Decomposition of Graphs into Two-Way Infinite Paths

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Abstract
We consider undirected graphs in which two vertices may be joined by more than one edge and in which a vertex may be joined to itself by one or more edges. G will always denote a graph. The set of vertices [edges] of G will be denoted by V(G) [E(G)]. G is finite or infinite according as V(G) ∪ E(G) is finite or infinite. The degree, d(ξ) or dG(ξ), of a vertex ξ of G is 2a + b, where a is the number of edges joining ξ to itself and b is the number of those joining ξ to other vertices.

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