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Continuum Theory and Graph Theory: Disconnection Numbers
Author(s) -
Nadler Sam B.
Publication year - 1993
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-47.1.167
Subject(s) - mathematics , connectivity , cardinality (data modeling) , degenerate energy levels , combinatorics , graph , mathematical proof , graph theory , metric space , discrete mathematics , connected component , compact space , polyhedron , pure mathematics , physics , computer science , geometry , quantum mechanics , data mining
Let X be a non‐degenerate compact connected metric space. It is proved that if there is an n , 2 ⩽ n ⩽ ℵ 0 , such that X − A is not connected for all A ⊂ X of cardinality n , then X is a graph (that is, a one‐dimensional compact connected polyhedron), and conversely. Furthermore, a formula involving only the edges and nodes of the graph X is found which calculates the smallest such n (which works for all A ⊂ X of cardinality n ). As one of the consequences, it is shown that these are exactly five compact connected metric spaces for which the smallest such n is 3. A number of other results, as well as some new proofs of known theorems, are obtained.