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DM

2011

2011

Let G be the set of ﬁnite graphs whose vertices belong to some ﬁxed countable set, and let ≡ be an equivalence relation on G. By the strengthening of ≡ we mean an equivalence relation ≡s such that G ≡s H, where G, H ∈ G, if for every F ∈ G, G ∪ F ≡ H ∪ F. The most important case that we study in this paper concerns equivalence relations deﬁned by graph properties. We write G ≡Φ H, where Φ is a graph property and G, H ∈ G, if either both G and H have the property Φ, or both do not have it. We characterize the strengthening of the relations ≡Φ for several graph properties Φ. For example, if Φ is the property of being a k-connected graph, we ﬁnd a polynomially veriﬁable (for k ﬁxed) condition that characterizes the pairs of graphs equivalent with respect to ≡Φ s . We obtain similar results when Φ is the property of being k-colorable, edge 2colorable, hamiltonian, or planar, and when Φ is the property of containing a subgraph isomorphic to a ...

Related Content

Added |
14 May 2011 |

Updated |
14 May 2011 |

Type |
Journal |

Year |
2011 |

Where |
DM |

Authors |
Zbigniew Lonc, Miroslaw Truszczynski |

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