Search

It has long been known that Feedback Vertex Set can be solved in time 2 O (w log w) n O (1) on n-vertex graphs of treewidth w, but it was only recently that this running time was improved to 2 O (w) n O (1) , that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class P of graphs, the Bounded P -Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether G contains a set S of at most k vertices such that each block of G - S has at most d vertices and is in P . Assuming that P is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of d: if P consists only of chordal graphs, then the problem can be solved in time 2 O (w d 2) n O (1) , if P contains a graph with an induced cycle of length ℓ ⩾ 4 , then the problem is not solvable in time 2 o (w log w) n O (1) even for fixed d = ℓ , unless the ETH fails. We also study a similar problem, called Bounded P -Component Vertex Deletion, where the target graphs have connected components of small size rather than blocks of small size, and we present analogous results. For this problem, we also show that if d is part of the input and P contains all chordal graphs, then it cannot be solved in time f (w) n o (w) for some function f, unless the ETH fails. [ABSTRACT FROM AUTHOR]

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring , the problem of determining whether a given graph has Grundy number at least k . We also study the variants Weak Grundy Coloring (where the coloring is not necessarily proper) and Connected Grundy Coloring (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that Grundy Coloring can be solved in time O ∗ ( 2 . 44 3 n ) and Weak Grundy Coloring in time O ∗ ( 2 . 71 6 n ) on graphs of order n . While Grundy Coloring and Weak Grundy Coloring are known to be solvable in time O ∗ ( 2 O ( w k ) ) for graphs of treewidth w (where k is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time O ∗ ( 2 o ( w log w ) ) . We also describe an O ∗ ( 2 2 O ( k ) ) algorithm for Weak Grundy Coloring , which is therefore FPT for the parameter k . Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for Grundy Coloring ). Although we do not know whether Grundy Coloring is in FPT , we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for Grundy Coloring and Weak Grundy Coloring on apex-minor graphs. In stark contrast with the two other problems, we show that Connected Grundy Coloring is NP -complete already for k = 7 colors. [ABSTRACT FROM AUTHOR]