Abstract sphere-of-influence graphs

  • Authors:

  • Frank Harary;Michael S. Jacobson;Marc J. Lipman;F. R. Mcmorris

  • Affiliations:

  • Department of Computer Science, New Mexico State University, Las Cruces, NM 88003, U.S.A.;Department of Mathematics, University of Louisville, Louisville, KY 40292, U.S.A.;Mathematical Sciences Division, Office of Naval Research, Arlington, VA 22217-5000, U.S.A.;Department of Mathematics, University of Louisville, Louisville, KY 40292, U.S.A.

  • Venue:
  • Mathematical and Computer Modelling: An International Journal
  • Year:
  • 1993

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Abstract

The SIG, or ''sphere-of-influence graph,'' G(S) of a set S of points in the plane was introduced by G. Toussaint. To each point of S assign an open ball centered at that point of radius equal to the smallest distance from that point to any other point of S. Then the vertex set of G(S) is S and two vertices x and y are adjacent whenever their open balls intersect. An abstract SIG is isomorphic to some G(S). It is verified that every path and every cycle is a SIG, and that every tree is an induced subgraph of some SIG. Corresponding but different results for the proximity graphs which use closed balls are derived. Several unsolved problems are explicitly stated.