Isolated Point

 

An isolated point of a graph is a node of degree 0 (Hartsfield and Ringel 1990, p. 8; Harary 1994, p. 15; D'Angelo and West 2000, p. 212; West 2000, p. 22). The number of -node graphs with no isolated points are 0, 1, 2, 7, 23, 122, 888, ... (OEIS A002494), the first few of which are illustrated above. The number of graphical partitions of length  is equal to the number of -node graphs that have no isolated points.

Connected graphs have no isolated points.

An isolated point on a curve is more commonly known as an acnode.

An isolated point of a discrete set  is a member of  (Krantz 1999, p. 63).

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REFERENCES

D'Angelo, J. P. and West, D. B. Mathematical Thinking: Problem-Solving and Proofs, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000.

Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.

Hartsfield, N. and Ringel, G. Pearls in Graph Theory: A Comprehensive Introduction. San Diego, CA: Academic Press, 1990.

Krantz, S. G. "Discrete Sets and Isolated Points." §4.6.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 63-64, 1999.

Sloane, N. J. A. Sequence A002494/M1762 in "The On-Line Encyclopedia of Integer Sequences."West, D. B. Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2000.

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