5,012 research outputs found
Dynamic Complexity of Planar 3-connected Graph Isomorphism
Dynamic Complexity (as introduced by Patnaik and Immerman) tries to express
how hard it is to update the solution to a problem when the input is changed
slightly. It considers the changes required to some stored data structure
(possibly a massive database) as small quantities of data (or a tuple) are
inserted or deleted from the database (or a structure over some vocabulary).
The main difference from previous notions of dynamic complexity is that instead
of treating the update quantitatively by finding the the time/space trade-offs,
it tries to consider the update qualitatively, by finding the complexity class
in which the update can be expressed (or made). In this setting, DynFO, or
Dynamic First-Order, is one of the smallest and the most natural complexity
class (since SQL queries can be expressed in First-Order Logic), and contains
those problems whose solutions (or the stored data structure from which the
solution can be found) can be updated in First-Order Logic when the data
structure undergoes small changes.
Etessami considered the problem of isomorphism in the dynamic setting, and
showed that Tree Isomorphism can be decided in DynFO. In this work, we show
that isomorphism of Planar 3-connected graphs can be decided in DynFO+ (which
is DynFO with some polynomial precomputation). We maintain a canonical
description of 3-connected Planar graphs by maintaining a database which is
accessed and modified by First-Order queries when edges are added to or deleted
from the graph. We specifically exploit the ideas of Breadth-First Search and
Canonical Breadth-First Search to prove the results. We also introduce a novel
method for canonizing a 3-connected planar graph in First-Order Logic from
Canonical Breadth-First Search Trees
Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces
The Subgraph Isomorphism problem asks, given a host graph G on n vertices and
a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P.
The restriction of this problem to planar graphs has often been considered.
After a sequence of improvements, the current best algorithm for planar graphs
is a linear time algorithm by Dorn (STACS '10), with complexity .
We generalize this result, by giving an algorithm of the same complexity for
graphs that can be embedded in surfaces of bounded genus. At the same time, we
simplify the algorithm and analysis. The key to these improvements is the
introduction of surface split decompositions for bounded genus graphs, which
generalize sphere cut decompositions for planar graphs. We extend the algorithm
for the problem of counting and generating all subgraphs isomorphic to P, even
for the case where P is disconnected. This answers an open question by Eppstein
(SODA '95 / JGAA '99)
Continuous optimization methods for the graph isomorphism problem
The graph isomorphism problem looks deceptively simple, but although
polynomial-time algorithms exist for certain types of graphs such as planar
graphs and graphs with bounded degree or eigenvalue multiplicity, its
complexity class is still unknown. Information about potential isomorphisms
between two graphs is contained in the eigenvalues and eigenvectors of their
adjacency matrices. However, symmetries of graphs often lead to repeated
eigenvalues so that associated eigenvectors are determined only up to basis
rotations, which complicates graph isomorphism testing. We consider orthogonal
and doubly stochastic relaxations of the graph isomorphism problem, analyze the
geometric properties of the resulting solution spaces, and show that their
complexity increases significantly if repeated eigenvalues exist. By
restricting the search space to suitable subspaces, we derive an efficient
Frank-Wolfe based continuous optimization approach for detecting isomorphisms.
We illustrate the efficacy of the algorithm with the aid of various highly
symmetric graphs
Elimination Distance to Bounded Degree on Planar Graphs
We study the graph parameter elimination distance to bounded degree, which was introduced by Bulian and Dawar in their study of the parameterized complexity of the graph isomorphism problem. We prove that the problem is fixed-parameter tractable on planar graphs, that is, there exists an algorithm that given a planar graph G and integers d and k decides in time f(k,d)? n^c for a computable function f and constant c whether the elimination distance of G to the class of degree d graphs is at most k
Graph Isomorphism for K_{3,3}-free and K_5-free graphs is in Log-space
Graph isomorphism is an important and widely studied computational problem with
a yet unsettled complexity.
However, the exact complexity is known for isomorphism of various classes of
graphs. Recently, cite{DLNTW09} proved that planar isomorphism is complete for log-space.
We extend this result %of cite{DLNTW09}
further to the classes of graphs which exclude or as
a minor, and give a log-space algorithm.
Our algorithm decomposes minor-free graphs into biconnected and those further into triconnected
components, which are known to be either planar or components cite{Vaz89}. This gives a triconnected
component tree similar to that for planar graphs. An extension of the log-space algorithm of cite{DLNTW09}
can then be used to decide the isomorphism problem.
For minor-free graphs, we consider -connected components.
These are either planar or isomorphic to the four-rung mobius ladder on vertices
or, with a further decomposition, one obtains planar -connected components cite{Khu88}.
We give an algorithm to get a unique
decomposition of minor-free graphs into bi-, tri- and -connected components,
and construct trees, accordingly.
Since the algorithm of cite{DLNTW09} does
not deal with four-connected component trees, it needs to be modified in a quite non-trivial way
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