Awesome graph parameters

TL;DR

This paper explores the relationship between graph parameters related to independent sets and cliques, classifying them as 'awesome' or 'awful' based on their boundedness properties and implications for graph classes.

Contribution

It introduces the concepts of 'awesome' and 'awful' parameters, identifies several parameters fitting these categories, and discusses their algorithmic and theoretical implications.

Findings

Identified several 'awesome' graph parameters.

Established criteria for 'awesomeness' and 'awfulness' of parameters.

Proposed open problems and applications related to these classifications.

Abstract

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