TY - GEN

T1 - Algorithmic aspects of the intersection and overlap numbers of a graph

AU - Hermelin, Danny

AU - Rizzi, Romeo

AU - Vialette, Stéphane

PY - 2012/1/1

Y1 - 2012/1/1

N2 - The intersection number of a graph G is the minimum size of a set S such that G is an intersection graph of some family of subsets F ⊆ 2 S. The overlap number of G is defined similarly, except that G is required to be an overlap graph of F. Computing the overlap number of a graph has been stated as an open problem in [B. Rosgen and L. Stewart, 2010, arXiv:1008.2170v2] and [D.W. Cranston, et al., J. Graph Theory., 2011]. In this paper we show two algorithmic aspects concerning both these graph invariants. On the one hand, we show that the corresponding optimization problems associated with these numbers are both APX-hard, where for the intersection number our results hold even for biconnected graphs of maximum degree 7, strengthening the previously known hardness result. On the other hand, we show that the recognition problem for any specific intersection graph class (e.g. interval, unit disc, . . . ) is easy when restricted to graphs with fixed intersection number.

AB - The intersection number of a graph G is the minimum size of a set S such that G is an intersection graph of some family of subsets F ⊆ 2 S. The overlap number of G is defined similarly, except that G is required to be an overlap graph of F. Computing the overlap number of a graph has been stated as an open problem in [B. Rosgen and L. Stewart, 2010, arXiv:1008.2170v2] and [D.W. Cranston, et al., J. Graph Theory., 2011]. In this paper we show two algorithmic aspects concerning both these graph invariants. On the one hand, we show that the corresponding optimization problems associated with these numbers are both APX-hard, where for the intersection number our results hold even for biconnected graphs of maximum degree 7, strengthening the previously known hardness result. On the other hand, we show that the recognition problem for any specific intersection graph class (e.g. interval, unit disc, . . . ) is easy when restricted to graphs with fixed intersection number.

UR - http://www.scopus.com/inward/record.url?scp=84871538590&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-35261-4_49

DO - 10.1007/978-3-642-35261-4_49

M3 - Conference contribution

AN - SCOPUS:84871538590

SN - 9783642352607

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 465

EP - 474

BT - Algorithms and Computation - 23rd International Symposium, ISAAC 2012, Proceedings

PB - Springer Verlag

T2 - 23rd International Symposium on Algorithms and Computation, ISAAC 2012

Y2 - 19 December 2012 through 21 December 2012

ER -