TY - GEN

T1 - On distance-d independent set and other problems in graphs with “few” minimal separators

AU - Montealegre, Pedro

AU - Todinca, Ioan

N1 - Publisher Copyright:
© Springer-Verlag GmbH Germany 2016.

PY - 2016

Y1 - 2016

N2 - Fomin and Villanger ([14], STACS 2010) proved that MAXIMUM INDEPENDENT SET, FEEDBACK VERTEX SET, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant t, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let Gpoly be the class of graphs with at most poly(n) minimal separators, for some polynomial poly. We show that the odd powers of a graph G have at most as many minimal separators as G. Consequently, DISTANCE-d INDEPENDENT SET, which consists in finding maximum set of vertices at pairwise distance at least d, is polynomial on Gpoly, for any even d. The problem is NP-hard on chordal graphs for any odd d ≥ 3 [12]. We also provide polynomial algorithms for CONNECTED VERTEX COVER AND CONNECTED FEEDBACK VERTEX SET on subclasses of Gpoly including chordal and circular-arc graphs, and we discuss variants of independent domination problems.

AB - Fomin and Villanger ([14], STACS 2010) proved that MAXIMUM INDEPENDENT SET, FEEDBACK VERTEX SET, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant t, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let Gpoly be the class of graphs with at most poly(n) minimal separators, for some polynomial poly. We show that the odd powers of a graph G have at most as many minimal separators as G. Consequently, DISTANCE-d INDEPENDENT SET, which consists in finding maximum set of vertices at pairwise distance at least d, is polynomial on Gpoly, for any even d. The problem is NP-hard on chordal graphs for any odd d ≥ 3 [12]. We also provide polynomial algorithms for CONNECTED VERTEX COVER AND CONNECTED FEEDBACK VERTEX SET on subclasses of Gpoly including chordal and circular-arc graphs, and we discuss variants of independent domination problems.

UR - http://www.scopus.com/inward/record.url?scp=84992752839&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-53536-3_16

DO - 10.1007/978-3-662-53536-3_16

M3 - Conference contribution

AN - SCOPUS:84992752839

SN - 9783662535356

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

SP - 183

EP - 194

BT - Graph-Theoretic Concepts in Computer Science - 42nd International Workshop, WG 2016, Revised Selected Papers

A2 - Heggernes, Pinar

PB - Springer Verlag

T2 - 42nd International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2016

Y2 - 22 June 2016 through 24 June 2016

ER -