiv:2010.08083v2
fatcat:oryutbr3ljc3hnyebu5mmjgtha
*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. ...##
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Packing bipartite graphs with covers of complete bipartite graphs

2014
*
Discrete Applied Mathematics
*

Let K k, be the complete bipartite

doi:10.1016/j.dam.2012.08.026
fatcat:6ku6d6cqu5gj5cpbq67lie4vj4
*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) → ...##
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Packing Bipartite Graphs with Covers of Complete Bipartite Graphs
[chapter]

2010
*
Lecture Notes in Computer Science
*

Let K k, be the complete bipartite

doi:10.1007/978-3-642-13073-1_25
fatcat:dkf43zipxzevrhggl4wtznv6ke
*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) → ...
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