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

Florent Foucaud, Reza Naserasr
2014 Lecture Notes in Computer Science  
We study the 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.  ... 
doi:10.1007/978-3-642-54423-1_46 fatcat:cudazm72vre5pjgoq3c24revui

Analogues of Cliques for (m, n)-Colored Mixed Graphs

Julien Bensmail, Christopher Duffy, Sagnik Sen
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?  ... 
doi:10.1007/s00373-017-1807-2 fatcat:b5s2qdfokfdj5g35ssakedgt3u

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

Pascal Ochem, Nazanin Movarraei
2017 Information Processing Letters  
Finally, we prove that deciding if the 2-edge-colored chromatic number 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] .  ... 
doi:10.1016/j.ipl.2017.02.009 fatcat:q3qusqx7jjayfb4urvcjjy7c44

Analogous to cliques for (m,n)-colored mixed graphs [article]

Julien Bensmail, Christopher Duffy, Sagnik Sen
2015 arXiv   pre-print
In this article, we mainly study different aspects 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.  ... 
arXiv:1411.7376v2 fatcat:lvryhvxwebfovk6k7de2x6kfyy

Homomorphism bounds of signed bipartite K4-minor-free graphs and edge-colorings of 2k-regular K4-minor-free multigraphs

Laurent Beaudou, Florent Foucaud, Reza Naserasr
2019 Discrete Applied Mathematics  
This supports a conjecture 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.  ... 
doi:10.1016/j.dam.2018.09.004 fatcat:zlnut5qj5faptddhasap5ae4pu

The complexity of signed graph and edge-coloured graph homomorphisms

Richard C. Brewster, Florent Foucaud, Pavol Hell, Reza Naserasr
2017 Discrete Mathematics  
We study two types 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.  ... 
doi:10.1016/j.disc.2016.08.005 fatcat:vjiixa4e7vh3podx4w2oaxmzdy

Counting Problems in Parameterized Complexity

Radu Curticapean, Michael Wagner
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.  ... 
doi:10.4230/lipics.ipec.2018.1 dblp:conf/iwpec/Curticapean18 fatcat:tdfs7zngzrgv5fixf2qawrccuy

Circular Chromatic Number of Signed Graphs

Reza Naserasr, Zhouningxin Wang, Xuding Zhu
2021 Electronic Journal of Combinatorics  
Various notions 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  ... 
doi:10.37236/9938 fatcat:5z46dr6kvncsxpsa6y4muo62qe

Concepts of signed graph coloring [article]

Eckhard Steffen, Alexander Vogel
2020 arXiv   pre-print
This paper surveys recent development 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] .  ... 
arXiv:1909.09381v2 fatcat:uveiuweurrblxavntkdulqhb2m

On the Links of Vertices in Simplicial d-Complexes Embeddable in the Euclidean 2d-Space

Salman Parsa
2017 Discrete & Computational Geometry  
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 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.  ... 
doi:10.1007/s00454-017-9936-1 fatcat:attp55r3kvah7efr2tjdmoihlm

Counting Subgraphs via Homomorphisms [chapter]

Omid Amini, Fedor V. Fomin, Saket Saurabh
2009 Lecture Notes in Computer Science  
We introduce a generic approach for counting subgraphs in a 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.  ... 
doi:10.1007/978-3-642-02927-1_8 fatcat:uw247ph62rbnjeggb2k3ysqq3y

Counting Subgraphs via Homomorphisms

Omid Amini, Fedor V. Fomin, Saket Saurabh
2012 SIAM Journal on Discrete Mathematics  
We introduce a generic approach for counting subgraphs in a 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.  ... 
doi:10.1137/100789403 fatcat:igr4ivf2z5gg3h3zabvnwnfifm

On a graph property generalizing planarity and flatness

Hein van der Holst, Rudi Pendavingh
2009 Combinatorica  
This parameter characterizes subgraphs 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).  ... 
doi:10.1007/s00493-009-2219-6 fatcat:7uzysghijzbbzm2t4qz4ygjo2m

On links of vertices in simplicial d-complexes embeddable in the euclidean 2d-space [article]

Salman Parsa
2020 arXiv   pre-print
These considerations lead us 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.  ... 
arXiv:1512.05164v8 fatcat:p5oh4vnaffad7grd77zx756r5y

Quasi-isometrically embedded subgroups of braid and diffeomorphism groups

John Crisp, Bert Wiest
2007 Transactions of the American Mathematical Society  
We show that a large class 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  ... 
doi:10.1090/s0002-9947-07-04332-2 fatcat:wfytxzvc2rfldmtp26d6ub6vui
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