diff --git a/arxiv_cbd.tex b/arxiv_cbd.tex index 55f0e8a5510ed0bdf6d4f285acdc06af511fd368..5f270e98135447f9beb848e238241edf681dcf6f 100644 --- a/arxiv_cbd.tex +++ b/arxiv_cbd.tex @@ -22,7 +22,7 @@ %\category{} %optional, e.g. invited paper -\relatedversion{A full version of the paper is available at \url{}} %optional, e.g. full version hosted on arXiv, HAL, or other respository/website +%\relatedversion{A full version of the paper is available at \url{}} %optional, e.g. full version hosted on arXiv, HAL, or other respository/website %\relatedversiondetails[linktext={opt. text shown instead of the URL}, cite=DBLP:books/mk/GrayR93]{Classification (e.g. Full Version, Extended Version, Previous Version}{URL to related version} %linktext and cite are optional %\supplement{}%optional, e.g. related research data, source code, ... hosted on a repository like zenodo, figshare, GitHub, ... @@ -100,24 +100,24 @@ This result has motivated numerous strengthenings and variations (see \cite{grap An attractive feature of touching representations is that it is possible to represent graph classes that are sparse (e.g., planar graphs, or more generally, graph classes with bounded expansion~\cite{nesbook}). This is in contrast to general intersection representations where the represented class always includes arbitrarily large cliques. -Of course, whether the class of touching graphs of objects from $\OO$ is sparse or not depends on the system $\OO$. +Of course, whether the class of touching graphs of objects from $\OO$ is sparse or not depends on the particular system $\OO$. For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of boxes in $\mathbb{R}^3$, where the vertices in one part are represented by $m\times 1\times 1$ boxes and the vertices of the other part are represented by $1\times n\times 1$ -boxes (throughout the paper, by a \emph{box} we mean an axis-aligned one, i.e., the Cartesian product of closed intervals of non-zero length). +boxes (throughout the paper, by \emph{box} we always mean \emph{axis-aligned box}, i.e., the Cartesian product of closed intervals of non-zero length). Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} noticed that this issue disappears if we forbid such a combination of -long and wide boxes, a condition which can be expressed as follows. For two boxes $B_1$ and $B_2$, we write $B_1 \sqsubseteq B_2$ if $B_2$ contains a translate of $B_1$. +long and wide boxes. This condition can be expressed as follows. For two boxes $B_1$ and $B_2$, we write $B_1 \sqsubseteq B_2$ if $B_2$ contains a translate of $B_1$. We say that $B_1$ and $B_2$ are \emph{comparable} if $B_1\sqsubseteq B_2$ or $B_2\sqsubseteq B_1$. A \emph{touching representation by comparable boxes} of a graph $G$ is a touching representation $f$ by boxes such that for every $u,v\in V(G)$, the boxes $f(u)$ and $f(v)$ are comparable. Let the \emph{comparable box dimension} $\cbdim(G)$ of a graph $G$ be the smallest integer $d$ such that $G$ has a touching representation by comparable boxes in $\mathbb{R}^d$. -Let us remark that the comparable box dimension of every graph $G$ is at most $|V(G)|$, see Section~\ref{sec-vertad} for details. -Then for a class $\GG$ of graphs, let $\cbdim(\GG)\colonequals\sup\{\cbdim(G):G\in\GG\}$. Note that $\cbdim(\GG)=\infty$ if the -comparable box dimension of graphs in $\GG$ is not bounded. +We remark that the comparable box dimension of every graph $G$ is at most $|V(G)|$, see Section~\ref{sec-vertad} for details. +Then, for a class $\GG$ of graphs, let $\cbdim(\GG)\colonequals\sup\{\cbdim(G):G\in\GG\}$. If the +comparable box dimension of graphs in $\GG$ is not bounded, we write $\cbdim(\GG)=\infty$. Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} proved some basic properties of this notion. In particular, they showed that if a class $\GG$ has finite comparable box dimension, then it has polynomial strong coloring numbers, which implies that $\GG$ has strongly sublinear separators. They also provided an example showing -that for many functions $h$, the class of graphs with strong coloring numbers bounded by $h$ has infinite +that, for many functions $h$, the class of graphs with strong coloring numbers bounded by $h$ has infinite comparable box dimension\footnote{In their construction $h(r)$ has to be at least 3, and has to tend to $+\infty$.}. Dvo\v{r}\'ak et al.~\cite{wcolig} proved that graphs of comparable box dimension $3$ have exponential weak coloring numbers, giving the first natural graph class with polynomial strong coloring numbers and superpolynomial weak coloring numbers @@ -133,7 +133,7 @@ The comparable box dimension of every proper minor-closed class of graphs is fin Additionally, we show that classes of graphs with finite comparable box dimension are fractionally treewidth-fragile. This gives arbitrarily precise approximation algorithms for all monotone maximization problems that are -expressible in terms of distances between the solution vertices and tractable on graphs of bounded treewidth~\cite{distapx} +expressible in terms of distances between the solution vertices and tractable on graphs of bounded treewidth~\cite{distapx}, or expressible in the first-order logic~\cite{logapx}. \section{Parameters} @@ -145,20 +145,20 @@ For any graph $G$, we have $\omega(G)\le 2^{\cbdim(G)}$. \end{lemma} \begin{proof} We may assume that $G$ has bounded comparable box dimension -witnessed by a representation $f$. To represent any clique $A = \{a_1,\ldots,a_w\}$ in $G$, the +witnessed by a box representation $f$. To represent any clique $A = \{a_1,\ldots,a_w\}$ in $G$, the corresponding boxes $f(a_1),\ldots,f(a_w)$ have pairwise non-empty intersections. Since axis-aligned boxes have the Helly property, there is a point $p \in \mathbb{R}^d$ contained in $f(a_1) \cap \cdots \cap -f(a_w)$. As each box is full-dimensional, its interior intersects at +f(a_w)$. As each box is full-dimensional, their interiors each intersect at least one of the $2^d$ orthants at $p$. At the same time, it follows from the definition of a touching representation that $f(a_1),\ldots,f(a_d)$ have pairwise disjoint interiors, and hence $w \leq 2^d$. \end{proof} Note that a clique with $2^d$ vertices has a touching representation by comparable boxes in $\mathbb{R}^d$, where each vertex is a hypercube defined as the Cartesian product of intervals of form $[-1,0]$ or $[0,1]$. -Together with Lemma~\ref{lemma-cliq}, it follows that $\cbdim(K_{2^d})=d$. +From this together with Lemma~\ref{lemma-cliq}, it follows that $\cbdim(K_{2^d})=d$. -In the following we consider the chromatic number $\chi(G)$, and two +The remaining bounds pertain to the chromatic number $\chi(G)$ of a graph $G$, and two of its variants. An \emph{acyclic coloring} (resp. \emph{star coloring}) of a graph $G$ is a proper coloring such that any two color classes induce a forest (resp. star forest, i.e., a forest in which each component is a star). The \emph{acyclic chromatic number} $\chi_a(G)$ (resp. \emph{star chromatic number} $\chi_s(G)$) of $G$ is the minimum number of colors in an acyclic (resp. star) @@ -175,11 +175,11 @@ Suppose that $G$ has comparable box dimension $d$ witnessed by a representation be the vertices of $G$ written so that $\vol(f(v_1)) \geq \ldots \geq \vol(f(v_n))$. Equivalently, we have $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. Now define a greedy coloring $c$ so that $c(v_i)$ is the smallest color such that $c(v_i)\neq c(v_j)$ for any $j<i$ for which either $v_jv_i\in E(G)$ or there -exists $m>j$ such that $v_jv_m,v_mv_i\in E(G)$. Note that this gives a star coloring, since a path on four vertices always contains a 3-vertex subpath of the form $v_{i_1}v_{i_2}v_{i_3}$ such that $i_1<i_2,i_3$ and our coloring procedure gives distinct colors to vertices forming such a path. +exists $m>j$ such that $v_jv_m,v_mv_i\in E(G)$. Note that this gives a star coloring, since a path on four vertices always contains a 3-vertex subpath of the form $v_{i_1}v_{i_2}v_{i_3}$ such that $i_1<i_2,i_3$, and our coloring procedure gives distinct colors to vertices forming such a path. It remains to bound the number of colors used. Suppose we are coloring $v_i$. We shall bound the number of vertices $v_j$ such that $j<i$ and such that there exists $m>i$ for which $v_jv_m,v_mv_i\in E(G)$. Let $B$ be the box obtained by scaling up $f(v_i)$ by a factor of 5 while keeping the same center. Since $f(v_m)\sqsubseteq f(v_i)\sqsubseteq f(v_j)$, there exists a translation $B_j$ of $f(v_i)$ -contained in $f(v_j)\cap B$ (see Figure~\ref{fig:lowercolcount}). Two boxes $B_{j}$ and $B_{j'}$ for $j\neq j'$ have disjoint interiors since their intersection is contained in the intersection of the touching boxes $f(v_{j})$ and $f(v_{j'})$, and their interiors are also disjoint from $f(v_i)\subset B$. Thus the number of such indices $j$ is at most $\vol(B)/\vol(f(v_i))-1=5^d-1$. +contained in $f(v_j)\cap B$ (see Figure~\ref{fig:lowercolcount}). Two boxes $B_{j}$ and $B_{j'}$ for $j\neq j'$ have disjoint interiors since their intersection is contained in the intersection of the touching boxes $f(v_{j})$ and $f(v_{j'})$, and their interiors are also disjoint from $f(v_i)\subset B$. Thus, the number of such indices $j$ is at most $\vol(B)/\vol(f(v_i))-1=5^d-1$. A similar argument shows that the number of indices $m$ such that $m<i$ and $v_mv_i\in E(G)$ is at most $3^d-1$. Consequently, the number of indices $j<i$ for which there exists $m$ such that $j<m<i$ and $v_jv_m,v_mv_i\in E(G)$ @@ -215,16 +215,14 @@ is at most $(3^d-1)^2$. This means that when choosing the color of $v_i$ greedil \section{Operations} -It is clear that given a touching representation of a graph $G$, one +It is clear that, given a touching representation of a graph $G$, one can easily obtains a touching representation by boxes of an induced subgraph $H$ of $G$ by simply deleting the boxes corresponding to the -vertices in $V(G)\setminus V(H)$. In this section we are going to -consider other basic operations on graphs. In the following, to describe -the boxes, we are going to use the Cartesian product $\times$ defined among boxes ($A\times B$ is the box whose projection on the first dimensions gives the box $A$, while the projection on the remaining dimensions gives the box $B$) or we are going to provide its projections for every dimension ($A[i]$ is the interval obtained from projecting $A$ on its $i^\text{th}$ dimension). +vertices in $V(G)\setminus V(H)$. We shall show that these representations also behave nicely under several other basic operations on graphs. To describe the boxes, we shall use the Cartesian product $\times$ defined among boxes of lower dimension (so that $A\times B$ is the box whose projection on some first number of dimensions gives the box $A$, while the projection on the remaining dimensions gives the box $B$), or specify its projections onto every dimension (and in this case write $A[i]$ for the interval obtained from projecting $A$ on its $i^\text{th}$ dimension). \subsection{Vertex addition}\label{sec-vertad} -Let us start with a simple lemma saying that the addition of a vertex +Let us start with a simple lemma which says that the addition of a vertex increases the comparable box dimension by at most one. In particular, this implies that $\cbdim(G)\le |V(G)|$. \begin{lemma}\label{lemma-apex} @@ -250,7 +248,7 @@ and $v_1$ is equal to or adjacent to $v_2$ in $H$. To obtain a touching representation of $G\boxtimes H$ it suffices to take a product of representations of $G$ and $H$, but the resulting representation may contain incomparable boxes. -Indeed, $\cbdim(G\boxtimes H)$ in general is not bounded by a function +Indeed, in general $\cbdim(G\boxtimes H)$ is not bounded by a function of $\cbdim(G)$ and $\cbdim(H)$; for example, every star has comparable box dimension at most two, but the strong product of the star $K_{1,n}$ with itself contains $K_{n,n}$ as an induced subgraph, and thus its comparable box dimension is at least $\Omega(\log n)$. @@ -264,16 +262,17 @@ of a single box; by scaling, we can without loss of generality assume this box i $\cbdim(G) + d_H$. \end{lemma} \begin{proof} - The proof simply consists in taking a product of the two + It suffices to take a product of the two representations. Indeed, consider a touching respresentation $g$ of $G$ by comparable boxes in $\mathbb{R}^{d_G}$, with $d_G=\cbdim(G)$, and the representation $h$ of $H$. Let us define a - representation $f$ of $G\boxtimes H$ in $\mathbb{R}^{d_G+d_H}$ as - follows. - \[f((u,v))[i]=\begin{cases} + representation $f$ of $G\boxtimes H$ in $\mathbb{R}^{d_G+d_H}$ by + \begin{equation*} + f((u,v))[i]=\begin{cases} g(u)[i]&\text{ if $i\le d_G$}\\ - h(v)[i-d_G]&\text{ if $i > d_G$} - \end{cases}\] + h(v)[i-d_G]&\text{ if $i > d_G$.} + \end{cases} + \end{equation*} Consider distinct vertices $(u,v)$ and $(u',v')$ of $G\boxtimes H$. The boxes $g(u)$ and $g(u')$ are comparable, say $g(u)\sqsubseteq g(u')$. Since $h(v')$ is a translation of $h(v)$, this implies that $f((u,v))\sqsubseteq f((u',v'))$. Hence, the boxes @@ -282,11 +281,11 @@ of a single box; by scaling, we can without loss of generality assume this box i The boxes of the representations $g$ and $h$ have pairwise disjoint interiors. Hence, if $u\neq u'$, then there exists $i\le d_G$ such that the interiors of the intervals $f((u,v))[i]=g(u)[i]$ and $f((u',v'))[i]=g(u')[i]$ are disjoint; - and if $v\neq v'$, then there exists $i\le d_H$ such that the interiors + if $v\neq v'$, then there exists $i\le d_H$ such that the interiors of the intervals $f((u,v))[i+d_G]=h(v)[i]$ and $f((u',v'))[i+d_G]=h(v')[i]$ are disjoint. Consequently, the interiors of boxes $f((u,v))$ and $f((u',v'))$ are pairwise disjoint. Moreover, if $u\neq u'$ and $uu'\not\in E(G)$, or if $v\neq v'$ and $vv'\not\in E(G)$, - then the intervals discussed above (not just their interiors) are disjoint for some $i$; + then the aforementioned intervals (not just their interiors) are disjoint for some $i$; hence, if $(u,v)$ and $(u',v')$ are not adjacent in $G\boxtimes H$, then $f((u,v))\cap f((u',v'))=\emptyset$. Therefore, $f$ is a touching representation of a subgraph of $G\boxtimes H$. @@ -301,8 +300,8 @@ of a single box; by scaling, we can without loss of generality assume this box i \subsection{Taking a subgraph} -The comparable box dimension of a subgraph of a graph $G$ may be larger than $\cbdim(G)$, see the end of this -section for an example. However, we show that the +The comparable box dimension of a subgraph of a graph $G$ may be larger than $\cbdim(G)$ (see the end of this +section for an example). However, we show that the comparable box dimension of a subgraph is at most exponential in the comparable box dimension of the whole graph. This is essentially Corollary~25 in~\cite{subconvex}, but since the setting is somewhat @@ -313,7 +312,7 @@ we provide details of the argument. If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+\frac12 \chi^2_s(G')$. \end{lemma} \begin{proof} -As we can remove the boxes that represent the vertices, we can assume $V(G')=V(G)$. +By removing boxes that represent vertices of $G$ that are not in $G'$, we may assume that $V(G')=V(G)$. Let $f$ be a touching representation of $G'$ by comparable boxes in $\mathbb{R}^d$, where $d=\cbdim(G')$. Let $\varphi$ be a star coloring of $G'$ using colors $\{1,\ldots,c\}$, where $c=\chi_s(G')$. @@ -350,8 +349,8 @@ Let us now combine Lemmas~\ref{lemma-chrom} and \ref{lemma-subg}. If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+2\cdot 81^{\cbdim(G')}\le 3\cdot 81^{\cbdim(G')}$. \end{corollary} -Let us remark that an exponential increase in the dimension is unavoidable: We have $\cbdim(K_{2^d})=d$, -but the graph obtained from $K_{2^d}$ by deleting a perfect matching has comparable box dimension $2^{d-1}$. Indeed, for every pair $u,v$ of non-adjacent vertices there is a specific dimension $i$ such that their boxes span intervals $[a,b]$ and $[c,d]$ with $b<c$, while for every other box in the representation their $i^\text{th}$ interval contains $[b,c]$. +An exponential increase in the dimension is unavoidable: we have $\cbdim(K_{2^d})=d$, +but the graph obtained from $K_{2^d}$ by deleting a perfect matching has comparable box dimension $2^{d-1}$. Indeed, for every pair $u,v$ of non-adjacent vertices there is a specific dimension $i$ such that their boxes span intervals $[a,b]$ and $[c,d]$ with $b<c$, while the $i^\text{th}$ interval of every other box in the representation contains $[b,c]$. \subsection{Clique-sums} @@ -363,8 +362,8 @@ edges of the resulting clique. A \emph{full clique-sum} is a clique-sum in which we keep all the edges of the resulting clique. The main issue to overcome in obtaining a representation for a (full) clique-sum is that the representations of $G_1$ and $G_2$ can be -``degenerate''. Consider e.g.\ the case that $G_1$ is represented by -unit squares arranged in a grid; in this case, there is no space to +``degenerate''. Consider, for example, the case where $G_1$ is represented by +unit squares arranged in a grid; here there is no space to attach $G_2$ at the cliques formed by four squares intersecting in a single corner. This can be avoided by increasing the dimension, but we need to be careful so that the dimension stays bounded even after @@ -408,7 +407,7 @@ $\emptyset$. Let $\ecbdim(G)$ be the minimum dimension such that $G$ has an $\emptyset$-clique-sum extendable touching representation by comparable boxes. -Let us remark that a clique-sum extendable representation in dimension $d$ implies +Let us remark that a clique-sum extendable representation in dimension $d$ implies the existence of such a representation in higher dimensions as well. \begin{lemma}\label{lemma-add} Let $G$ be a graph with a root clique $C^\star$ and let $h$ be @@ -456,8 +455,8 @@ behave well with respect to full clique-sums. without loss of generality, the \textbf{(vertices)} conditions are satisfied by setting $d_{v_i}=i$ for $i\in\{1,\ldots,k\}$ - We are now ready to define $h$. For $v\in V(G_1)$, we set $h(v)=h_1(v)$. - We now scale and translate $h_2$ to fit inside $h_1^\varepsilon(C_1)$. +To define $h$, for $v\in V(G_1)$, set $h(v)=h_1(v)$. + Then scale and translate $h_2$ to fit inside $h_1^\varepsilon(C_1)$. That is, we fix $\varepsilon>0$ small enough so that \begin{itemize} \item the conditions \textbf{(cliques)} hold for $h_1$, @@ -468,19 +467,19 @@ behave well with respect to full clique-sums. we set $h(v)[i]=p_1(C_1)[i] + \varepsilon h_2(v)[i]$ for $i\in\{1,\ldots,d\}$. Note that the condition (v2) for $h_2$ implies $h(v)\subset h_1^\varepsilon(C_1)$. Each clique $C$ of $H$ is a clique of $G_1$ or $G_2$. - If $C$ is a clique of $G_2$, we set $p(C)=p_1(C_1)+\varepsilon p_2(C)$, + If $C$ is a clique of $G_2$ then we set $p(C)=p_1(C_1)+\varepsilon p_2(C)$, otherwise we set $p(C)=p_1(C)$. In particular, for subcliques of $C_1=C^\star_2$, we use the former choice. Let us now check that $h$ is a $C^\star_1$-clique sum extendable - representation by comparable boxes. The fact that the boxes are + representation by comparable boxes. Firstly, the fact that the boxes are comparable follows from the fact that those of $h_1$ and $h_2$ - are comparable and from the scaling of $h_2$: By construction both + are comparable and from the scaling of $h_2$: by construction both $h_1(v) \sqsubseteq h_1(u)$ and $h_2(v) \sqsubseteq h_2(u)$ imply $h(v) \sqsubseteq h(u)$, and for any vertex $u\in V(G_1)$ and any vertex $v\in V(G_2) \setminus V(C^\star_2)$, we have $h(v) \subset h_1^\varepsilon(C_1) \sqsubseteq h(u)$. - We now check that $h$ is a contact representation of $G$. For $u,v + Next, we check that $h$ is a contact representation of $G$. For $u,v \in V(G_1)$ (resp. $u,v \in V(G_2) \setminus V(C^\star_2)$) it is clear that $h(u)$ and $h(v)$ have disjoint interiors, and that they intersect if and only if $h_1(u)$ and $h_1(v)$ intersect (resp. if @@ -498,7 +497,7 @@ behave well with respect to full clique-sums. by (c2) for $h_1$, we have $h_1^\varepsilon(C_1)[i]\subseteq h_1(u)[i]=h(u)[i]$. Since $h(v)[i]\subseteq h_1^\varepsilon(C_1)[i]$, it follows that $h(u)$ intersects $h(v)$. - Finally, let us consider the $C^\star_1$-clique-sum extendability. The \textbf{(vertices)} + Finally, we verify the conditions for $C^\star_1$-clique-sum extendability. The \textbf{(vertices)} conditions hold, since (v0) and (v1) are inherited from $h_1$, and (v2) is inherited from $h_1$ for $v\in V(G_1)\setminus V(C^\star_1)$ and follows from the fact that $h(v)\subseteq h_1^\varepsilon(C_1)\subset [0,1)^d$ @@ -509,7 +508,7 @@ behave well with respect to full clique-sums. On the other hand, if $C'$ is a clique of $G_1$ not contained in $C_1$, then there exists $v\in V(C')\setminus V(C_1)$, we have $p(C')=p_1(C')\in h_1(v)$, and $h_1(v)\cap h_1^\varepsilon(C_1)=\emptyset$ by (c2) for $h_1$. - Therefore, the mapping $p$ is injective, and thus for sufficiently small $\varepsilon'>0$, + Therefore, the mapping $p$ is injective, and thus for sufficiently small $\varepsilon'>0$ we have $h^{\varepsilon'}(C)\cap h^{\varepsilon'}(C')=\emptyset$ for any distinct cliques $C$ and $C'$ of $G$. @@ -557,11 +556,11 @@ the dimension by $\omega(G)$. \end{lemma} \begin{proof} - The proof is essentially the same as the one of + The proof is essentially the same as that of Lemma~\ref{lemma-apex}. Consider a $\emptyset$-clique-sum extendable touching representation $h'$ of $G\setminus V(C^\star)$ by comparable boxes in $\mathbb{R}^{d'}$, with $d' = \cbdim(G\setminus - V(C^\star))$, and let $V(C^\star) = \{v_1,\ldots,v_k\}$. We now construct + V(C^\star))$, and let $V(C^\star) = \{v_1,\ldots,v_k\}$. We construct the desired representation $h$ of $G$ as follows. For each vertex $v_i\in V(C^\star)$, let $h(v_i)$ be the box in $\mathbb{R}^d$ uniquely determined by the condition (v1) with $d_{v_i} = i$. For each vertex $u\in V(G)\setminus V(C^\star)$, @@ -589,8 +588,8 @@ the dimension by $\omega(G)$. For the \textbf{(cliques)} condition (c2), let us first consider a vertex $v\in V(G)\setminus V(C^\star)$ and a clique $C$ of $G$ containing $v$. In the dimensions $i\in\{1,\ldots,k\}$, we always have $h^\varepsilon(C)[i] \subseteq h(v)[i]$. Indeed, if $v_i \in V(C)$, then - $h^\varepsilon(C)[i] \subseteq [0,1/2] = h(v)[i]$, as in this case $v$ and $v_i$ are adjacent; - and if $v_i \notin V(C)$, then $h^\varepsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$. + $h^\varepsilon(C)[i] \subseteq [0,1/2] = h(v)[i]$, as in this case $v$ and $v_i$ are adjacent. + If instead $v_i \notin V(C)$, then $h^\varepsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$. By the property (c2) of $h'$, we have $h^\varepsilon(C)[i] \subseteq h(v)[i]$ for every $i>k$, except one, for which $h^\varepsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$. @@ -603,7 +602,7 @@ the dimension by $\omega(G)$. we have that $h^\varepsilon(C)[i] \cap h(v_i)[i] = [0,\varepsilon]\cap [-1,0] = \{0\}$, and $h^\varepsilon(C)[j] \subseteq [0,1] = h(v_i)[j]$ for any $j\neq i$. For a clique $C$ that does not contain $v_i$ we have that $h^\varepsilon(C)[i] \cap h(v_i)[i] \subset (0,1)\cap [-1,0] = \emptyset$. - Condition (c2) is thus fulfilled and this completes the proof of the lemma. + Condition (c2) is therefore fulfilled, which completes the proof of the lemma. \end{proof} @@ -621,7 +620,7 @@ of $\cbdim(G)$ and $\chi(G)$. sufficiently small real $\alpha>0$ we increase each box in $h$ by $2\alpha$ in every dimension, that is we replace $h(v)[i] = [a,b]$ by $[a-\alpha,b+\alpha]$ for each vertex $v$ and dimension - $i$. We choose $\alpha$ sufficiently small so that the boxes representing + $i$. Here, we choose $\alpha$ to be sufficiently small so that the boxes representing non-adjacent vertices remain disjoint, and thus the resulting representation $h_1$ is an intersection representation of the same graph $G$. Moreover, observe that for every clique $C$ of $G$, the intersection $I_C=\bigcap_{v\in V(C)} h_1(v)$ is @@ -631,19 +630,19 @@ of $\cbdim(G)$ and $\chi(G)$. Now we add $\chi(G)$ dimensions to make the representation touching again, and to ensure some space for the clique boxes - $h^\varepsilon(C)$. Formally we define $h_2$ as follows. + $h^\varepsilon(C)$. Formally we define $h_2$ as \[h_2(u)[i]=\begin{cases} h_1(u)[i]&\text{ if $i\le d$}\\ [1/5,3/5]&\text{ if $i>d$ and $c(u) < i-d$}\\ [0,2/5]&\text{ if $i>d$ and $c(u) = i-d$}\\ - [2/5,4/5]&\text{ otherwise (if $c(u) > i-d > 0$)} + [2/5,4/5]&\text{ otherwise (if $c(u) > i-d > 0$).} \end{cases}\] For any clique $C$ of $G$, let $c(C)$ denote the color set $\{c(u)\ |\ u\in V(C)\}$. We now set \[p_2(C)[i]=\begin{cases} p_1(C)[i] &\text{ if $i\le d$}\\ 2/5 &\text{ if $i>d$ and $i-d \in c(C)$}\\ - 1/2 &\text{ otherwise} + 1/2 &\text{ otherwise.} \end{cases} \] As $h_2$ is an extension of $h_1$, and as in each dimension $j>d$, @@ -671,7 +670,7 @@ of $\cbdim(G)$ and $\chi(G)$. \end{proof} A touching representation of axis-aligned boxes in $\mathbb{R}^d$ is said \emph{fully touching} if any two intersecting boxes intersect on a $(d-1)$-dimensional box. Note that the construction above is fully touching. -Indeed, two intersecting boxes corresponding to vertices $u,v$ of colors $c(u) < c(v)$, only touch at coordinate $2/5$ in the $(d+c(u))^\text{th}$ dimension, while they fully intersect in every other dimension. This observation with Lemma~\ref{lemma-chrom} lead to the following. +Indeed, two intersecting boxes corresponding to vertices $u,v$ of colors $c(u) < c(v)$, only touch at coordinate $2/5$ in the $(d+c(u))^\text{th}$ dimension, while they fully intersect in every other dimension. This observation with Lemma~\ref{lemma-chrom} leads to the following. \begin{corollary} \label{cor-fully-touching} Any graph $G$ has a fully touching representation of comparable axis-aligned boxes in $\mathbb{R}^d$, where $d= \cbdim(G) + 3^{\cbdim(G)}$. @@ -683,11 +682,11 @@ full clique-sums. \begin{corollary} \label{cor-csum} Let $\GG$ be a class of graphs of chromatic number at most $k$. If $\GG'$ is the class -of graphs obtained from $\GG$ by repeatedly performing full clique-sums, then -\[\cbdim(\GG')\le \cbdim(\GG) + 2k.\] +of all graphs that can be obtained from $\GG$ by repeatedly performing full clique-sums, then +$\cbdim(\GG')\le \cbdim(\GG) + 2k.$ \end{corollary} \begin{proof} -Suppose a graph $G$ is obtained from $G_1, \ldots, G_m\in\GG$ by performing full clique-sums. +Suppose a graph $G$ is obtained from $G_1, \ldots, G_m\in\GG$ by a sequence of full clique-sums. Without loss of generality, the labelling of the graphs is chosen so that we first perform the full clique-sum on $G_1$ and $G_2$, then on the resulting graph and $G_3$, and so on. Let $C^\star_1=\emptyset$ and for $i=2,\ldots,m$, let $C^\star_i$ be the root clique of $G_i$ on which it is @@ -697,7 +696,7 @@ where $d=\cbdim(\GG) + 2k$. Repeatedly applying Lemma~\ref{lem-cs}, we conclude $\cbdim(G)\le d$. \end{proof} -By Lemmas~\ref{lemma-chrom} and \ref{lemma-subg}, this gives the following bounds. +Putting the preceding corollary together with Lemma~\ref{lemma-chrom} and Lemma~\ref{lemma-subg}, we now have the following bounds. \begin{corollary}\label{cor-csump} Let $\GG$ be a class of graphs of comparable box dimension at most $d$. \begin{itemize} @@ -709,7 +708,7 @@ has comparable box dimension at most $1250^d$. \end{corollary} \begin{proof} The former bound directly follows from Corollary~\ref{cor-csum} and the bound on the chromatic number -from Lemma~\ref{lemma-chrom}. For the latter one, we need to bound the star chromatic number of $\GG'$. +from Lemma~\ref{lemma-chrom}. For the latter, we need to bound the star chromatic number of $\GG'$. Suppose a graph $G$ is obtained from $G_1, \ldots, G_m\in\GG$ by performing full clique-sums. For $i=1,\ldots, m$, suppose $G_i$ has an acyclic coloring $\varphi_i$ by at most $k$ colors. Note that the vertices of any clique get pairwise different colors, and thus by permuting the colors, @@ -783,7 +782,7 @@ and that for $j\neq i$, $p(C)[j]$ is in the interior of $h(v_i)[j]$, implying $h(v_i)[j] \cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon]$ for sufficiently small $\varepsilon>0$. \end{proof} The \emph{treewidth} $\tw(G)$ of a graph $G$ is the minimum $k$ such that $G$ is a subgraph of a $k$-tree. -Note that actually the bound on the comparable box dimension of Theorem~\ref{thm-ktree} +It is worth noting that the bound on the comparable box dimension of Theorem~\ref{thm-ktree} actually extends to graphs of treewidth at most $k$. \begin{corollary}\label{cor-tw} Every graph $G$ satisfies $\cbdim(G)\le\tw(G)+1$. @@ -841,9 +840,8 @@ Suppose $G$ is a connected planar graph and $v$ is a vertex of $G$. For an inte give each vertex at distance $d$ from $v$ the color $d\bmod k$. Then deleting the vertices of any of the $k$ colors results in a graph of treewidth at most $3k$. This fact (which follows from the result of Robertson and Seymour~\cite{rs3} on treewidth of planar graphs of bounded radius) is (in the modern terms) the basis of Baker's technique~\cite{baker1994approximation} -for design of approximation algorithms. However, even quite simple graph classes (e.g., strong products of three paths~\cite{gridtw}) -do not admit such a coloring (where the removal of any color class results in a graph of bounded treewidth). -However, a fractional version of this coloring concept is still very useful in the design of approximation algorithms~\cite{distapx} +for design of approximation algorithms. However, even quite simple graph classes, such as the strong products of three paths~\cite{gridtw}, do not admit such a coloring where the removal of any color class results in a graph of bounded treewidth. +Nonetheless, a fractional version of this coloring concept is still very useful in the design of approximation algorithms~\cite{distapx} and applies to much more general graph classes, including all graph classes with strongly sublinear separators and bounded maximum degree~\cite{twd}. We say that a class of graphs $\GG$ is \emph{fractionally treewidth-fragile} if there exists a function $f$ such that @@ -874,23 +872,26 @@ Our main result is that all graph classes of bounded comparable box dimension ar We will show the result in a more general setting, motivated by concepts from~\cite{subconvex} and by applications to related representations. The argument is motivated by the idea used in the approximation algorithms for disk graphs by Erlebach et al.~\cite{erlebach2005polynomial}. Before introducing this more general setting, and as a warm-up, let us outline -how to prove that disk graphs of thickness $t$ are fractionally treewidth-fragile. Consider first unit disk graphs. -By partitionning the plane with a random grid $\HH$, having squared cells of side-length $2k$, any unit disk has probability $1/2k$ -to intersect a vertical (resp. horizontal) line of the grid. By union bound, any disk has probability at most $1/k$ to intersect -the grid. Considering this probability distribution, let us now show that removing the disks intersected by the grid leads to a +how to prove that disk graphs of thickness $t$ are fractionally treewidth-fragile. + +We first consider unit disk graphs. +By partitioning the plane with a random grid $\HH$ with square cells of side-length $2k$, any unit disk has probability $1/2k$ +of intersecting a vertical (resp. horizontal) line of the grid. Using a union bound, any disk has probability at most $1/k$ of intersecting +the grid. Using this probability distribution, we show that removing the disks intersected by the grid leads to a unit disk graph of bounded treewidth. Indeed, in such a graph any connected component corresponds to unit disks contained in the -same cell of the grid. Such cell having area bounded by $4k^2$, there are at most $16tk^2/\pi$ disks contained in a cell. -The size of the connected components being bounded, so is the treewidth. Note that this distribution also works if we are given -disks whose diameter lie in a certain range. If any diameter $\delta$ is such that $1/c \le \delta \le 1$, then the same process -with a random grid of $2k\times 2k$ cells, ensures that any disk is deleted with probability at most $1/k$, while now the -connected components have size at most $4tc^2k^2/\pi$. Dealing with arbitrary disk graphs (with any diameter $\delta$ being in the range -$0< \delta \le 1$) requires to delete more disks. This is why each $(2k\times 2k)$-cell is now partitionned in a quadtree-like manner. +same cell of the grid. Each cell has area bounded by $4k^2$, so there are at most $16tk^2/\pi$ disks contained in a cell. +This bounds the size of any connected component, and so the treewidth is also bounded. + +Note that the above distribution also works if we are given +disks whose diameter lie in a certain range. That is, for any diameter $\delta$ with $1/c \le \delta \le 1$, applying the same process +with a random grid of $2k\times 2k$ cells ensures that any disk is deleted with probability at most $1/k$, and the +connected components have size at most $4tc^2k^2/\pi$. Dealing with arbitrary disk graphs (with any diameter $\delta$ in the range +$0< \delta \le 1$) necessitates deleting more disks. This can be handled by partitioning each $(2k\times 2k)$-cell in a quadtree-like manner. Now a disk with diameter between $\ell /2$ and $\ell$ (with $\ell =1/2^i$ for some integer $i\ge 0$) is deleted if it is not contained -in a $(2k\ell \times 2k\ell)$-cell of a quadtree. It is not hard to see that a disk is deleted with probability at most $1/k$. -To prove that the remaining graph has bounded treewidth one should consider the following tree decomposition $(T,\beta)$. The -tree $T$ is obtained by linking the roots of the quadtrees we used (as trees) to a new common root. -Then for a $(2k\ell \times 2k\ell)$-cell $C$, $\beta(C)$ contains all the disks of diameter at least $\ell/2$ intersecting $C$. -To see that such bag is bounded consider the $((2k+1)\ell \times (2k+1)\ell)$ square $C'$ centered on $C$, and note that any +in a $(2k\ell \times 2k\ell)$-cell of a quadtree. It is straightforward to see that each disk is deleted with probability at most $1/k$. +To prove that the remaining graph has bounded treewidth, one should consider the following tree decomposition $(T,\beta)$. Here, the tree $T$ is obtained by linking the roots of the quadtrees used (as trees) to a new common root. +Then for a $(2k\ell \times 2k\ell)$-cell $C$, $\beta(C)$ contains all disks of diameter at least $\ell/2$ intersecting $C$. +To see that such bag is bounded, consider the $((2k+1)\ell \times (2k+1)\ell)$ square $C'$ centered on $C$, and note that any disk in $\beta(C)$ intersects $C'$ on an area at least $\pi\ell^2/16$. This implies that $|\beta(C)| \le 16t(2k+1)^2 / \pi$. Let us now give a detailed proof in a more general setting. @@ -927,8 +928,8 @@ is fractionally treewidth-fragile, with a function $f(k) = O_{t,s,d}\bigl(k^{d}\ For a positive integer $k$, let $f(k)=(2ksd+2)^dst$. Let $(\iota,\omega)$ be an $s$-comparable envelope representation of a graph $G$ in $\mathbb{R}^d$ of thickness at most $t$, and let $v_1$, \ldots, $v_n$ be the corresponding ordering of the vertices of $G$. -Let us define $\ell_{i,j}\in \mathbb{R}^+$ for $i=1,\ldots, n$ and $j\in\{1,\ldots,d\}$ as an approximation of $ksd|\omega(v_i)[j]|$ such that $\ell_{i-1,j} / \ell_{i,j}$ is a positive integer. Formally -it is defined as follows. +Let us define $\ell_{i,j}\in \mathbb{R}^+$ for $i=1,\ldots, n$ and $j\in\{1,\ldots,d\}$ as an approximation of $ksd|\omega(v_i)[j]|$ such that $\ell_{i-1,j} / \ell_{i,j}$ is a positive integer. Formally, +it is defined by the following process. \begin{itemize} \item Let $\ell_{1,j}=ksd|\omega(v_1)[j]|$. \item For $i=2,\ldots, n$, let $\ell_{i,j} = \ell_{i-1,j}$, if @@ -938,17 +939,17 @@ it is defined as follows. \min\{\ell_{i-1,j}/b \ |\ b\in \mathbb{N}^+ \text{ and } \ell_{i-1,j}/b \ge ksd|\omega(v_i)[j]|\}$. \end{itemize} -Let the real $x_j\in [0,\ell_{1,j}]$ be chosen uniformly at random, +Choose $x_j\in [0,\ell_{1,j}]$ uniformly at random, and let $\HH^i_j$ be the set of hyperplanes in $\mathbb{R}^d$ consisting of the points whose $j$-th coordinate is equal to $x_j+m\ell_{i,j}$ for some $m\in\mathbb{Z}$. As $\ell_{i,j}$ is a multiple of $\ell_{i',j}$ whenever $i\le i'$, we have that $\HH^i_j \subseteq \HH^{i'}_j$ whenever $i\le i'$. For $i\in\{1,\ldots,n\}$, the \emph{$i$-grid} is $\HH^i=\bigcup_{j=1}^d \HH^i_j$, and we let the -$0$-grid $\HH^0=\emptyset$. Similarly as above we have that $\HH^i +$0$-grid $\HH^0=\emptyset$. Then, as above, we have that $\HH^i \subseteq \HH^{i'}$ whenever $i\le i'$. -We let $X\subseteq V(G)$ consist of the vertices $v_a\in V(G)$ such +Let $X\subseteq V(G)$ consist of the vertices $v_a\in V(G)$ such that the box $\omega(v_a)$ intersects some hyperplane $H\in \HH^a$, that is such that $x_j+m\ell_{a,j}\in \omega(v_a)[j]$, for some $j\in\{1,\ldots,d\}$ and some $m\in \mathbb{Z}$. First, let us argue @@ -962,13 +963,13 @@ Combining these inequalities, \[\frac{|\omega(v_a)[j]|}{\ell_{a,j}}\le \frac{s\omega(v_{a'})[j]}{ksd|\omega(v_{a'})[j]|}=\frac{1}{kd}.\] By the union bound, we conclude that $\text{Pr}[v_a\in X]\le 1/k$. -Let us now bound the treewidth of $G-X$. +We now bound the treewidth of $G-X$. For $a\ge 0$, an \emph{$a$-cell} is a maximal connected subset of $\mathbb{R}^d\setminus \bigl(\bigcup_{H\in \HH^a} H\bigr)$. A set $C\subseteq\mathbb{R}^d$ is a \emph{cell} if it is an $a$-cell for some $a\ge 0$. A cell $C$ is \emph{non-empty} if there exists $v\in V(G-X)$ such that $\iota(v)\subseteq C$. Note that there exists a rooted tree $T$ whose vertices are the non-empty cells and such that for $x,y\in V(T)$, we have $x\preceq y$ if and only if $x\subseteq y$. -For each non-empty cell $C$, let us define $\beta(C)$ as the set of vertices $v_i\in V(G-X)$ such that +For each non-empty cell $C$, define $\beta(C)$ to be the set of vertices $v_i\in V(G-X)$ such that $\iota(v)\cap C\neq\emptyset$ and $C$ is an $a$-cell for some $a\ge i$. Let us show that $(T,\beta)$ is a tree decomposition of $G-X$. For each $v_j\in V(G-X)$, the $j$-grid is disjoint from $\omega(v_j)$, @@ -977,10 +978,10 @@ We have $\omega(v_j)\cap \iota(v_i)\neq\emptyset$, and thus $\iota(v_i)\cap C\ne Finally, suppose that $v_j\in C'$ for some $C'\in V(T)$. Then $C'$ is an $a$-cell for some $a\ge j$, and since $\iota(v_j)\cap C'\neq\emptyset$ and $\iota(v_j)\subset C$, we conclude that $C'\subseteq C$, and consequently $C'\preceq C$. Moreover, any cell $C''$ such that $C'\preceq C''\preceq C$ (and thus $C'\subseteq C''\subseteq C$) is an $a'$-cell -for some $a'\ge j$ and $\iota(v_j)\cap C''\supseteq \iota(v_j)\cap C'\neq\emptyset$, implying $v_j\in\beta(C'')$. +for some $a'\ge j$ and $\iota(v_j)\cap C''\supseteq \iota(v_j)\cap C'\neq\emptyset$, which implies that $v_j\in\beta(C'')$. It follows that $\{C':v_j\in\beta(C')\}$ induces a connected subtree of $T$. -Finally, let us bound the width of the decomposition $(T,\beta)$. Let $C$ be a non-empty cell and let $a$ be maximum such that $C$ +Finally, we bound the width of the decomposition $(T,\beta)$. Let $C$ be a non-empty cell and let $a$ be maximum number for which $C$ is an $a$-cell. Then $C$ is an open box with sides of lengths $\ell_{a,1}$, \ldots, $\ell_{a,d}$. Consider $j\in\{1,\ldots,d\}$: \begin{itemize} \item If $a=1$, then $\ell_{a,j}=ksd |\omega(v_a)[j]|$. @@ -988,11 +989,11 @@ is an $a$-cell. Then $C$ is an open box with sides of lengths $\ell_{a,1}$, \ld \item If $a>1$ and $\ell_{a,j} < \ell_{a-1,j}$, then $\ell_{a-1,j}\ge b\times ksd|\omega(v_a)[j]|$ for some integer $b\ge 2$. Now let $b$ be the greatest such integer (that is such that $\ell_{a-1,j} < (b+1)\times ksd|\omega(v_a)[j]|$) and note that \[\ell_{a,j}=\frac{\ell_{a-1,j}}{b}<\tfrac{b+1}{b}ksd|\omega(v_a)[j]|<\tfrac{3}{2}ksd|\omega(v_a)[j]|.\] \end{itemize} -Hence, $\ell_{a,j}<2ksd |\omega(v_a)[j]|$. Let $C'$ be the box with the same center as $C$ and with $|C'[j]|=(2ksd+2)|\omega(v_a)[j]|$. +Hence, in all cases we have $\ell_{a,j}<2ksd |\omega(v_a)[j]|$. Let $C'$ be the box with the same center as $C$ and with $|C'[j]|=(2ksd+2)|\omega(v_a)[j]|$. For any $v_i\in \beta(C)\setminus\{v_a\}$, we have $i\le a$ and $\iota(v_i)\cap C\neq\emptyset$, and since $\omega(v_a)\sqsubseteq_s \iota(v_i)$, there exists a translation $B_i$ of $\omega(v_a)$ that intersects $C\cap \iota(v_i)$ and such that $\vol(B_i\cap\iota(v_i))\ge \tfrac{1}{s}\vol(\omega(v_a))$. Note that as $B_i$ intersects $C$, we have that $B_i\subseteq C'$. -Since the representation has thickness at most $t$, +Using the initial assumption that the representation has thickness at most $t$, we now have \begin{align*} \vol(C')&\ge \vol\left(C'\cap \bigcup_{v_i\in \beta(C)\setminus\{v_a\}} \iota(v_i)\right)\\ &\ge \vol\left(\bigcup_{v_i\in \beta(C)\setminus\{v_a\}} B_i\cap\iota(v_i)\right)\\