TY - GEN
T1 - Parameterized algorithms for recognizing monopolar and 2-subcolorable graphs
AU - Kanj, Iyad
AU - Komusiewicz, Christian
AU - Sorge, Manuel
AU - Van Leeuwen, Erik Jan
N1 - Publisher Copyright:
© Iyad Kanj, Christian Komusiewicz, Manuel Sorge, and Erik Jan van Leeuwen.
PY - 2016/6/1
Y1 - 2016/6/1
N2 - We consider the recognition problem for two graph classes that generalize split and unipolar graphs, respectively. First, we consider the recognizability of graphs that admit a monopolar partition: a partition of the vertex set into sets A, B such that G[A] is a disjoint union of cliques and G[B] an independent set. If in such a partition G[A] is a single clique, then G is a split graph. We show that in O(2k·k3·(|V(G)| + |E(G)|)) time we can decide whether G admits a monopolar partition (A, B) where G[A] has at most k cliques. This generalizes the linear-time algorithm for recognizing split graphs corresponding to the case when k = 1. Second, we consider the recognizability of graphs that admit a 2-subcoloring: a partition of the vertex set into sets A, B such that each of G[A] and G[B] is a disjoint union of cliques. If in such a partition G[A] is a single clique, then G is a unipolar graph. We show that in O(k2k+2·(|V(G)|2 + |V(G)|·|E(G)|)) time we can decide whether G admits a 2-subcoloring (A, B) where G[A] has at most k cliques. This generalizes the polynomial-time algorithm for recognizing unipolar graphs corresponding to the case when k = 1. We also show that in O∗ (4k) time we can decide whether G admits a 2-subcoloring (A, B) where G[A] and G[B] have at most k cliques in total. To obtain the first two results above, we formalize a technique, which we dub inductive recognition, that can be viewed as an adaptation of iterative compression to recognition problems. We believe that the formalization of this technique will prove useful in general for designing parameterized algorithms for recognition problems. Finally, we show that, unless the Exponential Time Hypothesis fails, no subexponential-time algorithms for the above recognition problems exist, and that, unless P=NP, no generic fixed-parameter algorithm exists for the recognizability of graphs whose vertex set can be bipartitioned such that one part is a disjoint union of k cliques.
AB - We consider the recognition problem for two graph classes that generalize split and unipolar graphs, respectively. First, we consider the recognizability of graphs that admit a monopolar partition: a partition of the vertex set into sets A, B such that G[A] is a disjoint union of cliques and G[B] an independent set. If in such a partition G[A] is a single clique, then G is a split graph. We show that in O(2k·k3·(|V(G)| + |E(G)|)) time we can decide whether G admits a monopolar partition (A, B) where G[A] has at most k cliques. This generalizes the linear-time algorithm for recognizing split graphs corresponding to the case when k = 1. Second, we consider the recognizability of graphs that admit a 2-subcoloring: a partition of the vertex set into sets A, B such that each of G[A] and G[B] is a disjoint union of cliques. If in such a partition G[A] is a single clique, then G is a unipolar graph. We show that in O(k2k+2·(|V(G)|2 + |V(G)|·|E(G)|)) time we can decide whether G admits a 2-subcoloring (A, B) where G[A] has at most k cliques. This generalizes the polynomial-time algorithm for recognizing unipolar graphs corresponding to the case when k = 1. We also show that in O∗ (4k) time we can decide whether G admits a 2-subcoloring (A, B) where G[A] and G[B] have at most k cliques in total. To obtain the first two results above, we formalize a technique, which we dub inductive recognition, that can be viewed as an adaptation of iterative compression to recognition problems. We believe that the formalization of this technique will prove useful in general for designing parameterized algorithms for recognition problems. Finally, we show that, unless the Exponential Time Hypothesis fails, no subexponential-time algorithms for the above recognition problems exist, and that, unless P=NP, no generic fixed-parameter algorithm exists for the recognizability of graphs whose vertex set can be bipartitioned such that one part is a disjoint union of k cliques.
KW - Fixed-parameter algorithms
KW - Monopolar graphs
KW - Split graphs
KW - Subcolorings
KW - Unipolar graphs
KW - Vertex-partition problems
UR - http://www.scopus.com/inward/record.url?scp=85011967271&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SWAT.2016.14
DO - 10.4230/LIPIcs.SWAT.2016.14
M3 - Conference contribution
AN - SCOPUS:85011967271
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 14.1-14.14
BT - 15th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2016
A2 - Pagh, Rasmus
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 15th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2016
Y2 - 22 June 2016 through 24 June 2016
ER -