TY - GEN

T1 - Bounded representations of interval and proper interval graphs

AU - Balko, Martin

AU - Klavík, Pavel

AU - Otachi, Yota

PY - 2013/12/1

Y1 - 2013/12/1

N2 - Klavík et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals ℒv and ℛv called bounds. We ask whether there exists a bounded representation in which each interval Iv has its left endpoint in ℒv and its right endpoint in . We show that the problem can be solved in linear time for interval graphs and in quadratic time for proper interval graphs. Robert's Theorem states that the classes of proper interval graphs and unit interval graphs are equal. Surprisingly, the bounded representation problem is polynomially solvable for proper interval graphs and NP-complete for unit interval graphs [Klavík et al., arxiv:1207.6960]. So unless P = NP, the proper and unit interval representations behave very differently. The bounded representation problem belongs to a wider class of restricted representation problems. These problems are generalizations of the well-understood recognition problem, and they ask whether there exists a representation of G satisfying some additional constraints. The bounded representation problems generalize many of these problems.

AB - Klavík et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals ℒv and ℛv called bounds. We ask whether there exists a bounded representation in which each interval Iv has its left endpoint in ℒv and its right endpoint in . We show that the problem can be solved in linear time for interval graphs and in quadratic time for proper interval graphs. Robert's Theorem states that the classes of proper interval graphs and unit interval graphs are equal. Surprisingly, the bounded representation problem is polynomially solvable for proper interval graphs and NP-complete for unit interval graphs [Klavík et al., arxiv:1207.6960]. So unless P = NP, the proper and unit interval representations behave very differently. The bounded representation problem belongs to a wider class of restricted representation problems. These problems are generalizations of the well-understood recognition problem, and they ask whether there exists a representation of G satisfying some additional constraints. The bounded representation problems generalize many of these problems.

UR - http://www.scopus.com/inward/record.url?scp=84893361089&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-45030-3_50

DO - 10.1007/978-3-642-45030-3_50

M3 - Conference contribution

AN - SCOPUS:84893361089

SN - 9783642450297

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

SP - 535

EP - 546

BT - Algorithms and Computation - 24th International Symposium, ISAAC 2013, Proceedings

T2 - 24th International Symposium on Algorithms and Computation, ISAAC 2013

Y2 - 16 December 2013 through 18 December 2013

ER -