Reconfiguration on sparse graphs

A vertex-subset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions S and T of size k, whether it is possible to transform S into T by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We study reconfiguration variants of two classical vertex-subset problems, namely Independent Set and Dominating Set. We denote the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete on graphs of bounded bandwidth and W[1]-hard parameterized by k on general graphs. We show that ISR is fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that ISR is fixed-parameter tractable on planar graphs and graphs of bounded degree. For DSR, we show the problem fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes graphs of bounded degeneracy and nowhere-dense graphs

A linear kernel for planar red-blue dominating set

In the Red-Blue Dominating Set problem, we are given a bipartite graph G = (V B ∪ V R , E) and an integer k, and asked whether G has a subset D ⊆ V B of at most k 'blue' vertices such that each 'red' vertex from V R is adjacent to a vertex in D. We provide the first explicit linear kernel for this problem on planar graphs

Maximum K-vertex covers for some classes of graphs.

Leung Chi Wai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2005.Includes bibliographical references (leaves 52-57).Abstracts in English and Chinese.Abstract --- p.iAcknowledgement --- p.iiiChapter 1 --- Introduction --- p.1Chapter 1.1 --- Motivations --- p.1Chapter 1.2 --- Related work --- p.3Chapter 1.2.1 --- Fixed-parameter tractability --- p.3Chapter 1.2.2 --- Maximum k-vertex cover --- p.4Chapter 1.2.3 --- Dominating set --- p.4Chapter 1.3 --- Overview of the thesis --- p.5Chapter 2 --- Preliminaries --- p.6Chapter 2.1 --- Notation and definitions --- p.6Chapter 2.1.1 --- Basic definitions --- p.6Chapter 2.1.2 --- Partial t-trees --- p.7Chapter 2.1.3 --- Cographs --- p.9Chapter 2.1.4 --- Chordal graphs and interval graphs --- p.11Chapter 2.2 --- Upper bound --- p.12Chapter 2.3 --- Extension method --- p.14Chapter 3 --- Planar Graphs --- p.17Chapter 3.1 --- Trees --- p.17Chapter 3.2 --- Partial t-trees --- p.23Chapter 3.3 --- Planar graphs --- p.30Chapter 4 --- Perfect Graphs --- p.34Chapter 4.1 --- Maximum k-vertex cover in cographs --- p.34Chapter 4.2 --- Maximum dominating k-set in interval graphs --- p.39Chapter 4.3 --- Maximum k-vertex subgraph in chordal graphs --- p.46Chapter 4.3.1 --- Maximum k-vertex subgraph in partial t- trees --- p.46Chapter 4.3.2 --- Maximum k-vertex subgraph in chordal graphs --- p.47Chapter 5 --- Concluding Remarks --- p.49Chapter 5.1 --- Summary of results --- p.49Chapter 5.2 --- Open problems --- p.5