14 research outputs found
Boxicity of Line Graphs
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) <=
2\Delta(\lceil log_2(log_2(\Delta)) \rceil + 3) + 1, where \Delta denotes the
maximum degree of G. Since \Delta <= 2(\chi - 1), for any line graph G with
chromatic number \chi, box(G) = O(\chi log_2(log_2(\chi))). 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 Sivadasan in [L. Sunil Chandran and Naveen
Sivadasan. The cubicity of Hypercube Graphs. Discrete Mathematics,
308(23):5795-5800, 2008]. The above results are consequences of bounds that we
obtain for the boxicity of fully subdivided graphs (a graph which can be
obtained by subdividing every edge of a graph exactly once).Comment: 14 page
On Generalizations of Pairwise Compatibility Graphs
A graph is a PCG if there exists an edge-weighted tree such that each
leaf of the tree is a vertex of the graph, and there is an edge in
if and only if the weight of the path in the tree connecting and
lies within a given interval. PCGs have different applications in phylogenetics
and have been lately generalized to multi-interval-PCGs. In this paper we
define two new generalizations of the PCG class, namely k-OR-PCGs and
k-AND-PCGs, that are the classes of graphs that can be expressed as union and
intersection, respectively, of PCGs. The problems we consider can be also
described in terms of the \emph{covering number} and the \emph{intersection
dimension} of a graph with respect to the PCG class. In this paper we
investigate how the classes of PCG, multi-interval-PCG, OR-PCG and AND-PCG are
related to each other and to other graph classes known in the literature. In
particular, we provide upper bounds on the minimum for which an arbitrary
graph belongs to k-interval-PCG, k-OR-PCG and k-AND-PCG classes.
Furthermore, for particular graph classes, we improve these general bounds.
Moreover, we show that, for every integer , there exists a bipartite graph
that is not in the k-interval-PCG class, proving that there is no finite
for which the k-interval-PCG class contains all the graphs. Finally, we use a
Ramsey theory argument to show that for any , there exist graphs that are
not in k-AND-PCG, and graphs that are not in k-OR-PCG
An Efficient Representation for Filtrations of Simplicial Complexes
A filtration over a simplicial complex is an ordering of the simplices of
such that all prefixes in the ordering are subcomplexes of . Filtrations
are at the core of Persistent Homology, a major tool in Topological Data
Analysis. In order to represent the filtration of a simplicial complex, the
entire filtration can be appended to any data structure that explicitly stores
all the simplices of the complex such as the Hasse diagram or the recently
introduced Simplex Tree [Algorithmica '14]. However, with the popularity of
various computational methods that need to handle simplicial complexes, and
with the rapidly increasing size of the complexes, the task of finding a
compact data structure that can still support efficient queries is of great
interest.
In this paper, we propose a new data structure called the Critical Simplex
Diagram (CSD) which is a variant of the Simplex Array List (SAL) [Algorithmica
'17]. Our data structure allows one to store in a compact way the filtration of
a simplicial complex, and allows for the efficient implementation of a large
range of basic operations. Moreover, we prove that our data structure is
essentially optimal with respect to the requisite storage space. Finally, we
show that the CSD representation admits fast construction algorithms for Flag
complexes and relaxed Delaunay complexes.Comment: A preliminary version appeared in SODA 201
An Efficient Representation for Filtrations of Simplicial Complexes
International audienceA filtration over a simplicial complex K is an ordering of the simplices of K such that all prefixes in the ordering are subcomplexes of K. Filtrations are at the core of Persistent Homology, a major tool in Topo-logical Data Analysis. In order to represent the filtration of a simplicial complex, the entire filtration can be appended to any data structure that explicitly stores all the simplices of the complex such as the Hasse diagram or the recently introduced Simplex Tree [Algorithmica '14]. However, with the popularity of various computational methods that need to handle simplicial complexes, and with the rapidly increasing size of the complexes, the task of finding a compact data structure that can still support efficient queries is of great interest. This direction has been recently pursued for the case of maintaining simplicial complexes. For instance, Boissonnat et al. [Algorithmica '17] considered storing the simplices that are maximal with respect to inclusion and Attali et al. [IJCGA '12] considered storing the simplices that block the expansion of the complex. Nevertheless, so far there has been no data structure that compactly stores the filtration of a simplicial complex, while also allowing the efficient implementation of basic operations on the complex. In this paper, we propose a new data structure called the Critical Simplex Diagram (CSD) which is a variant of the Simplex Array List (SAL) [Algorithmica '17]. Our data structure allows one to store in a compact way the filtration of a simplicial complex, and allows for the efficient implementation of a large range of basic operations. Moreover, we prove that our data structure is essentially optimal with respect to the requisite storage space. Finally, we show that the CSD representation admits fast construction algorithms for Flag complexes and relaxed Delaunay complexes
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum