It is the object of this paper to study the topological propertms of fimte graphs that can be embedded in the n-dimensional integral lattice (denoted N") The basic notion of deletabllity of a node is first introduced A node is deletable with respect to a graph if certain computable conditions are satisfied oil its neighborhood. An equivalence relation oll graphs called reducibility and denoted by ',N" is then defined m terms of deletablhty, and it IS shown that (a) most important topological properties of the graph (homotogy, homology, and cohomology groups) are ~-invanants; (b) for graphs embedded Jn N ~, different knot types belong to different ~-eqmvalenee classes; (c) it is decidable whether two graphs are reducible to each other in N 2 but th~s problem is undecldable in N" for n >_ 4 Finally, it is shown that two different methods of approximating an n-dimensional closed manifold with boundary by a graph of the type studied in this paper lead to graphs whose corresponding homology groups are isomorphic.]KEY WORDS AND PHRASES: computational topology, computational geometry, quantlzed spaces, picture processing, pattern recognition CR CATEGORIES; 3.63, 5.32
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