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The Complexity of Homomorphisms of Signed Graphs and Signed Constraint Satisfaction
[chapter]

2014
*
Lecture Notes in Computer Science
*

We study the

doi:10.1007/978-3-642-54423-1_46
fatcat:cudazm72vre5pjgoq3c24revui
*complexity**of*deciding whether a given*signed**graph*admits a*homomorphism**to*a fixed target*signed**graph*[H, Σ], i.e. the (H, Σ)-Coloring problem. ... Extending the notion*of*usual*graph**homomorphisms*,*homomorphisms**of**signed**graphs*were introduced, and have lead*to*some extensions and strengthenings in the theory*of**graph*colorings and*homomorphisms*... Only recently, the development*of*the theory*of**homomorphisms**of**signed**graphs*has begun, see [10, 14] . This paper is the first study*of*the*complexity**of**signed**graph**homomorphisms*. ...##
###
Analogues of Cliques for (m, n)-Colored Mixed Graphs

2017
*
Graphs and Combinatorics
*

*complexity*

*of*a decision problem related

*to*(0, 2)-cliques. ... A

*homomorphism*

*of*an (m, n)-colored mixed

*graph*G

*to*an (m, n)-colored mixed

*graph*H is a vertex mapping such that if uv is an arc (edge)

*of*color c in G, then f (u)f (v) is also an arc (edge)

*of*color ... Question 1 . 1 . 11 Given an undirected simple

*graph*, what is the

*complexity*

*of*deciding if it is the underlying

*graph*

*of*a

*signed*clique? ...

##
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Oriented, 2-edge-colored, and 2-vertex-colored homomorphisms

2017
*
Information Processing Letters
*

Finally, we prove that deciding if the 2-edge-colored chromatic number

doi:10.1016/j.ipl.2017.02.009
fatcat:q3qusqx7jjayfb4urvcjjy7c44
*of*a 2-edge-colored*graph*is at most 4 is NP-complete, even if restricted*to*2-connected subcubic bipartite*planar**graphs*with arbitrarily ... Both reductions are rather long and follow the reduction*to*the case*of*oriented*homomorphism*in "*Graphs*and*homomorphisms*" by Hell and Nešetřil. ... If a 2-edge-colored core H contains a monochromatic odd*cycle**of**sign*s, then the*homomorphism**to*H restricted*to*input*graphs*containing only edges*of**sign*s is NP-complete [6] . ...##
###
Analogous to cliques for (m,n)-colored mixed graphs
[article]

2015
*
arXiv
*
pre-print

In this article, we mainly study different aspects

arXiv:1411.7376v2
fatcat:lvryhvxwebfovk6k7de2x6kfyy
*of*"cliques" for*signed*(*graphs*with positive or negative*signs*assigned*to*each edge) and switchable*signed**graphs*(equivalence class*of**signed**graph*... Vertex coloring*of*a*graph*G with n-colors can be equivalently thought*to*be a*graph**homomorphism*(edge preserving vertex mapping)*of*G*to*the complete*graph*K_n*of*order n. ... We will avoid the definitions related*to**homomorphisms**of**signed**graphs*as it will make this article unnesessarily complicated. ...##
###
Homomorphism bounds of signed bipartite K4-minor-free graphs and edge-colorings of 2k-regular K4-minor-free multigraphs

2019
*
Discrete Applied Mathematics
*

This supports a conjecture

doi:10.1016/j.dam.2018.09.004
fatcat:zlnut5qj5faptddhasap5ae4pu
*of*Guenin claiming that every*signed*bipartite*planar**graph**of*unbalanced-girth 2k admits a*homomorphism**to*SPC(2k) (this would be an extension*of*the four-color theorem). ... A*homomorphism**of*(G,Σ)*to*(H,Π) is a*homomorphism**of*G*to*H which preserves the balance*of*closed walks. ... Acknowledgments We thank the referees for carefully reading the original submission and helping*to*improve the presentation. ...##
###
The complexity of signed graph and edge-coloured graph homomorphisms

2017
*
Discrete Mathematics
*

We study two types

doi:10.1016/j.disc.2016.08.005
fatcat:vjiixa4e7vh3podx4w2oaxmzdy
*of**homomorphisms**of*a*signed**graph*(G,Σ)*to*a*signed**graph*(H,Π): ec-*homomorphisms*and s-*homomorphisms*. ... We study*homomorphism*problems*of**signed**graphs*from a computational point*of*view. ... Another line*of*research is*to*study (H, Π)-Colouring for special instance restrictions, such as*signed**graphs*whose underlying*graph*is*planar*or has bounded degree. ...##
###
Counting Problems in Parameterized Complexity

2019
*
International Symposium on Parameterized and Exact Computation
*

*graphs*, and counting perfect matchings and Hamiltonian

*cycles*in well-structured

*graphs*. ... After an introduction

*to*the peculiarities

*of*counting

*complexity*, we survey the parameterized approach

*to*counting problems, with a focus on two topics

*of*recent interest: Counting small patterns in large ... ]

*to*classify the

*complexity*

*of*counting

*homomorphism*variants such as locally injective

*homomorphisms*. ...

##
###
Circular Chromatic Number of Signed Graphs

2021
*
Electronic Journal of Combinatorics
*

Various notions

doi:10.37236/9938
fatcat:5z46dr6kvncsxpsa6y4muo62qe
*of*coloring*of**signed**graphs*have been studied. In this paper, we extend circular coloring*of**graphs**to**signed**graphs*. ... A*signed**graph*is a pair $(G, \sigma)$, where $G$ is a*graph*(loops and multi edges allowed) and $\sigma: E(G) \*to*\{+, -\}$ is a signature which assigns*to*each edge*of*$G$ a*sign*. ... The*complexity**of**signed**graph*and edge-coloured*graph**homomorphisms*. Discrete Math., 340(2) (2017), 315-331. References ...##
###
Concepts of signed graph coloring
[article]

2020
*
arXiv
*
pre-print

This paper surveys recent development

arXiv:1909.09381v2
fatcat:uveiuweurrblxavntkdulqhb2m
*of*concepts related*to*coloring*of**signed**graphs*. Various approaches are presented and discussed. ... One*of*them pointed out that there is a short survey on some aspects*of*coloring*signed**graphs*by Lynn Takeshita [45] . ... In general, the problem "Does a*signed**graph*[G, σ ′ ] admit a*homomorphism**to*[H, σ]?" is not easy*to*solve and its*complexity*has been studied e.g. in [4, 5, 14] . ...##
###
On the Links of Vertices in Simplicial d-Complexes Embeddable in the Euclidean 2d-Space

2017
*
Discrete & Computational Geometry
*

In short, we show that in any such

doi:10.1007/s00454-017-9936-1
fatcat:attp55r3kvah7efr2tjdmoihlm
*complex*, for any three vertices, the intersection*of*the link-*complexes**of*the vertices is linklessly embeddable in p2d´1q-dimensional euclidean space. ... Moreover, the bound is also true for the size*of*d-*complexes*linklessly embeddable in p2d'1q-dimensional space. ... Acknowledgements The author is indebted*to*Herbert Edelsbrunner for bringing the Lemma 2*to*his notice. ...##
###
Counting Subgraphs via Homomorphisms
[chapter]

2009
*
Lecture Notes in Computer Science
*

We introduce a generic approach for counting subgraphs in a

doi:10.1007/978-3-642-02927-1_8
fatcat:uw247ph62rbnjeggb2k3ysqq3y
*graph*. The main idea is*to*relate counting subgraphs*to*counting*graph**homomorphisms*. ... polynomial, the classical algorithm*of*Kohn, Gottlieb, Kohn, and Karp for counting Hamiltonian*cycles*, Ryser's formula for counting perfect matchings*of*a bipartite*graph*, and color coding based algorithms ... Acknowledgement Many thanks*to*László Lovász for answering our questions on*graph**homomorphisms*. ...##
###
Counting Subgraphs via Homomorphisms

2012
*
SIAM Journal on Discrete Mathematics
*

We introduce a generic approach for counting subgraphs in a

doi:10.1137/100789403
fatcat:igr4ivf2z5gg3h3zabvnwnfifm
*graph*. The main idea is*to*relate counting subgraphs*to*counting*graph**homomorphisms*. ... polynomial, the classical algorithm*of*Kohn, Gottlieb, Kohn, and Karp for counting Hamiltonian*cycles*, Ryser's formula for counting perfect matchings*of*a bipartite*graph*, and color coding based algorithms ... Acknowledgement Many thanks*to*László Lovász for answering our questions on*graph**homomorphisms*. ...##
###
On a graph property generalizing planarity and flatness

2009
*
Combinatorica
*

This parameter characterizes subgraphs

doi:10.1007/s00493-009-2219-6
fatcat:7uzysghijzbbzm2t4qz4ygjo2m
*of*paths, outerplanar*graphs*,*planar**graphs*, and*graphs*that have a flat embedding as those*graphs*G with σ(G) ≤ 1, 2, 3, and 4, respectively. ... Among several other theorems, we show that if H is a minor*of*G, then σ(H) ≤ σ(G), that σ(Kn) = n−1, and that if H is the suspension*of*G, then σ(H) = σ(G)+1. ... One step in proving this, is*to*show that*planar**graphs*G have σ(G) ≤ 3. A*planar**graph*G clearly has an even mapping in R 2 . Hence I(z) = 0 for each symmetric 2-*cycle*z*of*D(G). ...##
###
On links of vertices in simplicial d-complexes embeddable in the euclidean 2d-space
[article]

2020
*
arXiv
*
pre-print

These considerations lead us

arXiv:1512.05164v8
fatcat:p5oh4vnaffad7grd77zx756r5y
*to*a new upper bound on the total number*of*d-simplices in an embeddable*complex*in 2d-space with n vertices, improving known upper bounds, for all d ≥ 2. ... In short, we show that in any such*complex*, for any three vertices, the intersection*of*the link-*complexes**of*the vertices is linklessly embeddable in the (2d-1)-dimensional euclidean space. ... Acknowledgements The author is indebted*to*Herbert Edelsbrunner for bringing Lemma 2*to*his notice. ...##
###
Quasi-isometrically embedded subgroups of braid and diffeomorphism groups

2007
*
Transactions of the American Mathematical Society
*

We show that a large class

doi:10.1090/s0002-9947-07-04332-2
fatcat:wfytxzvc2rfldmtp26d6ub6vui
*of*right-angled Artin groups (in particular, those with*planar*complementary defining*graph*) can be embedded quasi-isometrically in pure braid groups and in the group Diff(D ... 2 , ∂D 2 , vol)*of*area preserving diffeomorphisms*of*the disk fixing the boundary (with respect*to*the L 2 -norm metric); this extends results*of*Benaim and Gambaudo who gave quasi-isometric embeddings ... We also thank Benson Farb for pointing out*to*us the work*of*Januszkiewicz andŚwiatkowski [7] and its relevance*to*the present work, and in particular the existence*of*hyperbolic 4-manifold subgroups ...
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