Boundedness and Separation Between Induced and Non-Induced Covering Numbers

Source abstract

There are four covering numbers $\mathrm{c}_g^{\mathcal{G}}(H),\mathrm{c}_u^{\mathcal{G}}(H),\mathrm{c}_l^{\mathcal{G}}(H),\mathrm{c}_f^{\mathcal{G}}(H)$, each of which measures in a slightly different way how well the edges of a graph $H$ (called a host) can be covered with graphs of a class $\mathcal{G}$ (called a guest class). If we require the graphs of $\mathcal{G}$ to correspond to induced subgraphs of $H$, we obtain an induced variant $\mathrm{ic}_x^{\mathcal{G}}$ for each covering number $\mathrm{c}_x^{\mathcal{G}}$ which satisfies $\mathrm{c}_x^{\mathcal{G}}(H) \leq \mathrm{ic}_x^{\mathcal{G}}(H)$ for every graph $H$. Yet, in general $\mathrm{ic}_x^{\mathcal{G}}$ cannot be bounded in terms of $\mathrm{c}_x^{\mathcal{G}}$. If there exists for a guest class $\mathcal{G}$ and a host class $\mathcal{H}$ a function $f$ such that $\mathrm{ic}_x^{\mathcal{G}}(H) \leq f(\mathrm{c}_x^{\mathcal{G}}(H))$ for every graph $H \in \mathcal{H}$, we call $f$ a binding function. Within this work, we study for which structural properties of a guest class $\mathcal{G}$ and a host class $\mathcal{H}$ such binding functions exist. We consider guest classes $\mathcal{G}$ that are monotone, hereditary, component-closed or neither, and have bounded maximum average degree, bounded chromatic number or neither. The host classes $\mathcal{H}$ we consider have bounded treewidth, exclude some minor, have bounded maximum average degree, bounded chromatic number, or none of these properties. For $219$ out of the $240$ possible $3$-tuples of properties for $\mathcal{G}$ and $\mathcal{H}$ and covering numbers we either provide a binding function or an example where no such function exists. In particular, we show that such binding functions always exist for hereditary guest classes $\mathcal{G}$ of bounded maximum average degree for three of the four covering numbers, but may not for the fourth kind.