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Cycle transversals in bounded degree graphs

Abstract : In this work we investigate the algorithmic complexity of computing a minimum C(k)-transversal, i.e., a subset of vertices that intersects all the chordless cycles with k vertices of the input graph, for a fixed k \textgreater= 3. For graphs of maximum degree at most three, we prove that this problem is polynomial-time solvable when k \textless= 4, and NP-hard otherwise. For graphs of maximum degree at most four, we prove that this problem is NP-hard for any fixed k \textgreater= 3. We also discuss polynomial-time approximation algorithms for computing C(3)-transversals in graphs of maximum degree at most four, based on a new decomposition theorem for such graphs that leads to useful reduction rules.
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Marina Groshaus, Pavol Hell, Sulamita Klein, Loana Tito Nogueira, Fábio Protti. Cycle transversals in bounded degree graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2011, Vol. 13 no. 1 (1), pp.45--66. ⟨10.46298/dmtcs.533⟩. ⟨hal-00990477⟩

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