A box is the cartesian product of real intervals, which are either bounded or equal to $\mathbb{R}$. A box is said to be $d$-local if at most $d$ of the intervals are bounded. In this paper, we investigate the recently introduced local boxicity of a graph $G$, which is the minimum $d$ such that $G$ can be represented as the intersection of $d$-local boxes in some dimension. We prove that all graphs of maximum degree $Δ$ have local boxicity $O(Δ)$, while almost all graphs of maximum degree $Δ$ have local boxicity $Ω(Δ)$, improving known upper and lower bounds. We also give improved bounds on the local boxicity as a function of the number of edges or the genus. Finally, we investigate local boxicity through the lens of chromatic graph theory. We prove that the family of graphs of local boxicity at most 2 is $χ$-bounded, which means that the chromatic number of the graphs in this class can be bounded by a function of their clique number. This extends a classical result on graphs of boxicity at most 2.
The boxicity of a graph $G$ is the minimum non-negative integer $k$ such that $G$ can be isomorphic to the intersection graph of a family of boxes in Euclidean $k$-space, where a box in Euclidean $k$-space is the Cartesian product of $k$ closed intervals on the real line. In this note, we define the fractional boxicity of a graph as the optimum value of the linear relaxation of a covering problem with respect to boxicity, which gives a lower bound for its boxicity. We show that the fractional boxicity of a graph is at least the lower bounds for boxicity given by Adiga et al. in 2014. We also p…
The boxicity of a graph $G$ is the least integer $d$ such that $G$ has an intersection model of axis-aligned $d$-dimensional boxes. Boxicity, the problem of deciding whether a given graph $G$ has boxicity at most $d$, is NP-complete for every fixed $d \ge 2$. We show that boxicity is fixed-parameter tractable when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Adiga et al., that boxicity is fixed-parameter tractable in the vertex cover number. Moreover, we show that boxicity admits an additive $1$-approximation when parameterized by the p…
The boxicity of a graph $G=(V,E)$ is the least integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap ... \cap E_k$. Scheinerman proved in 1984 that outerplanar graphs have boxicity at most two and Thomassen proved in 1986 that planar graphs have boxicity at most three. In this note we prove that the boxicity of toroidal graphs is at most 7, and that the boxicity of graphs embeddable in a surface $Σ$ of genus $g$ is at most $5g+3$. This result yields improved bounds on the dimension of the adjacency poset of graphs on surfaces.
The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity.
The boxicity of a graph $G=(V,E)$ is the smallest integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap \cdots \cap E_k$. In the first part of this note, we prove that every graph on $m$ edges has boxicity $O(\sqrt{m \log m})$, which is asymptotically best possible. We use this result to study the connection between the boxicity of graphs and their Colin de Verdière invariant, which share many similarities. Known results concerning the two parameters suggest that for any graph $G$, the boxicity of $G$ is at most the Colin de Verdière invari…
An axis-parallel $b$-dimensional box is a Cartesian product $R_1\times R_2\times...\times R_b$ where $R_i$ is a closed interval of the form $[a_i,b_i]$ on the real line. For a graph $G$, its \emph{boxicity} box(G) is the minimum dimension $b$, such that $G$ is representable as the intersection graph of boxes in $b$-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which …
In this paper, we relate the seemingly unrelated concepts of treewidth and boxicity. Our main result is that, for any graph G, boxicity(G) <= treewidth(G) + 2. We also show that this upper bound is (almost) tight. Our result leads to various interesting consequences, like bounding the boxicity of many well known graph classes, such as chordal graphs, circular arc graphs, AT-free graphs, co--comparability graphs etc. All our bounds are shown to be tight up to small constant factors. An algorithmic consequence of our result is a linear time algorithm to construct a box representation for graphs …
The three well-known graph classes, planar graphs (P), series-parallel graphs(SP) and outer planar graphs(OP) satisfy the following proper inclusion relation: OP C SP C P. It is known that box(G) <= 3 if G belongs to P and box(G) <= 2 if G belongs to OP. Thus it is interesting to decide whether the maximum possible value of the boxicity of series-parallel graphs is 2 or 3. In this paper we construct a series-parallel graph with boxicity 3, thus resolving this question. Recently Chandran and Sivadasan showed that for any G, box(G) <= treewidth(G)+2. They conjecture that for any k, there exists …
The boxicity of a graph G, denoted as box(G) is defined as the minimum integer t such that G is an intersection graph of axis-parallel t-dimensional boxes. A graph G is a k-leaf power if there exists a tree T such that the leaves of the tree correspond to the vertices of G and two vertices in G are adjacent if and only if their corresponding leaves in T are at a distance of at most k. Leaf powers are a subclass of strongly chordal graphs and are used in the construction of phylogenetic trees in evolutionary biology. We show that for a k-leaf power G, box(G)\leq k-1. We also show the tightness …
For a graph $G$, its \emph{cubicity} $cub(G)$ is the minimum dimension $k$ such that $G$ is representable as the intersection graph of (axis--parallel) cubes in $k$--dimensional space. Chandran, Mannino and Oriolo showed that for a $d$--dimensional hypercube $H_d$, $\frac{d-1}{\log d} \le cub(H_d) \le 2d$. In this paper, we show that $cub(H_d) = Θ(\frac{d}{\log d})$.The parameter \emph{boxicity} generalizes cubicity: the boxicity $box(G)$ of a graph $G$ is defined as the minimum dimension $k$ such that $G$ is representable as the intersection graph of axis parallel boxes in $k$ dimensional spa…
In this short note, we relate the boxicity of graphs (and the dimension of posets) with their generalized coloring parameters. In particular, together with known estimates, our results imply that any graph with no $K_t$-minor can be represented as the intersection of $O(t^2\log t)$ interval graphs (improving the previous bound of $O(t^4)$), and as the intersection of $\tfrac{15}2 t^2$ circular-arc graphs.
An axis-parallel $d$--dimensional box is a Cartesian product $R_1 \times R_2 \times ... \times R_d$ where $R_i$ (for $1 \le i \le d$) is a closed interval of the form $[a_i, b_i]$ on the real line. For a graph $G$, its \emph{boxicity} $\boxi(G)$ is the minimum dimension $d$, such that $G$ is representable as the intersection graph of (axis--parallel) boxes in $d$--dimensional space. The concept of boxicity finds applications in various areas such as ecology, operation research etc. We show that for any graph $G$ with maximum degree $Δ$, $\boxi(G) \le 2 Δ^2 + 2$. That the bound does not depend …
A k-dimensional box is the Cartesian product R_1 x R_2 x ... x R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G) is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph that is not isomorphic to K_4, then box(G)=2. In fact, we prove the stronger result that if G is a planar graph formed …
A box in Euclidean $k$-space is the Cartesian product of $k$ closed intervals on the real line. The boxicity of a graph $G$, denoted by $\text{box}(G)$, is the minimum nonnegative integer $k$ such that $G$ can be isomorphic to the intersection graph of a family of boxes in Euclidean $k$-space. In this paper, we present a sufficient condition for a graph $G$ under which $\text{box}(G)\leq χ(G)$ holds, where $χ(G)$ denotes the chromatic number of $G$. Bhowmick and Chandran (2010) proved that $\text{box}(G)\leq χ(G)$ holds for a graph $G$ with no asteroidal triples. We prove that $\text{box}(G)\l…
A $k$-dimensional box is the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $box(G)$, is the minimum integer $k$ such that $G$ is the intersection graph of a collection of $k$-dimensional boxes. A unit cube in $k$-dimensional space or a $k$-cube is defined as the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line of the form $[a_i, a_{i}+1]$. The {\it cubicity} of $G$, denoted as $cub(G)$, is the minimum $k$ such that $G$ i…
An axis-parallel $d$-dimensional box is a cartesian product $I_1\times I_2\times \dots \times I_b$ where $I_i$ is a closed sub-interval of the real line. For a graph $G = (V,E)$, the $boxicity \ of \ G$, denoted by $\text{box}(G)$, is the minimum dimension $d$ such that $G$ is the intersection graph of a family $(B_v)_{v\in V}$ of $d$-dimensional boxes in $\mathbb R^d$. Let $k$ and $n$ be two positive integers such that $n\geq 2k+1$. The $Kneser \ graph$ $Kn(k,n)$ is the graph with vertex set given by all subsets of $\{1,2,\dots,n\}$ of size $k$ where two vertices are adjacent if their corresp…
An axis-parallel b-dimensional box is a Cartesian product $R_1 \times R_2 \times ... \times R_b$ where each $R_i$ (for $1 \leq i \leq b$) is a closed interval of the form $[a_i,b_i]$ on the real line. The boxicity of any graph $G$, box(G) is the minimum positive integer b such that G can be represented as the intersection graph of axis parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product $R_1 \times R_2\times ... \times R_b$, where each $R_i$ (for $1 \leq i \leq b$) is a closed interval of the form [$a_i$,$a_i$+1] on the real line. When the boxes are restricted to be axis-…
Let $G$ be a simple, undirected, finite graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-dimensional box is a Cartesian product of closed intervals $[a_1,b_1]\times [a_2,b_2]\times...\times [a_k,b_k]$. The {\it boxicity} of $G$, $\boxi(G)$ is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional boxes, i.e. each vertex is mapped to a $k$-dimensional box and two vertices are adjacent in $G$ if and only if their corresponding boxes intersect. Let $\poset=(S,P)$ be a poset where $S$ is the ground set and $P$ is a reflexive, anti-symmetric and…
The 'boxicity' ('cubicity') of a graph G is the minimum natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes (axis-parallel unit cubes) in $R^k$. In this article, we give estimates on the boxicity and the cubicity of Cartesian, strong and direct products of graphs in terms of invariants of the component graphs. In particular, we study the growth, as a function of $d$, of the boxicity and the cubicity of the $d$-th power of a graph with respect to the three products. Among others, we show a surprising result that the boxicity and the cubici…
The boxicity of a graph $G$, denoted by $box(G)$, is the least positive integer $\ell$ such that $G$ can be isomorphic to the intersection graph of a family of boxes in Euclidean $\ell$-space, where box in an Euclidean $\ell$-space is the Cartesian product of $\ell$ closed intervals on the real line. Let $k$ and $d$ be two positive integers with $k\geq 2d$. The circulant graph $G_k^d$ is the graph with vertices set $V(G_k^d)=\{a_0, a_1,\ldots, a_{k-1}\}$ and edge set $E(G_k^d)=\{a_i a_j | \ d\leq |i-j|\leq k-d\}$. Denote $χ(G)$ the chromatic number of a graph $G$. In \cite{Aki} Akira Kamibeppu…
A box in Euclidean $k$-space is the Cartesian product $I_1\times I_2\times \cdots \times I_k$, where $I_j$ is a closed interval on the real line. The boxicity of a graph $G$, denoted by $\text{box}(G)$, is the minimum nonnegative integer $k$ such that $G$ can be isomorphic to the intersection graph of a family of boxes in Euclidean $k$-space. Mycielski introduced an interesting graph operation that extends a graph $G$ to a new graph $M(G)$, called the Mycielski graph of $G$. In this paper, we observe behavior of the boxicity of Mycielski graphs. The inequality $\text{box}(M(G))\geq \text{box}(…
We prove that the dimension of every poset whose comparability graph has maximum degree $Δ$ is at most $Δ\log^{1+o(1)} Δ$. This result improves on a 30-year old bound of Füredi and Kahn, and is within a $\log^{o(1)}Δ$ factor of optimal. We prove this result via the notion of boxicity. The "boxicity" of a graph $G$ is the minimum integer $d$ such that $G$ is the intersection graph of $d$-dimensional axis-aligned boxes. We prove that every graph with maximum degree $Δ$ has boxicity at most $Δ\log^{1+o(1)} Δ$, which is also within a $\log^{o(1)}Δ$ factor of optimal. We also show that the maximum …
The boxicity (respectively cubicity) of a graph $G$ is the minimum non-negative integer $k$, such that $G$ can be represented as an intersection graph of axis-parallel $k$-dimensional boxes (respectively $k$-dimensional unit cubes) and is denoted by $box(G)$ (respectively $cub(G)$). It was shown by Adiga and Chandran (Journal of Graph Theory, 65(4), 2010) that for any graph $G$, $cub(G) \le$ box$(G) \left \lceil \log_2 α\right \rceil$, where $α= α(G)$ is the cardinality of the maximum independent set in $G$. In this note we show that $cub(G) \le 2 \left \lceil \log_2 χ(G) \right \rceil box(G) …
The boxicity $\operatorname{box}(H)$ of a graph $H$ is the smallest integer $d$ such that $H$ is the intersection of $d$ interval graphs, or equivalently, that $H$ is the intersection graph of axis-aligned boxes in $\mathbb{R}^d$. These intersection representations can be interpreted as covering representations of the complement $H^c$ of $H$ with co-interval graphs, that is, complements of interval graphs. We follow the recent framework of global, local and folded covering numbers (Knauer and Ueckerdt, Discrete Mathematics 339 (2016)) to define two new parameters: the local boxicity $\operator…
The \emph{local boxicity} of a graph $G$, denoted by $lbox(G)$, is the minimum positive integer $l$ such that $G$ can be obtained using the intersection of $k$ (, where $k \geq l$,) interval graphs where each vertex of $G$ appears as a non-universal vertex in at most $l$ of these interval graphs. Let $G$ be a graph on $n$ vertices having $m$ edges. Let $Δ$ denote the maximum degree of a vertex in $G$. We show that, (i) $lbox(G) \leq 2^{13\log^{*}Δ} Δ$. There exist graphs of maximum degree $Δ$ having a local boxicity of $Ω(\fracΔ{\logΔ})$. (ii) $lbox(G) \in O(\frac{n}{\log{n}})$. There exist gr…
A $k$-dimensional box is the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $box(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of a collection of $k$-dimensional boxes: that is two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. Let $G$ be a circular arc graph with maximum degree $Δ$. We show that if $Δ<\lfloor \…
Boxicity of a graph $G(V,$ $E)$, denoted by $box(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of axis parallel boxes in $\mathbb{R}^k$. The problem of computing boxicity is inapproximable even for graph classes like bipartite, co-bipartite and split graphs within $O(n^{1 - ε})$-factor, for any $ε>0$ in polynomial time unless $NP=ZPP$. We give FPT approximation algorithms for computing the boxicity of graphs, where the parameter used is the vertex or edge edit distance of the given graph from families of graphs of bounded boxicity. This can be seen …
Boxicity of a graph $G(V,E)$ is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional axis parallel rectangles in $\mathbf{R}^k$. Equivalently, it is the minimum number of interval graphs on the vertex set $V$ such that the intersection of their edge sets is $E$. It is known that boxicity cannot be approximated even for graph classes like bipartite, co-bipartite and split graphs below $O(n^{0.5 - ε})$-factor, for any $ε>0$ in polynomial time unless $NP=ZPP$. Till date, there is no well known graph class of unbounded boxicity for which even an $n^…
Boxicity of a graph H, denoted by box(H), is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in R^k. In this paper, we show that for a line graph G of a multigraph, box(G) <= 2Δ(\lceil log_2(log_2(Δ)) \rceil + 3) + 1, where Δdenotes the maximum degree of G. Since Δ<= 2(χ- 1), for any line graph G with chromatic number χ, box(G) = O(χlog_2(log_2(χ))). For the d-dimensional hypercube H_d, we prove that box(H_d) >= (\lceil log_2(log_2(d)) \rceil + 1)/2. The question of finding a non-trivial lower bound for box(H_d) was left open by Chandran and Siva…
In this paper, we study two applications of graph minor reduction. In the first part of the paper, we introduce a variant of the boxicity, called strong boxicity, where the rectangular representation satisfies an additional condition that each rectangle contains at least one point not present in any other rectangle. We show how the strong boxicity of a graph~\(G\) can be estimated in terms of the strong boxicity of a minor~\(H\) and the number of edit operations needed to obtain~\(H\) from~\(G.\) In the second part of the paper, we consider false data injection (attack) in a flow graph~\(G\) a…
An axis parallel $d$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-dimensional boxes. An axis parallel unit cube in $d$-dimensional space or a $d$-cube is defined as the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$. The {\it cubicity} of $G$, denot…
A $k$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of a collection of $k$-dimensional boxes. A unit cube in $k$-dimensional space or a $k$-cube is defined as the Cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$. The {\it cubicity} of $G$, denoted as $\cub(G)$, is the minimu…
In this document, we study the scope of the following graph model: each vertex is assigned to a box in a metric space and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in the Euclidean line. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model…
Let $G(V,E)$ be a simple, undirected graph where $V$ is the set of vertices and $E$ is the set of edges. A $b$-dimensional cube is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval of unit length on the real line. The \emph{cubicity} of $G$, denoted by $\cub(G)$ is the minimum positive integer $b$ such that the vertices in $G$ can be mapped to axis parallel $b$-dimensional cubes in such a way that two vertices are adjacent in $G$ if and only if their assigned cubes intersect. Suppose $S(m)$ denotes a star graph on $m+1$ nodes. We define \emph{claw nu…
We introduce the notion of \emph{stab number} and \emph{exact stab number} of rectangle intersection graphs, otherwise known as graphs of boxicity at most 2. A graph $G$ is said to be a \emph{$k$-stabbable rectangle intersection graph}, or \emph{$k$-SRIG} for short, if it has a rectangle intersection representation in which $k$ horizontal lines can be chosen such that each rectangle is intersected by at least one of them. If there exists such a representation with the additional property that each rectangle intersects exactly one of the $k$ horizontal lines, then the graph $G$ is said to be a …
A $k$-box $B=(R_1,...,R_k)$, where each $R_i$ is a closed interval on the real line, is defined to be the Cartesian product $R_1\times R_2\times ...\times R_k$. If each $R_i$ is a unit length interval, we call $B$ a $k$-cube. Boxicity of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $k$ such that $G$ is an intersection graph of $k$-boxes. Similarly, the cubicity of $G$, denoted as $\cubi(G)$, is the minimum integer $k$ such that $G$ is an intersection graph of $k$-cubes. It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan: Representing graphs as the interse…
Graph product is a fundamental tool with rich applications in both graph theory and theoretical computer science. It is usually studied in the form $f(G*H)$ where $G$ and $H$ are graphs, * is a graph product and $f$ is a graph property. For example, if $f$ is the independence number and * is the disjunctive product, then the product is known to be multiplicative: $f(G*H)=f(G)f(H)$. In this paper, we study graph products in the following non-standard form: $f((G\oplus H)*J)$ where $G$, $H$ and $J$ are graphs, $\oplus$ and * are two different graph products and $f$ is a graph property. We show t…
