On graph equivalences preserved under extensions

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 �...

• Type: Journal
• Year: 2011
• Where: DM
• Authors: Zbigniew Lonc, Miroslaw Truszczynski
• Keywords: DM 2011, Education, Equivalence Relations, Graph Properties, Strengthening