7 research outputs found
Graph Isomorphism for unit square graphs
In the past decades for more and more graph classes the Graph Isomorphism
Problem was shown to be solvable in polynomial time. An interesting family of
graph classes arises from intersection graphs of geometric objects. In this
work we show that the Graph Isomorphism Problem for unit square graphs,
intersection graphs of axis-parallel unit squares in the plane, can be solved
in polynomial time. Since the recognition problem for this class of graphs is
NP-hard we can not rely on standard techniques for geometric graphs based on
constructing a canonical realization. Instead, we develop new techniques which
combine structural insights into the class of unit square graphs with
understanding of the automorphism group of such graphs. For the latter we
introduce a generalization of bounded degree graphs which is used to capture
the main structure of unit square graphs. Using group theoretic algorithms we
obtain sufficient information to solve the isomorphism problem for unit square
graphs.Comment: 31 pages, 6 figure
Graph isomorphism for (H1,H2)-free graphs : an almost complete dichotomy.
We almost completely resolve the computational complexity
of Graph Isomorphism for classes of graphs characterized by two forbidden
induced subgraphs H1 and H2. Schweitzer settled the complexity of
this problem restricted to (H1;H2)-free graphs for all but a nite number
of pairs (H1;H2), but without explicitly giving the number of open cases.
Grohe and Schweitzer proved that Graph Isomorphism is polynomialtime
solvable on graph classes of bounded clique-width. By combining
known results with a number of new results, we reduce the number of
open cases to seven. By exploiting the strong relationship between Graph
Isomorphism and clique-width, we simultaneously reduce the number of
open cases for boundedness of clique-width for (H1;H2)-free graphs to
ve
An Improved Isomorphism Test for Bounded-Tree-Width Graphs
We give a new fpt algorithm testing isomorphism of n-vertex graphs of tree width k in time 2^{k polylog(k)} poly n, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time 2^{O(k^5 log k)}poly n. Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree width k. Our algorithm then makes heavy use of the group theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism test. In fact, we even use Babai\u27s algorithm as a black box in one place. We give a second algorithm which, at the price of a slightly worse run time 2^{O(k^2 log k)}poly n, avoids the use of Babai\u27s algorithm and, more importantly, has the additional benefit that it can also be used as a canonization algorithm
The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs
The Weisfeiler-Leman procedure is a widely-used approach for graph
isomorphism testing that works by iteratively computing an
isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool
in structural graph theory, which is often exploited in approaches to tackle
the graph isomorphism problem, is the decomposition into 2- and 3-connected
components.
We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly
computes the decomposition of a graph into its 3-connected components. Thus,
the dimension of the algorithm needed to distinguish two given graphs is at
most the dimension required to distinguish the corresponding decompositions
into 3-connected components (assuming it is at least 2).
This result implies that for k >= 2, the k-dimensional algorithm
distinguishes k-separators, i.e., k-tuples of vertices that separate the graph,
from other vertex k-tuples. As a byproduct, we also obtain insights about the
connectivity of constituent graphs of association schemes.
In an application of the results, we show the new upper bound of k on the
Weisfeiler-Leman dimension of graphs of treewidth at most k. Using a
construction by Cai, F\"urer, and Immerman, we also provide a new lower bound
that is asymptotically tight up to a factor of 2.Comment: 30 pages, 4 figures, full version of a paper accepted at MFCS 201
Hypergraph Isomorphism for Groups with Restricted Composition Factors
We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices V and a permutation group ? over domain V, and asking whether there is a permutation ? ? ? that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on d points, this problem can be solved in time (n+m)^O((log d)^c) for some absolute constant c where n denotes the number of vertices and m the number of hyperedges. In particular, this gives the currently fastest isomorphism test for hypergraphs in general. The previous best algorithm for the above problem due to Schweitzer and Wiebking (STOC 2019) runs in time n^O(d)m^O(1).
As an application of this result, we obtain, for example, an algorithm testing isomorphism of graphs excluding K_{3,h} as a minor in time n^O((log h)^c). In particular, this gives an isomorphism test for graphs of Euler genus at most g running in time n^O((log g)^c)