1,236 research outputs found
Subgraph Complementation
A subgraph complement of the graph G is a graph obtained from G by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph G and graph class G, is there a subgraph complement of G which is in G? We show that this problem can be solved in polynomial time for various choices of the graphs class G, such as bipartite, d-degenerate, or cographs. We complement these results by proving that the problem is NP-complete when G is the class of regular graphs.publishedVersio
Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions
In the companion paper [Linear rank-width of distance-hereditary graphs I. A
polynomial-time algorithm, Algorithmica 78(1):342--377, 2017], we presented a
characterization of the linear rank-width of distance-hereditary graphs, from
which we derived an algorithm to compute it in polynomial time. In this paper,
we investigate structural properties of distance-hereditary graphs based on
this characterization.
First, we prove that for a fixed tree , every distance-hereditary graph of
sufficiently large linear rank-width contains a vertex-minor isomorphic to .
We extend this property to bigger graph classes, namely, classes of graphs
whose prime induced subgraphs have bounded linear rank-width. Here, prime
graphs are graphs containing no splits. We conjecture that for every tree ,
every graph of sufficiently large linear rank-width contains a vertex-minor
isomorphic to . Our result implies that it is sufficient to prove this
conjecture for prime graphs.
For a class of graphs closed under taking vertex-minors, a graph
is called a vertex-minor obstruction for if but all of
its proper vertex-minors are contained in . Secondly, we provide, for
each , a set of distance-hereditary graphs that contains all
distance-hereditary vertex-minor obstructions for graphs of linear rank-width
at most . Also, we give a simpler way to obtain the known vertex-minor
obstructions for graphs of linear rank-width at most .Comment: 38 pages, 13 figures, 1 table, revised journal version. A preliminary
version of Section 5 appeared in the proceedings of WG1
Graph Concatenation for Quantum Codes
Graphs are closely related to quantum error-correcting codes: every
stabilizer code is locally equivalent to a graph code, and every codeword
stabilized code can be described by a graph and a classical code. For the
construction of good quantum codes of relatively large block length,
concatenated quantum codes and their generalizations play an important role. We
develop a systematic method for constructing concatenated quantum codes based
on "graph concatenation", where graphs representing the inner and outer codes
are concatenated via a simple graph operation called "generalized local
complementation." Our method applies to both binary and non-binary concatenated
quantum codes as well as their generalizations.Comment: 26 pages, 12 figures. Figures of concatenated [[5,1,3]] and [[7,1,3]]
are added. Submitted to JM
Inferring Network Mechanisms: The Drosophila melanogaster Protein Interaction Network
Naturally occurring networks exhibit quantitative features revealing
underlying growth mechanisms. Numerous network mechanisms have recently been
proposed to reproduce specific properties such as degree distributions or
clustering coefficients. We present a method for inferring the mechanism most
accurately capturing a given network topology, exploiting discriminative tools
from machine learning. The Drosophila melanogaster protein network is
confidently and robustly (to noise and training data subsampling) classified as
a duplication-mutation-complementation network over preferential attachment,
small-world, and other duplication-mutation mechanisms. Systematic
classification, rather than statistical study of specific properties, provides
a discriminative approach to understand the design of complex networks.Comment: 19 pages, 5 figure
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
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