50 research outputs found
Boxicity and Cubicity of Product Graphs
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 . 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 , of the boxicity and the cubicity of the
-th power of a graph with respect to the three products. Among others, we
show a surprising result that the boxicity and the cubicity of the -th
Cartesian power of any given finite graph is in and
, respectively. On the other hand, we show that there
cannot exist any sublinear bound on the growth of the boxicity of powers of a
general graph with respect to strong and direct products.Comment: 14 page
Boxicity and Cubicity of Asteroidal Triple free graphs
An axis parallel -dimensional box is the Cartesian product where each is a closed interval on the real line.
The {\it boxicity} of a graph , denoted as \boxi(G), is the minimum
integer such that can be represented as the intersection graph of a
collection of -dimensional boxes. An axis parallel unit cube in
-dimensional space or a -cube is defined as the Cartesian product where each is a closed interval on the
real line of the form . The {\it cubicity} of , denoted as
\cub(G), is the minimum integer such that can be represented as the
intersection graph of a collection of -cubes.
Let denote a star graph on nodes. We define {\it claw number} of
a graph as the largest positive integer such that is an induced
subgraph of and denote it as \claw.
Let be an AT-free graph with chromatic number and claw number
\claw. In this paper we will show that \boxi(G) \leq \chi(G) and this bound
is tight. We also show that \cub(G) \leq \boxi(G)(\ceil{\log_2 \claw} +2)
\chi(G)(\ceil{\log_2 \claw} +2). If is an AT-free graph having
girth at least 5 then \boxi(G) \leq 2 and therefore \cub(G) \leq
2\ceil{\log_2 \claw} +4.Comment: 15 pages: We are replacing our earlier paper regarding boxicity of
permutation graphs with a superior result. Here we consider the boxicity of
AT-free graphs, which is a super class of permutation graph
An upper bound for Cubicity in terms of Boxicity
AbstractAn axis-parallel b-dimensional box is a Cartesian product R1×R2×⋯×Rb where each Ri (for 1≤i≤b) is a closed interval of the form [ai,bi] 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 R1×R2×⋯×Rb, where each Ri (for 1≤i≤b) is a closed interval of the form [ai,ai+1] on the real line. When the boxes are restricted to be axis-parallel cubes in b-dimension, the minimum dimension b required to represent the graph is called the cubicity of the graph (denoted by cub(G)). In this paper we prove that cub(G)≤⌈log2n⌉box(G), where n is the number of vertices in the graph. We also show that this upper bound is tight.Some immediate consequences of the above result are listed below: 1.Planar graphs have cubicity at most 3⌈log2n⌉.2.Outer planar graphs have cubicity at most 2⌈log2n⌉.3.Any graph of treewidth tw has cubicity at most (tw+2)⌈log2n⌉. Thus, chordal graphs have cubicity at most (ω+1)⌈log2n⌉ and circular arc graphs have cubicity at most (2ω+1)⌈log2n⌉, where ω is the clique number.The above upper bounds are tight, but for small constant factors
Cubicity of interval graphs and the claw number
Let be a simple, undirected graph where is the set of vertices
and is the set of edges. A -dimensional cube is a Cartesian product
, where each is a closed interval of
unit length on the real line. The \emph{cubicity} of , denoted by \cub(G)
is the minimum positive integer such that the vertices in can be mapped
to axis parallel -dimensional cubes in such a way that two vertices are
adjacent in if and only if their assigned cubes intersect. Suppose
denotes a star graph on nodes. We define \emph{claw number} of
the graph to be the largest positive integer such that is an induced
subgraph of . It can be easily shown that the cubicity of any graph is at
least \ceil{\log_2\psi(G)}.
In this paper, we show that, for an interval graph
\ceil{\log_2\psi(G)}\le\cub(G)\le\ceil{\log_2\psi(G)}+2. Till now we are
unable to find any interval graph with \cub(G)>\ceil{\log_2\psi(G)}. We also
show that, for an interval graph , \cub(G)\le\ceil{\log_2\alpha}, where
is the independence number of . Therefore, in the special case of
, \cub(G) is exactly \ceil{\log_2\alpha}.
The concept of cubicity can be generalized by considering boxes instead of
cubes. A -dimensional box is a Cartesian product , where each is a closed interval on the real
line. The \emph{boxicity} of a graph, denoted , is the minimum
such that is the intersection graph of -dimensional boxes. It is clear
that box(G)\le\cub(G). From the above result, it follows that for any graph
, \cub(G)\le box(G)\ceil{\log_2\alpha}
Cubicity, Degeneracy, and Crossing Number
A -box , where each is a closed interval on the
real line, is defined to be the Cartesian product . If each is a unit length interval, we call a
-cube. Boxicity of a graph , denoted as \boxi(G), is the minimum
integer such that is an intersection graph of -boxes. Similarly, the
cubicity of , denoted as \cubi(G), is the minimum integer such that
is an intersection graph of -cubes.
It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan:
Representing graphs as the intersection of axis-parallel cubes. MCDES-2008,
IISc Centenary Conference, available at CoRR, abs/cs/ 0607092, 2006.] that, for
a graph with maximum degree , \cubi(G)\leq \lceil 4(\Delta +1)\log
n\rceil. In this paper, we show that, for a -degenerate graph ,
\cubi(G) \leq (k+2) \lceil 2e \log n \rceil. Since is at most
and can be much lower, this clearly is a stronger result. This bound is tight.
We also give an efficient deterministic algorithm that runs in time
to output a dimensional cube representation
for .
An important consequence of the above result is that if the crossing number
of a graph is , then \boxi(G) is . This bound is tight up to a factor of .
We also show that, if has vertices, then \cubi(G) is .
Using our bound for the cubicity of -degenerate graphs we show that
cubicity of almost all graphs in model is ,
where denotes the average degree of the graph under consideration.Comment: 21 page