1,158 research outputs found
Nucleation-free rigidity
When all non-edge distances of a graph realized in as a {\em
bar-and-joint framework} are generically {\em implied} by the bar (edge)
lengths, the graph is said to be {\em rigid} in . For ,
characterizing rigid graphs, determining implied non-edges and {\em dependent}
edge sets remains an elusive, long-standing open problem.
One obstacle is to determine when implied non-edges can exist without
non-trivial rigid induced subgraphs, i.e., {\em nucleations}, and how to deal
with them.
In this paper, we give general inductive construction schemes and proof
techniques to generate {\em nucleation-free graphs} (i.e., graphs without any
nucleation) with implied non-edges. As a consequence, we obtain (a) dependent
graphs in that have no nucleation; and (b) nucleation-free {\em
rigidity circuits}, i.e., minimally dependent edge sets in . It
additionally follows that true rigidity is strictly stronger than a tractable
approximation to rigidity given by Sitharam and Zhou
\cite{sitharam:zhou:tractableADG:2004}, based on an inductive combinatorial
characterization.
As an independently interesting byproduct, we obtain a new inductive
construction for independent graphs in . Currently, very few such inductive
constructions are known, in contrast to
Using twins and scaling to construct cospectral graphs for the normalized Laplacian
The spectrum of the normalized Laplacian matrix cannot determine the number
of edges in a graph, however finding constructions of cospectral graphs with
differing number of edges has been elusive. In this paper we use basic
properties of twins and scaling to show how to construct such graphs. We also
give examples of families of graphs which are cospectral with a subgraph for
the normalized Laplacian matrix
Subgraph covers -- An information theoretic approach to motif analysis in networks
Many real world networks contain a statistically surprising number of certain
subgraphs, called network motifs. In the prevalent approach to motif analysis,
network motifs are detected by comparing subgraph frequencies in the original
network with a statistical null model. In this paper we propose an alternative
approach to motif analysis where network motifs are defined to be connectivity
patterns that occur in a subgraph cover that represents the network using
minimal total information. A subgraph cover is defined to be a set of subgraphs
such that every edge of the graph is contained in at least one of the subgraphs
in the cover. Some recently introduced random graph models that can incorporate
significant densities of motifs have natural formulations in terms of subgraph
covers and the presented approach can be used to match networks with such
models. To prove the practical value of our approach we also present a
heuristic for the resulting NP-hard optimization problem and give results for
several real world networks.Comment: 10 pages, 7 tables, 1 Figur
Evolutionary accessibility of mutational pathways
Functional effects of different mutations are known to combine to the total
effect in highly nontrivial ways. For the trait under evolutionary selection
(`fitness'), measured values over all possible combinations of a set of
mutations yield a fitness landscape that determines which mutational states can
be reached from a given initial genotype. Understanding the accessibility
properties of fitness landscapes is conceptually important in answering
questions about the predictability and repeatability of evolutionary
adaptation. Here we theoretically investigate accessibility of the globally
optimal state on a wide variety of model landscapes, including landscapes with
tunable ruggedness as well as neutral `holey' landscapes. We define a
mutational pathway to be accessible if it contains the minimal number of
mutations required to reach the target genotype, and if fitness increases in
each mutational step. Under this definition accessibility is high, in the sense
that at least one accessible pathwayexists with a substantial probability that
approaches unity as the dimensionality of the fitness landscape (set by the
number of mutational loci) becomes large. At the same time the number of
alternative accessible pathways grows without bound. We test the model
predictions against an empirical 8-locus fitness landscape obtained for the
filamentous fungus \textit{Aspergillus niger}. By analyzing subgraphs of the
full landscape containing different subsets of mutations, we are able to probe
the mutational distance scale in the empirical data. The predicted effect of
high accessibility is supported by the empirical data and very robust, which we
argue to reflect the generic topology of sequence spaces.Comment: 16 pages, 4 figures; supplementary material available on reques
Linear-Time Algorithms for Finding Tucker Submatrices and Lekkerkerker-Boland Subgraphs
Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs
for the class of interval graphs. We give a linear-time algorithm to find one
in any graph that is not an interval graph. Tucker characterized the minimal
forbidden submatrices of binary matrices that do not have the consecutive-ones
property. We give a linear-time algorithm to find one in any binary matrix that
does not have the consecutive-ones property.Comment: A preliminary version of this work appeared in WG13: 39th
International Workshop on Graph-Theoretic Concepts in Computer Scienc
Characterising and recognising game-perfect graphs
Consider a vertex colouring game played on a simple graph with
permissible colours. Two players, a maker and a breaker, take turns to colour
an uncoloured vertex such that adjacent vertices receive different colours. The
game ends once the graph is fully coloured, in which case the maker wins, or
the graph can no longer be fully coloured, in which case the breaker wins. In
the game , the breaker makes the first move. Our main focus is on the
class of -perfect graphs: graphs such that for every induced subgraph ,
the game played on admits a winning strategy for the maker with only
colours, where denotes the clique number of .
Complementing analogous results for other variations of the game, we
characterise -perfect graphs in two ways, by forbidden induced subgraphs
and by explicit structural descriptions. We also present a clique module
decomposition, which may be of independent interest, that allows us to
efficiently recognise -perfect graphs.Comment: 39 pages, 8 figures. An extended abstract was accepted at the
International Colloquium on Graph Theory (ICGT) 201
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