@article{oai:kyutech.repo.nii.ac.jp:00007781,
 author = {Eto,  Hiroshi and Guo,  Fengrui and Miyano, Eiji and 宮野, 英次},
 journal = {Journal of Combinatorial Optimization},
 month = {Jan},
 note = {The paper studies a generalization of the INDEPENDENT SET problem (IS for short). A distance-d independent set for an integer d≥2 in an unweighted graph G=(V,E) is a subset S⊆V of vertices such that for any pair of vertices u,v∈S, the distance between u and v is at least d in G. Given an unweighted graph G and a positive integer k, the DISTANCE-d INDEPENDENT SET problem (D d IS for short) is to decide whether G contains a distance-d independent set S such that |S|≥k. D2IS is identical to the original IS. Thus D2IS is NP-complete even for planar graphs, but it is in P for bipartite graphs and chordal graphs. In this paper we investigate the computational complexity of D d IS, its maximization version MaxD d IS, and its parameterized version ParaD d IS(k), where the parameter is the size of the distance-d independent set: (1) We first prove that for any ε>0 and any fixed integer d≥3, it is NP-hard to approximate MaxD d IS to within a factor of n1/2−ε for bipartite graphs of n vertices, and for any fixed integer d≥3, ParaD d IS(k) is W[1]-hard for bipartite graphs. Then, (2) we prove that for every fixed integer d≥3, D d IS remains NP-complete even for planar bipartite graphs of maximum degree three. Furthermore, (3) we show that if the input graph is restricted to chordal graphs, then D d IS can be solved in polynomial time for any even d≥2, whereas D d IS is NP-complete for any odd d≥3. Also, we show the hardness of approximation of MaxD d IS and the W[1]-hardness of ParaD d IS(k) on chordal graphs for any odd d≥3.},
 pages = {88--99},
 title = {Distance-d independent set problems for bipartite and chordal graphs},
 volume = {27},
 year = {2013},
 yomi = {ミヤノ, エイジ}
}