Awesome graph parameters

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.