69 research outputs found
Structural parameterizations for boxicity
The boxicity of a graph is the least integer such that has an
intersection model of axis-aligned -dimensional boxes. Boxicity, the problem
of deciding whether a given graph has boxicity at most , is NP-complete
for every fixed . 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 -approximation when
parameterized by the pathwidth of the input graph.
Finally, we provide evidence in favor of a conjecture of Adiga et al. that
boxicity remains NP-complete when parameterized by the treewidth.Comment: 19 page
Boxicity of Series Parallel 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 a k-tree with
boxicity k+1. (This would show that their upper bound is tight but for an
additive factor of 1, since the treewidth of any k-tree equals k.) The
series-parallel graph we construct in this paper is a 2-tree with boxicity 3
and is thus a first step towards proving their conjecture.Comment: 10 pages, 0 figure
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
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
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
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