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 -