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Unit disk graphs. (English) Zbl 0739.05079
Any given $$n$$ points in the plane form the vertices of some graph by the convention that distinct points are adjacent whenever their distance is at most 2. The resulting graph is called a unit disk graph, since it is the intersection graph of the unit disks around the given $$n$$ points.
It is shown that certain hard decision problems remain NP-complete when restricted to unit disk graphs, even when the position of the points is given. These problems are CHROMATIC NUMBER, INDEPENDENT SET, and several others.
On the other hand, a maximum cardinality clique in unit disk graphs can be found in polynomial time when the position of the points is given.

##### MSC:
 05C85 Graph algorithms (graph-theoretic aspects) 05C99 Graph theory
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##### References:
 [1] Clark, B.N., Unit disk graphs, () · Zbl 0739.05079 [2] Edmonds, J.; Karp, R.M., Theoretical improvements in algorithmic efficiency for network flow problems, J. ACM, 19, 248-264, (1972) · Zbl 0318.90024 [3] Garey, M.R.; Johnson, D.S.; Stockmeyer, L., Some simplified NP-complete graph problems, Theoret. comput. sci., 1, 237-267, (1976) · Zbl 0338.05120 [4] Garey, M.R.; Johnson, D.S., The rectilinear Steiner tree problem is NP-complete, SIAM J. appl. math., 32, 826-834, (1977) · Zbl 0396.05009 [5] Garey, M.R.; Johnson, D.S., Computers and intractability: A guide to the theory of NP-completeness, (1979), Freeman New York · Zbl 0411.68039 [6] Golumbic, M.C., Algorithmic graph theory and perfect graphs, (1980), Academic Press New York · Zbl 0541.05054 [7] Hale, W.K., Frequency assignment: theory and applications, Proc IEEE, 68, 1497-1514, (1980) [8] Itai, A.; Papadimitriou, C.H.; Szwarcfiter, J.L., Hamilton paths in grid graphs, SIAM J. comput., 11, 676-686, (1982) · Zbl 0506.05043 [9] Johnson, D.S., The NP-completeness column: an ongoing guide, J. algorithms, 3, 182-195, (1982) · Zbl 0494.68049 [10] Johnson, D.S., The NP-completeness column: an ongoing guide, J. algorithms, 6, 434-451, (1985) · Zbl 0608.68032 [11] Johnson, D.S., The NP-completeness column: an ongoing guide, J. algorithms, 8, 438-448, (1987) · Zbl 0633.68022 [12] Kammerlander, K., C 900 — an advanced mobile radio telephone system with optimum frequency utilization, IEEE trans. selected areas in communication, 2, 589-597, (1984) [13] Karp, R.M., Reducibility among combinatorial problems, (), 85-104 · Zbl 0366.68041 [14] J. Kilian, personal communication. [15] Lichtenstein, D., Planar formulae and their uses, SIAM J. comput., 11, 329-343, (1982) · Zbl 0478.68043 [16] Masuyama, S.; Ibaraki, T.; Hasegawa, T., The computational complexity of the M-center problems in the plane, Trans. IECE Japan, E64, 57-64, (1981) [17] Mohring, R.H., Algorithmic aspects of comparability graphs and interval graphs, (), 41-101 [18] Toregas, C.; Swain, R.; Revelle, C.; Bergeman, L., The location of emergency service facilities, Oper. res., 19, 1363-1373, (1971) · Zbl 0224.90048 [19] Valiant, L.G., Universality considerations in VLSI circuits, IEEE trans. computers, 30, 135-140, (1981) · Zbl 0455.94046 [20] Yeh, Y.; Wilson, J.; Schwartz, S.C., Outage probability in mobile telephony with directive antennas and macrodiversity, IEEE trans. selected areas in communication, 2, 507-511, (1984)
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