Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs

We show that the class of chordal claw-free graphs admits LREC=-definable canonization. LREC= is a logic that extends first-order logic with counting by an operator that allows it to formalize a limited form of recursion. This operator can be evaluated in logarithmic space. It follows that there exists a logarithmic-space canonization algorithm for the class of chordal claw-free graphs, and that LREC= captures logarithmic space on this graph class. Since LREC= is contained in fixed-point logic with counting, we also obtain that fixed-point logic with counting captures polynomial time on the class of chordal claw-free graphs

Complexity of Grundy coloring and its variants

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 $O^*(2.443^n)$ and WEAK GRUNDY COLORING in time $O^*(2.716^n)$ on graphs of order $n$. While GRUNDY COLORING and WEAK GRUNDY COLORING are known to be solvable in time $O^*(2^{O(wk)})$ 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.Comment: 24 pages, 7 figures. This version contains some new results and improvements. A short paper based on version v2 appeared in COCOON'1

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