Minors.

comparable-box-dimension.tex

@@ -395,6 +395,60 @@ $\cbdim(G)\le 5\cdot 81^{6^{\cbdim(\GG)}}$.
 
 \section{The product structure and minor-closed classes}
 
+Let $G\boxtimes H$ denote the \emph{strong product} of the graphs $G$ and $H$, i.e., the graph with vertex
+set $V(G)\times V(H)$ and with distinct vertices $(u_1,v_1)$ and $(u_2,v_2)$ adjacent if and only if
+either $u_1=u_2$ or $u_1u_2\in E(G)$ and either $v_1=v_2$ or $v_1v_2\in E(G)$.
+Dujmovi{\'c} et al.~\cite{DJM+} proved the following result.
+\begin{theorem}\label{thm-prod}
+Any graph $G$ is a subgraph of the strong product of a path, a graph of threewidth at most $t$, and $K_m$, where
+\begin{itemize}
+\item $t=3$ and $m=3$ if $G$ is planar, and
+\item $t=4$ and $m=\max(2g,3)$ if $G$ has Euler genus at most $g$.
+\end{itemize}
+Moreover, for every $k$, there exists $t$ such that if $K_k\not\preceq_m G$, then $G$ is a subgraph of a clique-sum
+for graphs that can be obtained from the strong product of a path and a graph of treewidth at most $t$ by adding
+at most $t$ apex vertices.
+\end{theorem}
+
+The connection to the comparable box dimension comes from the following observation.
+\begin{lemma}\label{lemma-ps}
+The strong product of a path $P$, a graph $T$ of treewidth at most $t$, and $K_m$ has
+comparable box dimension at most $t+2+\lceil \log_2 m\rceil$.
+\end{lemma}
+\begin{proof}
+Without loss of generality, we can assume that $k=\log_2 m$ is an integer and the
+vertices of $K_m$ are elements of $\{0,1\}^k$.  Moreover, we can assume that $V(P)=\{1,\ldots,n\}$
+with $ij\in E(P)$ iff $|i-j|=1$.  Let $h$ be the touching representation of $T$ by
+hypercubes in $\mathbb{R}^{t+1}$ obtained by Lemma~\ref{lemma-tw}.
+
+The representation $f$ of $P\boxtimes T\boxtimes K_m$ in $\mathbb{R}^{t+2+k}$
+is obtained as follows: For $p\in V(P)$, $v\in V(T)$, and $(x_1,\ldots, x_k)\in V(K_m)$,
+we set
+$$f(p,v,x_1, \ldots, x_k)[j]=\begin{cases}
+f(v)[j]&\text{ for $j=1,\ldots, t+1$}\\
+[p,p+1]&\text{ if $j=t+2$}\\
+[x_{j-t-2},x_{j-t-2}+1]&\text{ if $j>t+2$.}
+\end{cases}$$
+Clearly, this is a touching representation by comparable boxes.
+\end{proof}
+
+Combining Theorem~\ref{thm-prod}, Lemma~\ref{lemma-ps}, and the results of the previous section,
+we obtain the following corollary, which in particular implies Theorem~\ref{thm-minor}.
+
+\begin{corollary}\label{cor-minor}
+For every graph $G$ of Euler genus $g$, there exists a supergraph $G'$ of $G$ such that
+$\cbdim(G')\le 6+\lceil \log_2 \max(2g,3)\rceil$.  Consequently,
+$$\cbdim(G)\le 5\cdot 81^7 \cdot (2g+3)^{\log_2 81}.$$
+Moreover, for every $k$ there exists $d$ such that if $K_k\not\preceq_m(G)$, then
+$\cbdim(G)\le d$.
+\end{corollary}
+Note that the graph obtained from $K_{2n}$ by deleting a perfect matching has Euler genus $\Theta(n^2)$
+and comparable box dimension $n$, showing that the dependence of the comparable box dimension cannot be
+subpolynomial (though the degree $\log_2 81$ of the polynomial established in Corollary~\ref{cor-minor}
+certainly can be improved).  The dependence of the comparable box dimension on the size of the forbidden minor that we
+established is not explicit, as Theorem~\ref{thm-prod} is based on the structure theorem of Robertson and Seymour~\cite{robertson2003graph}.
+It would be interesting to prove this part of Corollary~\ref{cor-minor} without using the structure theorem.
+
 \subsection*{Acknowledgments}
 This research was carried out at the workshop on Geometric Graphs and Hypergraphs organized by Yelena Yuditsky and Torsten Ueckerdt
 in September 2021.  We would like to thank the organizers and all participants for creating a friendly and productive environment.