1,241 research outputs found
Enumeration of chordal planar graphs and maps
We determine the number of labelled chordal planar graphs with n vertices, which is asymptotically for a constant and . We also determine the number of rooted simple chordal planar maps with n edges, which is asymptotically , where , , and s is an algebraic number of degree 12. The proofs are based on combinatorial decompositions and singularity analysis. Chordal planar graphs (or maps) are a natural example of a subcritical class of graphs in which the class of 3-connected graphs is relatively rich. The 3-connected members are precisely chordal triangulations, those obtained starting from by repeatedly adding vertices adjacent to an existing triangular face.We gratefully acknowledge earlier discussions on this project with Erkan Narmanli. M.N. was supported by grants MTM2017-82166-P and PID2020-113082GB-I00, the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). C.R. was supported by the grant Beatriu de Pinós BP2019, funded by the H2020 COFUND project No 801370 and AGAUR (the Catalan agency for management of university and research grants), and the grant PID2020-113082GB-I00 of the Spanish Ministry of Science and Innovation.Peer ReviewedPostprint (author's final draft
Partitioning Perfect Graphs into Stars
The partition of graphs into "nice" subgraphs is a central algorithmic
problem with strong ties to matching theory. We study the partitioning of
undirected graphs into same-size stars, a problem known to be NP-complete even
for the case of stars on three vertices. We perform a thorough computational
complexity study of the problem on subclasses of perfect graphs and identify
several polynomial-time solvable cases, for example, on interval graphs and
bipartite permutation graphs, and also NP-complete cases, for example, on grid
graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
Slimness of graphs
Slimness of a graph measures the local deviation of its metric from a tree
metric. In a graph , a geodesic triangle with
is the union of three shortest
paths connecting these vertices. A geodesic triangle is
called -slim if for any vertex on any side the
distance from to is at most , i.e. each path
is contained in the union of the -neighborhoods of two others. A graph
is called -slim, if all geodesic triangles in are
-slim. The smallest value for which is -slim is
called the slimness of . In this paper, using the layering partition
technique, we obtain sharp bounds on slimness of such families of graphs as (1)
graphs with cluster-diameter of a layering partition of , (2)
graphs with tree-length , (3) graphs with tree-breadth , (4)
-chordal graphs, AT-free graphs and HHD-free graphs. Additionally, we show
that the slimness of every 4-chordal graph is at most 2 and characterize those
4-chordal graphs for which the slimness of every of its induced subgraph is at
most 1
On The Center Sets and Center Numbers of Some Graph Classes
For a set of vertices and the vertex in a connected graph ,
is called the -eccentricity of in
. The set of vertices with minimum -eccentricity is called the -center
of . Any set of vertices of such that is an -center for some
set of vertices of is called a center set. We identify the center sets
of certain classes of graphs namely, Block graphs, , , wheel
graphs, odd cycles and symmetric even graphs and enumerate them for many of
these graph classes. We also introduce the concept of center number which is
defined as the number of distinct center sets of a graph and determine the
center number of some graph classes
The Irreducible Spine(s) of Undirected Networks
Using closure concepts, we show that within every undirected network, or
graph, there is a unique irreducible subgraph which we call its "spine". The
chordless cycles which comprise this irreducible core effectively characterize
the connectivity structure of the network as a whole. In particular, it is
shown that the center of the network, whether defined by distance or
betweenness centrality, is effectively contained in this spine. By counting the
number of cycles of length 3 <= k <= max_length, we can also create a kind of
signature that can be used to identify the network. Performance is analyzed,
and the concepts we develop are illurstrated by means of a relatively small
running sample network of about 400 nodes.Comment: Submitted to WISE 201
- …