# Minimum dominating sets of intervals on lines

## Abstract

We study the problem of computing minimum dominating sets of *n* intervals on lines in three cases: (1) the lines intersect at a single point, (2) all lines except one are parallel, and (3) one line with *t* weighted points on it and the minimum dominating set must maximize the weight sum of the weighted points covered. We propose polynomial-time algorithms for the first two problems, which are special cases of the minimum dominating set problem for path graphs which is known to be NP-hard. The third problem requires identifying the structure of minimum dominating sets of intervals on a line so as to be able to select one that maximizes the weight sum of the weighted points covered. Assuming that presorting has been performed, the first problem has an *O(n)* time solution, while the second and the third problems are solved by dynamic programming algorithms, requiring *O*(*n* log *n*) and *O*(*n+t*) time, respectively.

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## References

- [BG88]A. Bertossi and A. Gori,
*Total domination and irredundance in weighted interval graphs*, SIAM J. Discrete Math., 3 (1988), pp. 317–327.CrossRefGoogle Scholar - [BJ82]K. Booth and J. Johnson,
*Dominating sets in chordal graphs*, SIAM Journal on Computing, 11 (1982), pp. 191–199.CrossRefGoogle Scholar - [Bra87]A. Brandtstaedt,
*The computational complexity of feedback vertex set, hamiltonian circuit, dominating set, steiner tree, and bandwidth on special perfect graphs*, J. Inf. Process. Cybern., 23 (1987), pp. 471–477.Google Scholar - [GT85]H. Gabow and R. Tarjan,
*A linear-time algorithm for a special case of disjoint set union*, Journal of Computer and System Sciences, (1985), pp. 209–221.Google Scholar - [Gol80]
- [McC85]E. McCreight,
*Priority search trees*, SIAM Journal on Computing, 14 (1985), pp. 257–276.CrossRefMathSciNetGoogle Scholar - [RR88]G. Ramalingam and P. Rangan,
*A unified approach to domination problems on interval graphs*, Information Processing Letters, 27 (1988), pp. 271–274.CrossRefGoogle Scholar - [Tar83]