8 research outputs found
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
Boxicity and topological invariants
The boxicity of a graph is the smallest integer for which there
exist interval graphs , , such that . In the first part of this note, we prove that every graph on
edges has boxicity , which is asymptotically best
possible. We use this result to study the connection between the boxicity of
graphs and their Colin de Verdi\`ere invariant, which share many similarities.
Known results concerning the two parameters suggest that for any graph , the
boxicity of is at most the Colin de Verdi\`ere invariant of , denoted by
. We observe that every graph has boxicity , while there are graphs with boxicity . In the second part of this note, we focus on graphs embeddable on a
surface of Euler genus . We prove that these graphs have boxicity
, while some of these graphs have boxicity . This improves the previously best known upper and lower bounds.
These results directly imply a nearly optimal bound on the dimension of the
adjacency poset of graphs on surfaces.Comment: 6 page
Boxicity of graphs on surfaces
The boxicity of a graph is the least integer for which there
exist interval graphs , , such that . 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 of
genus is at most . This result yields improved bounds on the
dimension of the adjacency poset of graphs on surfaces.Comment: 9 pages, 2 figure
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . This connection helps us in borrowing
results and techniques from the extensive literature on boxicity to study the
concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to
WG-2014. Some results proved in this paper are also present in
arXiv:1212.6756. arXiv admin note: substantial text overlap with
arXiv:1212.675
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. model is O(davlogn)