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Connectedness

In: Transfiniteness

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  • Armen H. Zemanian

    (State University of New York at Stony Brook, Department of Electrical Engineering)

Abstract

A key difference between conventional infinite graphs and transfinite graphs is that in conventional graphs two nodes are either connected through a finite path (i.e., a path of finitely many branches) or not connected at all, whereas in transfinite graphs two nodes may be connected through a transfinite path (i.e., one having infinitely many branches) but not through any finite path. In fact, for transfinite graphs there is a hierarchy of connectedness concepts, which is indexed by the countable ordinals. Thus, we speak of two nodes being “ρ-connected” when the two nodes are connected through a two-ended α -path for some α no larger than ρ. As ρ increases,ρ- connectedness weakens in the sense that, when ρ1

Suggested Citation

  • Armen H. Zemanian, 1996. "Connectedness," Springer Books, in: Transfiniteness, chapter 0, pages 47-78, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-0767-2_3
    DOI: 10.1007/978-1-4612-0767-2_3
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