48 research outputs found
On the Complexity of Role Colouring Planar Graphs, Trees and Cographs
We prove several results about the complexity of the role colouring problem.
A role colouring of a graph is an assignment of colours to the vertices of
such that two vertices of the same colour have identical sets of colours in
their neighbourhoods. We show that the problem of finding a role colouring with
colours is NP-hard for planar graphs. We show that restricting the
problem to trees yields a polynomially solvable case, as long as is either
constant or has a constant difference with , the number of vertices in the
tree. Finally, we prove that cographs are always -role-colourable for
and construct such a colouring in polynomial time
Guessing Numbers of Odd Cycles
For a given number of colours, , the guessing number of a graph is the
base logarithm of the size of the largest family of colourings of the
vertex set of the graph such that the colour of each vertex can be determined
from the colours of the vertices in its neighbourhood. An upper bound for the
guessing number of the -vertex cycle graph is . It is known that
the guessing number equals whenever is even or is a perfect
square \cite{Christofides2011guessing}. We show that, for any given integer
, if is the largest factor of less than or equal to
, for sufficiently large odd , the guessing number of with
colours is . This answers a question posed by
Christofides and Markstr\"{o}m in 2011 \cite{Christofides2011guessing}. We also
present an explicit protocol which achieves this bound for every . Linking
this to index coding with side information, we deduce that the information
defect of with colours is for sufficiently
large odd . Our results are a generalisation of the case which was
proven in \cite{bar2011index}.Comment: 16 page
Detection of Core-Periphery Structure in Networks Using Spectral Methods and Geodesic Paths
We introduce several novel and computationally efficient methods for
detecting "core--periphery structure" in networks. Core--periphery structure is
a type of mesoscale structure that includes densely-connected core vertices and
sparsely-connected peripheral vertices. Core vertices tend to be well-connected
both among themselves and to peripheral vertices, which tend not to be
well-connected to other vertices. Our first method, which is based on
transportation in networks, aggregates information from many geodesic paths in
a network and yields a score for each vertex that reflects the likelihood that
a vertex is a core vertex. Our second method is based on a low-rank
approximation of a network's adjacency matrix, which can often be expressed as
a tensor-product matrix. Our third approach uses the bottom eigenvector of the
random-walk Laplacian to infer a coreness score and a classification into core
and peripheral vertices. We also design an objective function to (1) help
classify vertices into core or peripheral vertices and (2) provide a
goodness-of-fit criterion for classifications into core versus peripheral
vertices. To examine the performance of our methods, we apply our algorithms to
both synthetically-generated networks and a variety of networks constructed
from real-world data sets.Comment: This article is part of EJAM's December 2016 special issue on
"Network Analysis and Modelling" (available at
https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/issue/journal-ejm-volume-27-issue-6/D245C89CABF55DBF573BB412F7651ADB
Logarithmic Weisfeiler--Leman and Treewidth
In this paper, we show that the -dimensional Weisfeiler--Leman
algorithm can identify graphs of treewidth in rounds. This
improves the result of Grohe & Verbitsky (ICALP 2006), who previously
established the analogous result for -dimensional Weisfeiler--Leman. In
light of the equivalence between Weisfeiler--Leman and the logic (Cai, F\"urer, & Immerman, Combinatorica 1992), we obtain an
improvement in the descriptive complexity for graphs of treewidth .
Precisely, if is a graph of treewidth , then there exists a
-variable formula in with
quantifier depth that identifies up to isomorphism
On the Last New Vertex Visited by a Random Walk in a Directed Graph
Consider a simple graph in which a random walk begins at a given vertex. It
moves at each step with equal probability to any neighbor of its current
vertex, and ends when it has visited every vertex. We call such a random walk a
random cover tour. It is well known that cycles and complete graphs have the
property that a random cover tour starting at any vertex is equally likely to
end at any other vertex. Ronald Graham asked whether there are any other graphs
with this property. In 1993, L\'aszlo Lov\'asz and Peter Winkler showed that
cycles and complete graphs are the only undirected graphs with this property.
We strengthen this result by showing that cycles and complete graphs (with all
edges considered bidirected) are the only directed graphs with this property
Subgraph complementation and minimum rank
Any finite simple graph can be represented by a collection
of subsets of such that if and only if and
appear together in an odd number of sets in . Let denote
the minimum cardinality of such a collection. This invariant is equivalent to
the minimum dimension of a faithful orthogonal representation of over
and is closely connected to the minimum rank of . We show
that when
is odd, or when is a forest. Otherwise,
. Furthermore, we show that the following
are equivalent for any graph with at least one edge: i.
; ii. the adjacency matrix of
is the unique matrix of rank which fits
over ; iii. there is a minimum collection as
described in which every vertex appears an even number of times; and iv. for
every component of , . We also show that, for these graphs, is
twice the minimum number of tricliques whose symmetric difference of edge sets
is . Additionally, we provide a set of upper bounds on in terms of
the order, size, and vertex cover number of . Finally, we show that the
class of graphs with is hereditary and finitely defined. For odd
, the sets of minimal forbidden induced subgraphs are the same as those for
the property , and we exhibit this set
for