Unique factorization of compositive hereditary graph properties
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Abstract
A graph property is any class of graphs that is closed under isomorphisms. A graph property P is hereditary if it is closed under taking subgraphs; it is compositive if for any graphs G 1,G 2 ∈ P there exists a graph G ∈ P containing both G 1 and G 2 as subgraphs.
- 1.
for each i, the graph induced by the i-th non-empty partition part is in P i , and
- 2.
for each i and j with i ≠ j, there is no edge between the i-th and j-th parts if υ i and υ j are non-adjacent vertices in H.
If a graph property P is H-factorizable over ℙ and we know the graph properties P 1, ..., P n , then we write P = H[P 1, ..., P n ]. In such a case, the presentation H[P 1, ..., P n ] is called a factorization of P over ℙ. This concept generalizes graph homomorphisms and (P 1, ..., P n )-colorings.
In this paper, we investigate all H-factorizations of a graph property P over the class of all hereditary compositive graph properties for finite graphs H. It is shown that in many cases there is exactly one such factorization.
Keywords
Graph property hereditary compositive property unique factorization minimal forbidden graphs reducibilityMR(2000) Subject Classification
05C75 05C15 05C35Preview
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References
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