2012
DOI: 10.1007/s00224-012-9412-5
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Partition Into Triangles on Bounded Degree Graphs

Abstract: We consider the PARTITION INTO TRIANGLES problem on bounded degree graphs. We show that this problem is polynomial-time solvable on graphs of maximum degree three by giving a linear-time algorithm. We also show that this problem becomes N P-complete on graphs of maximum degree four. Moreover, we show that there is no subexponential-time algorithm for this problem on graphs of maximum degree four unless the Exponential-Time Hypothesis fails. However, the PARTITION INTO TRIANGLES problem on graphs of maximum deg… Show more

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Cited by 29 publications
(4 citation statements)
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“…Note that in strong contrast to this, K 3 -Partition is linear-time solvable on graphs with maximum degree three [25]. Furthermore, we show P 3 -Partition to be NP-hard on chordal graphs, while K 3 -Partition is known to be polynomial-time solvable in this case [12].…”
Section: Star Partitionmentioning
confidence: 60%
See 1 more Smart Citation
“…Note that in strong contrast to this, K 3 -Partition is linear-time solvable on graphs with maximum degree three [25]. Furthermore, we show P 3 -Partition to be NP-hard on chordal graphs, while K 3 -Partition is known to be polynomial-time solvable in this case [12].…”
Section: Star Partitionmentioning
confidence: 60%
“…For every fixed s ≥ 2, Star Partition is NP-complete on bipartite graphs [7]. Partitioning into triangles (K 3 ), that is, K 3 -Partition, is polynomial-time solvable on chordal graphs [12] and linear-time solvable on graphs of maximum degree three [25].…”
Section: Star Partitionmentioning
confidence: 99%
“…Despite the hardness results, this is a very fast exponential-time algorithm for an -hard problem and we are not aware of another natural -hard problem that has no subexponential-time algorithm (unless the ETH fails) but admits an exponential-time algorithm with running time near O * (1.0099 n ) . We remark that the authors of [20] provided an O * (1.02220 n )-time algorithm for an -hard problem (the Partition Into Triangles problem restricted on graphs of maximum degree 4) that cannot be solved in subexponential-time (assuming ETH), and ask for such rare -hard problems with very fast exponential-time algorithms. Our results add an infinite family of such natural problems into this list.…”
Section: Our Contributionsmentioning
confidence: 99%
“…Guruswam et al [12] showed that MTP remains NP-hard on chordal, planar, line, and total graphs. Moreover, MTP has been proved to be APX-hard even on graphs with maximum degree 4 [19,26]. Chlebík and Chlebíková [7] showed that MTP is NP-hard to approximate better than 0.9929.…”
Section: Related Workmentioning
confidence: 99%