Graph homomorphisms: structure and symmetry

Hahn, Geňa; Tardif, Claude

doi:10.1007/978-94-015-8937-6_4

Cited by 126 publications

(142 citation statements)

References 97 publications

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“…A review of graph homomorphisms, especially with applications to colorings, is in Hahn and Tardif [11]. These studies do not include studies of homomorphisms with property (3).…”

Section: Introductionmentioning

confidence: 99%

CSD Homomorphisms between Phylogenetic Networks

Willson

2012

IEEE/ACM Trans. Comput. Biol. and Bioinf.

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Abstract-Since Darwin, species trees have been used as a simplified description of the relationships which summarize the complicated network N of reality. Recent evidence of hybridization and lateral gene transfer, however, suggest that there are situations where trees are inadequate. Consequently it is important to determine properties that characterize networks closely related to N and possibly more complicated than trees but lacking the full complexity of N . A connected surjective digraph map (CSD) is a map f from one network N to another network M such that every arc is either collapsed to a single vertex or is taken to an arc, such that f is surjective, and such that the inverse image of a vertex is always connected. CSD maps are shown to behave well under composition. It is proved that if there is a CSD map from N to M , then there is a way to lift an undirected version of M into N , often with added resolution. A CSD map from N to M puts strong constraints on N . In general, it may be useful to study classes of networks such that, for any N , there exists a CSD map from N to some standard member of that class.

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“…A review of graph homomorphisms, especially with applications to colorings, is in Hahn and Tardif [11]. These studies do not include studies of homomorphisms with property (3).…”

Section: Introductionmentioning

confidence: 99%

CSD Homomorphisms between Phylogenetic Networks

Willson

2012

IEEE/ACM Trans. Comput. Biol. and Bioinf.

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“…, p}, i.e., V (K p,q ) = The fractional chromatic number of K p,q is clearly at most p/q, and it is not hard to show that it is actually equal to p/q. The definition of the fractional chromatic number yields the following proposition which can also be found, e.g., in [11]. We include a short proof for the sake of completeness.…”

Section: Universal Graphsmentioning

confidence: 75%

Extending Fractional Precolorings

Kráľ¹,

Krnc²,

Kupec³

et al. 2012

SIAM J. Discrete Math.

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For every d ≥ 3 and k ∈ {2} ∪ [3, ∞), we determine the smallest ε such that every fractional (k + ε)-precoloring of vertices at mutual distance at least d of a graph G with fractional chromatic number equal to k can be extended to a proper fractional (k + ε)-coloring of G. Our work complements the analogous results of Albertson for ordinary colorings and those of Albertson and West for circular colorings.

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“…Welzl [21] showed that the core of a vertex-transitive graph is vertex-transitive; see [11,Theorem 3.7]. More general statements are true; for example, the properties of edge-transitivity and non-edge-transitivity are inherited by cores as well.…”

Section: Cores and Hullsmentioning

confidence: 99%

Cores of Symmetric Graphs

Cameron

Kazanidis

2008

J. Aust. Math. Soc.

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The core of a graph is the smallest graph that is homomorphically equivalent to (that is, there exist homomorphisms in both directions). The core of is unique up to isomorphism and is an induced subgraph of . We give a construction in some sense dual to the core. The hull of a graph is a graph containing as a spanning subgraph, admitting all the endomorphisms of , and having as core a complete graph of the same order as the core of . This construction is related to the notion of a synchronizing permutation group, which arises in semigroup theory; we provide some more insight by characterizing these permutation groups in terms of graphs. It is known that the core of a vertex-transitive graph is vertex-transitive. In some cases we can make stronger statements: for example, if is a nonedge-transitive graph, we show that either the core of is complete, or is its own core. Rank-three graphs are non-edge-transitive. We examine some families of these to decide which of the two alternatives for the core actually holds. We will see that this question is very difficult, being equivalent in some cases to unsolved questions in finite geometry (for example, about spreads, ovoids and partitions into ovoids in polar spaces).2000 Mathematics subject classification: primary 05C25; secondary 05C15, 05B25.

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Graph homomorphisms: structure and symmetry

Cited by 126 publications

References 97 publications

CSD Homomorphisms between Phylogenetic Networks

CSD Homomorphisms between Phylogenetic Networks

Extending Fractional Precolorings

Cores of Symmetric Graphs

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