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

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

Induced Minor Free Graphs: Isomorphism and Clique-width

Given two graphs $G$ and $H$, we say that $G$ contains $H$ as an induced minor if a graph isomorphic to $H$ can be obtained from $G$ 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 $H$ that Graph Isomorphism is polynomial-time solvable on $H$-induced-minor-free graphs or that it is GI-complete. Additionally, we classify those graphs $H$ for which $H$-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 $k$ and two graphs $G_1,G_2$, either concludes that one of these graphs has treewidth at least $k$, or determines whether $G_1$ and $G_2$ are isomorphic. The running time of the algorithm on an $n$-vertex graph is $2^{O(k^5\log k)}\cdot n^5$, 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 $2^{O(k^5\log k)}\cdot n^5$ time that, for a given graph $G$ on $n$ vertices, either concludes that the treewidth of $G$ is at least $k$, or: * finds in an isomorphic-invariant way a graph $\mathfrak{c}(G)$ that is isomorphic to $G$; * finds an isomorphism-invariant construction term --- an algebraic expression that encodes $G$ together with a tree decomposition of $G$ of width $O(k^4)$. Hence, the isomorphism test reduces to verifying whether the computed isomorphic copies or the construction terms for $G_1$ and $G_2$ are equal.Comment: Full version of a paper presented at FOCS 201

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 ${\omega}$ 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

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 $2^{O(k)} O(n)$. 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)

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