Journal of Symbolic Logic

Twilight Graphs

J. C. E. Dekker

Abstract

This paper deals primarily with countable, simple, connected graphs and the following two conditions which are trivially satisfied if the graphs are finite: (a) there is an edge-recognition algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to decide whether they are adjacent, (b) there is a shortest path algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to find a minimal path joining them. A graph $G = \langle\eta, \eta\rangle$ with $\eta$ as set of vertices and $\eta$ as set of edges is an $\alpha$-graph if it satisfies (a); it is an $\omega$-graph if it satisfies (b). $G$ is called r.e. (isolic) if the sets $\nu$ and $\eta$ are r.e. (isolated). While every $\omega$-graph is an $\alpha$-graph, the converse is false, even if $G$ is r.e. or isolic. Several basic properties of finite graphs do not generalize to $\omega$-graphs. For example, an $\omega$-tree with more than one vertex need not have two pendant vertices, but may have only one or none, since it may be a 1-way or 2-way infinite path. Nevertheless, some elementary propositions for finite graphs $G = \langle\nu, \eta\rangle$ with $n = \operatorname{card}(\nu), e = \operatorname{card}(\eta)$, do generalize to isolic $\omega$-graphs, e.g., $n - 1 \leq e \leq n(n - 1)/2$. Let $N$ and $E$ be the recursive equivalence types of $\nu$ and $\eta$ respectively. Then we have for an isolic $\alpha$-tree $G = \langle\nu, \eta\rangle: N = E + 1$ iff $G$ is an $\omega$-tree. The theorem that every finite graph has a spanning tree has a natural, effective analogue for $\omega$-graphs. The structural difference between isolic $\alpha$-graphs and isolic $\omega$-graphs will be illustrated by: (i) every infinite graph is isomorphic to some isolic $\alpha$-graph; (ii) there is an infinite graph which is not isomorphic to any isolic $\omega$-graph. An isolic $\alpha$-graph is also called a twilight graph. There are $c$ such graphs, $c$ denoting the cardinality of the continuum.

Article information

Source
J. Symbolic Logic, Volume 46, Issue 3 (1981), 539-571.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183740830

Mathematical Reviews number (MathSciNet)
MR627906

Zentralblatt MATH identifier
0466.03015

JSTOR