2,863 research outputs found
Reduction Techniques for Graph Isomorphism in the Context of Width Parameters
We study the parameterized complexity of the graph isomorphism problem when
parameterized by width parameters related to tree decompositions. We apply the
following technique to obtain fixed-parameter tractability for such parameters.
We first compute an isomorphism invariant set of potential bags for a
decomposition and then apply a restricted version of the Weisfeiler-Lehman
algorithm to solve isomorphism. With this we show fixed-parameter tractability
for several parameters and provide a unified explanation for various
isomorphism results concerned with parameters related to tree decompositions.
As a possibly first step towards intractability results for parameterized graph
isomorphism we develop an fpt Turing-reduction from strong tree width to the a
priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure
Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs
The Graph Isomorphism problem restricted to graphs of bounded treewidth or
bounded tree distance width are known to be solvable in polynomial time
[Bod90],[YBFT99]. We give restricted space algorithms for these problems
proving the following results: - Isomorphism for bounded tree distance width
graphs is in L and thus complete for the class. We also show that for this kind
of graphs a canon can be computed within logspace. - For bounded treewidth
graphs, when both input graphs are given together with a tree decomposition,
the problem of whether there is an isomorphism which respects the
decompositions (i.e. considering only isomorphisms mapping bags in one
decomposition blockwise onto bags in the other decomposition) is in L. - For
bounded treewidth graphs, when one of the input graphs is given with a tree
decomposition the isomorphism problem is in LogCFL. - As a corollary the
isomorphism problem for bounded treewidth graphs is in LogCFL. This improves
the known TC1 upper bound for the problem given by Grohe and Verbitsky
[GroVer06].Comment: STACS conference 2010, 12 page
Induced Minor Free Graphs: Isomorphism and Clique-width
Given two graphs and , we say that contains as an induced
minor if a graph isomorphic to can be obtained from by a sequence of
vertex deletions and edge contractions. We study the complexity of Graph
Isomorphism on graphs that exclude a fixed graph as an induced minor. More
precisely, we determine for every graph that Graph Isomorphism is
polynomial-time solvable on -induced-minor-free graphs or that it is
GI-complete. Additionally, we classify those graphs for which
-induced-minor-free graphs have bounded clique-width. These two results
complement similar dichotomies for graphs that exclude a fixed graph as an
induced subgraph, minor, or subgraph.Comment: 16 pages, 5 figures. An extended abstract of this paper previously
appeared in the proceedings of the 41st International Workshop on
Graph-Theoretic Concepts in Computer Science (WG 2015
Towards an Isomorphism Dichotomy for Hereditary Graph Classes
In this paper we resolve the complexity of the isomorphism problem on all but
finitely many of the graph classes characterized by two forbidden induced
subgraphs. To this end we develop new techniques applicable for the structural
and algorithmic analysis of graphs. First, we develop a methodology to show
isomorphism completeness of the isomorphism problem on graph classes by
providing a general framework unifying various reduction techniques. Second, we
generalize the concept of the modular decomposition to colored graphs, allowing
for non-standard decompositions. We show that, given a suitable decomposition
functor, the graph isomorphism problem reduces to checking isomorphism of
colored prime graphs. Third, we extend the techniques of bounded color valence
and hypergraph isomorphism on hypergraphs of bounded color size as follows. We
say a colored graph has generalized color valence at most k if, after removing
all vertices in color classes of size at most k, for each color class C every
vertex has at most k neighbors in C or at most k non-neighbors in C. We show
that isomorphism of graphs of bounded generalized color valence can be solved
in polynomial time.Comment: 37 pages, 4 figure
Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth
We give a fixed-parameter tractable algorithm that, given a parameter and
two graphs , either concludes that one of these graphs has treewidth
at least , or determines whether and are isomorphic. The running
time of the algorithm on an -vertex graph is ,
and this is the first fixed-parameter algorithm for Graph Isomorphism
parameterized by treewidth.
Our algorithm in fact solves the more general canonization problem. We namely
design a procedure working in time that, for a
given graph on vertices, either concludes that the treewidth of is
at least , or: * finds in an isomorphic-invariant way a graph
that is isomorphic to ; * finds an isomorphism-invariant
construction term --- an algebraic expression that encodes together with a
tree decomposition of of width .
Hence, the isomorphism test reduces to verifying whether the computed
isomorphic copies or the construction terms for and are equal.Comment: Full version of a paper presented at FOCS 201
Grad and Classes with Bounded Expansion II. Algorithmic Aspects
Classes of graphs with bounded expansion are a generalization of both proper
minor closed classes and degree bounded classes. Such classes are based on a
new invariant, the greatest reduced average density (grad) of G with rank r,
∇r(G). These classes are also characterized by the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. These results lead to several new linear time algorithms, such as an
algorithm for counting all the isomorphs of a fixed graph in an input graph or
an algorithm for checking whether there exists a subset of vertices of a priori
bounded size such that the subgraph induced by this subset satisfies some
arbirtrary but fixed first order sentence. We also show that for fixed p,
computing the distances between two vertices up to distance p may be performed
in constant time per query after a linear time preprocessing. We also show,
extending several earlier results, that a class of graphs has sublinear
separators if it has sub-exponential expansion. This result result is best
possible in general
Shrub-depth: Capturing Height of Dense Graphs
The recent increase of interest in the graph invariant called tree-depth and
in its applications in algorithms and logic on graphs led to a natural
question: is there an analogously useful "depth" notion also for dense graphs
(say; one which is stable under graph complementation)? To this end, in a 2012
conference paper, a new notion of shrub-depth has been introduced, such that it
is related to the established notion of clique-width in a similar way as
tree-depth is related to tree-width. Since then shrub-depth has been
successfully used in several research papers. Here we provide an in-depth
review of the definition and basic properties of shrub-depth, and we focus on
its logical aspects which turned out to be most useful. In particular, we use
shrub-depth to give a characterization of the lower levels of the
MSO1 transduction hierarchy of simple graphs
Computing Shrub-Depth Decompositions
Shrub-depth is a width measure of graphs which, roughly speaking, corresponds to the smallest depth of a tree into which a graph can be encoded. It can be thought of as a low-depth variant of clique-width (or rank-width), similarly as treedepth is a low-depth variant of treewidth. We present an fpt algorithm for computing decompositions of graphs of bounded shrub-depth. To the best of our knowledge, this is the first algorithm which computes the decomposition directly, without use of rank-width decompositions and FO or MSO logic
Approximately Counting Embeddings into Random Graphs
Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic)
copies of H contained in a graph G. We investigate the fundamental problem of
estimating C_H(G). Previous results cover only a few specific instances of this
general problem, for example, the case when H has degree at most one
(monomer-dimer problem). In this paper, we present the first general subcase of
the subgraph isomorphism counting problem which is almost always efficiently
approximable. The results rely on a new graph decomposition technique.
Informally, the decomposition is a labeling of the vertices such that every
edge is between vertices with different labels and for every vertex all
neighbors with a higher label have identical labels. The labeling implicitly
generates a sequence of bipartite graphs which permits us to break the problem
of counting embeddings of large subgraphs into that of counting embeddings of
small subgraphs. Using this method, we present a simple randomized algorithm
for the counting problem. For all decomposable graphs H and all graphs G, the
algorithm is an unbiased estimator. Furthermore, for all graphs H having a
decomposition where each of the bipartite graphs generated is small and almost
all graphs G, the algorithm is a fully polynomial randomized approximation
scheme.
We show that the graph classes of H for which we obtain a fully polynomial
randomized approximation scheme for almost all G includes graphs of degree at
most two, bounded-degree forests, bounded-length grid graphs, subdivision of
bounded-degree graphs, and major subclasses of outerplanar graphs,
series-parallel graphs and planar graphs, whereas unbounded-length grid graphs
are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition
3.
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