Digital metrics: A graph-theoretical approach

Authors:
Frank Harary;Robert A Melter;Ioan Tomescu
Affiliations:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA;Department of Mathematics, Southampton College of Long Island University, Southampton, NY 11968, USA;Faculty of Mathematics, University of Bucharest, Bucharest, Romania
Venue:
Pattern Recognition Letters
Year:
1984

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Abstract

Consider the following two graphs M and N, both with vertex set Z x Z, where Z is the set of all integers. In M, two vertices are adjacent when their euclidean distance is 1, while in N, adjacency is obtained when the distance is either 1 or @/2. By definition, H is a metric subgraph of the graph G if the distance between any two points of H is the same as their distance in G. We determine all the metric subgraphs of M and N. The graph-theoretical distances in M and N are equal respectively to the city block and chessboard matrics used in pattern recognition.