Hereditary property

In mathematics, a hereditary property is a property of an object, that inherits to all its subobjects, where the term subobject depends on the context. These properties are particularly considered in topology and graph theory.

Contents 1 In topology 2 In graph theory 2.1 Monotone property 3 In model theory 4 In matroid theory 5 See also 6 References

In topology

In topology, a topological property is said to be hereditary if whenever a space has that property, then so does every subspace of it. If the latter is true only for closed subspaces, then the property is called weakly hereditary.

For example, second countability and metrisability are hereditary properties. Sequentiality and Hausdorff compactness are weakly hereditary, but not hereditary.^[1] Connectivity is not weakly hereditary.

In graph theory

In graph theory, a hereditary property is a property of a graph which also holds for (is "inherited" by) its induced subgraphs.^[2] Alternately, a hereditary property is preserved by the removal of vertices.

In some cases, the term "hereditary" has been defined with reference to graph minors, but this is more properly called a minor-hereditary property. The Robertson–Seymour theorem implies that a minor-hereditary property may be characterized in terms of a finite set of forbidden minors.

Monotone property

There is no consensus for the meaning of "monotone property" in graph theory. Examples of definitions are:

• Preserved by the removal of edges.^[3]
• Preserved by the removal of edges and vertices (i.e., the property should hold for all subgraphs).^[2]
• Preserved by the addition of edges and vertices (i.e., the property should hold for all supergraphs).^[4]
• Preserved by the addition of edges.^[5] (This meaning is used in the statement of the Aanderaa–Karp–Rosenberg conjecture.)

The complementary property of a property that is preserved by the removal of edges is preserved under the addition of edges. Hence some authors avoid this ambiguity by saying a property A is monotone if A or A^C (the complement of A) is monotone.^[6] Some authors choose to resolve this by using the term increasing monotone for properties preserved under the addition of some object, and decreasing monotone for those preserved under the removal of the same object.

In model theory

In model theory and universal algebra, a class K of structures of a given signature is said to have the hereditary property if every substructure of a structure in K is again in K. A variant of this definition is used in connection with Fraïssé's theorem: A class K of finitely generated structures has the hereditary property if every finitely generated substructure is again in K. See age.

In matroid theory

In a matroid, every subset of an independent set is again independent. This is also sometimes called the hereditary property.

See also

• Hereditarily finite set
• Hereditarily countable set

References

1. ^ *Goreham, Anthony, "Sequential Convergence in Topological Spaces
2. ^ ^{a b} Alon, Noga; Shapira, Asaf (2008). "Every monotone graph property is testable". SIAM Journal on Computing 38 (2): 505–522. doi:10.1137/050633445. http://www.math.tau.ac.il/~nogaa/PDFS/monotone1.pdf. 
3. ^ King, R. (1990), A lower bound for the recognition of digraph properties, Combinatorica, vol 10, 53–59
4. ^ http://www.cs.ucsc.edu/~optas/papers/k-col-threshold.pdf
5. ^ Spinrad, J. (2003), Efficient Graph Representations, AMS Bookstore, ISBN 0821828150, p9.
6. ^ Ashish Goel; Sanatan Rai; Bhaskar Krishnamachari (2003). "Monotone properties of random geometric graphs have sharp thresholds". The Annals of Applied Probability 15 (4): 2535–2552. arXiv:math.PR/0310232. doi:10.1214/105051605000000575.