Homomorphism–homogeneous graphs

Rusinov, Momchil; Schweitzer, Pascal

doi:10.1002/jgt.20478

Cited by 32 publications

(40 citation statements)

References 12 publications

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“…By the connected C-MH characterization, all components of each graph in the list are C-MH. By Proposition 3, Corollary 5, Lemmas 8,9,11,12,[14][15][16][17][18][19], and the accompanying remarks, all components of the graphs on the list are pairwise C-MH-symmetric, so every graph on the list is C-MH. Now, we prove the converse.…”

Section: Proof Of the Characterizationmentioning

confidence: 99%

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Characterization of the finite C-MH-homogeneous graphs

Rollick

2015

Discrete Mathematics

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a b s t r a c tA graph G is C-MH-homogeneous, or C-MH, if every monomorphism between finite connected induced subgraphs of G extends to a homomorphism from G into itself. Similarly, G is C-IH if every isomorphism between finite connected induced subgraphs of G extends to a homomorphism from G into itself. In this paper, the finite C-MH graphs are characterized and a new family of finite C-IH graphs is discussed.

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Section: Proof Of the Characterizationmentioning

confidence: 99%

“…With these new definitions in place, the task of characterizing these families began. For instance, an investigation into countable MM, MH, and HH graphs was carried out by Rusinov and Schweitzer in [12].…”

Section: Introductionmentioning

confidence: 99%

Characterization of the finite C-MH-homogeneous graphs

Rollick

2015

Discrete Mathematics

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“…The motivations for Mašulović's investigations come from clone theory [20], and in fact some of his work on the topic of homomorphism-homogeneity preempted the formal definition by Cameron and Nešetřil. In [24], Rusinov and Schweitzer investigate countable homomorphism-homogeneous graphs (in particular those that are MM, MH, HH), obtaining nice results answering questions posed in [2].…”

Section: Introductionmentioning

confidence: 99%

Connected-Homomorphism-Homogeneous Graphs

Lockett

2014

J. Graph Theory

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A relational structure is (connected-)homogeneous if every isomorphism between finite (connected) substructures extends to an automorphism of the structure. We investigate notions which generalise (connected-)homogeneity, where "isomorphism" may be replaced by "homomorphism" or "monomorphism" in the definition. Specifically, we study the classes of finite connected-homomorphism-homogeneous graphs, with the aim of producing classifications. The main result is a classification of the finite C-HH graphs, where a graph G is C-HH if every homomorphism from a finite connected induced subgraph of G into G extends to an endomorphism of G. The finite C-II (connected-homogeneous) graphs were classified by Gardiner in 1976, and from this we obtain classifications of the finite C-HI and C-MI finite graphs. Although not all the classes of finite connected-homomorphism-homogeneous graphs are completely characterised, we may still obtain the final hierarchy picture for these classes.

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“…In several recent papers [2,5,7,9] homomorphism-homogeneous relational structures such as graphs, tournaments and partially ordered sets have been described.…”

Section: Introductionmentioning

confidence: 99%

“…[5,6]). What makes the classification problem particularly interesting is a result presented in [9] where the authors show that deciding homomorphism-homogeneity for finite graphs with loops allowed is coNP-complete. Hence, there exist classes of finite structures where no feasible characterization of homomorphism-homogeneous objects is possible (unless P = coNP).…”

Section: Introductionmentioning

confidence: 99%

On the Complexity of Deciding Homomorphism-Homogeneity for Finite Algebras

Mašulović

2013

Int. J. Algebra Comput.

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Communicated by R. McKenzieIn 2006, Cameron and Nešetřil introduced the following variant of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely generated substructures of the structure extends to an endomorphism of the structure. In several recent papers homomorphism-homogeneous objects in some well-known classes of algebras have been described (e.g. monounary algebras and lattices), while finite homomorphism-homogeneous groups were described in 1979 under the name of finite quasi-injective groups. In this paper we show that, in general, deciding homomorphismhomogeneity for finite algebras with finitely many fundamental operations and with at least one at least binary fundamental operation is coNP-complete. Therefore, unless P = coNP, there is no feasible characterization of finite homomorphism-homogeneous algebras of this kind.

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Homomorphism–homogeneous graphs

Cited by 32 publications

References 12 publications

Characterization of the finite C-MH-homogeneous graphs

Characterization of the finite C-MH-homogeneous graphs

Connected-Homomorphism-Homogeneous Graphs

On the Complexity of Deciding Homomorphism-Homogeneity for Finite Algebras

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