Awesome graph parameters

TL;DR

The paper introduces the framework of awesomeness to study when a parameter’s α-variant being bounded implies a bound by a function of the clique number, and conversely. It formalizes hyperparameters, modulators, and a Ramsey-based toolkit to relate α-ρ boundedness with clique-boundedness, proving that treedepth and pathwidth are awful while several parameters (e.g., vertex cover, feedback vertex set, and odd cycle transversal) are awesome. The authors derive algorithmic consequences, notably polynomial-time solvability of Maximum Weight Independent Set in broad hereditary classes with clique-bounded ρ, via modulators and inherited properties. They provide concrete constructions and transformations (like the s-claw substitution) to separate α-pathwidth from td and pw, and discuss a rich landscape of open questions around weak awesomeness and other hyperparameterisations. The work lays a unified, tool-driven approach to diagnose when independence-variants yield tractability through clique-boundedness and where they fail.

Abstract

For a graph $G$, we denote by $α(G)$ the size of a maximum independent set and by $ω(G)$ the size of a maximum clique in $G$. Our paper lies on the edge of two lines of research, related to $α$ and $ω$, respectively. One of them studies $α$-variants of graph parameters, such as $α$-treewidth or $α$-degeneracy. The second line deals with graph classes where some parameters are bounded by a function of $ω(G)$. A famous example of this type is the family of $χ$-bounded classes, where the chromatic number $χ(G)$ is bounded by a function of $ω(G)$. A Ramsey-type argument implies that if the $α$-variant of a graph parameter $ρ$ is bounded by a constant in a class $\mathcal{G}$, then $ρ$ is bounded by a function of $ω$ in $\mathcal{G}$. If the reverse implication also holds, we say that $ρ$ is awesome. Otherwise, we say that $ρ$ is awful. In the present paper, we identify a number of awesome and awful graph parameters, derive some algorithmic applications of awesomeness, and propose a number of open problems related to these notions.

Figures (2)

• Figure 1: The diagram illustrates various awesome and awful graph parameters (with respect to their canonical hyperparameterisations, as defined in \ref{['sec:hyperparameterisations']}), along with some of the relations between them. An arrow from a parameter $\rho_1$ to a parameter $\rho_2$ indicates that boundedness of $\rho_2$ in a class of graphs implies boundedness of $\rho_1$ in that class. For clarity, not all such relations are shown in the figure.
• Figure 2: Inclusions of class properties described in terms of a basic hyperparameter $\rho$. An arc from $A$ to $B$ means that $A \subseteq B$. Unlabelled arcs represent inclusions that hold for all basic hyperparameters and follow directly from the definitions. The left-to-right inclusions (marked by $\star$) hold for inheritable hyperparameters and follow from \ref{['thm:alpha-rho at most alpha-rhocm']}.

Theorems & Definitions (72)

• Proposition 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Theorem 4.1
• proof
• Remark 4.2