Grundy Distinguishes Treewidth from Pathwidth

Structural graph parameters, such as treewidth, pathwidth, and clique-width, are a central topic of study in parameterized complexity. A main aim of research in this area is to understand the "price of generality" of these widths: as we transition from more restrictive to more general notions, which are the problems that see their complexity status deteriorate from fixed-parameter tractable to intractable? This type of question is by now very well-studied, but, somewhat strikingly, the algorithmic frontier between the two (arguably) most central width notions, treewidth and pathwidth, is still not understood: currently, no natural graph problem is known to be W-hard for one but FPT for the other. Indeed, a surprising development of the last few years has been the observation that for many of the most paradigmatic problems, their complexities for the two parameters actually coincide exactly, despite the fact that treewidth is a much more general parameter. It would thus appear that the extra generality of treewidth over pathwidth often comes "for free". Our main contribution in this paper is to uncover the first natural example where this generality comes with a high price. We consider Grundy Coloring, a variation of coloring where one seeks to calculate the worst possible coloring that could be assigned to a graph by a greedy First-Fit algorithm. We show that this well-studied problem is FPT parameterized by pathwidth; however, it becomes significantly harder (W[1]-hard) when parameterized by treewidth. Furthermore, we show that Grundy Coloring makes a second complexity jump for more general widths, as it becomes para-NP-hard for clique-width. Hence, Grundy Coloring nicely captures the complexity trade-offs between the three most well-studied parameters. Completing the picture, we show that Grundy Coloring is FPT parameterized by modular-width.Comment: To be published in proceedings of ESA 202

On the diameter and zero forcing number of some graph classes in the Johnson, Grassmann and Hamming association scheme

We establish the diameter of generalized Grassmann graphs and the zero forcing number of some generalized Johnson graphs, generalized Grassmann graphs and the Hamming graphs. Our work extends several previously known results

Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games)

In this article, we study directed graphs (digraphs) with a coloring constraint due to Von Neumann and related to Nim-type games. This is equivalent to the notion of kernels of digraphs, which appears in numerous fields of research such as game theory, complexity theory, artificial intelligence (default logic, argumentation in multi-agent systems), 0-1 laws in monadic second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead to numerous difficult questions (in the sense of NP-completeness, #P-completeness). However, we show here that it is possible to use a generating function approach to get new informations: we use technique of symbolic and analytic combinatorics (generating functions and their singularities) in order to get exact and asymptotic results, e.g. for the existence of a kernel in a circuit or in a unicircuit digraph. This is a first step toward a generatingfunctionology treatment of kernels, while using, e.g., an approach "a la Wright". Our method could be applied to more general "local coloring constraints" in decomposable combinatorial structures.Comment: Presented (as a poster) to the conference Formal Power Series and Algebraic Combinatorics (Vancouver, 2004), electronic proceeding

Grundy dominating sequences and zero forcing sets

In a graph $G$ a sequence $v_1,v_2,\dots,v_m$ of vertices is Grundy dominating if for all $2\le i \le m$ we have $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ and is Grundy total dominating if for all $2\le i \le m$ we have $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. The length of the longest Grundy (total) dominating sequence has been studied by several authors. In this paper we introduce two similar concepts when the requirement on the neighborhoods is changed to $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ or $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. In the former case we establish a strong connection to the zero forcing number of a graph, while we determine the complexity of the decision problem in the latter case. We also study the relationships among the four concepts, and discuss their computational complexities