We present the implementation of an algorithm for graph isomorphism testing, based on ideas about number of walks (of sufficiently large length) between vertices. The algorithm is expanded for strongly regular graphs (SRG-s) by testing the local complements and values of determinants of their adjacency matrices. All known non-isomorphic SRG-s (with no more than 64 vertices) are distinguishable by this method.
Properties of the `$k$-equivalent' graph families constructed in Cai, Fürer and Immerman, and Evdokimov and Ponomarenko are analysed relative the the recursive $k$-dim WL method. An extension to the recursive $k$-dim WL method is presented that is shown to efficiently characterise all such types of `counterexample' graphs, under certain assumptions. These assumptions are shown to hold in all known cases.
A polynomial algorithm for graphs' isomorphism testing is constructed in assumption that there exists a corresponding polynomial algorithm for graphs with trivial automorphism group.
Let ${\frak K}$ be a class of combinatorial objects invariant with respect to a given regular cyclic group. It is proved that the isomorphism of any two objects $X,Y\in{\frak K}$ can be tested in polynomial time in sizes of $X$ and $Y$.
We construct a polynomial-time algorithm that given a graph $X$ with $4p$ vertices ($p$ is prime), finds (if any) a Cayley representation of $X$ over the group $C_2\times C_2\times C_p$. This result, together with the known similar result for circulant graphs, shows that recognising and testing isomorphism of Cayley graphs over an abelian group of order $4p$ can be done in polynomial time.
We give an algorithm that, for every fixed k, decides isomorphism of graphs of rank width at most k in polynomial time. As the clique width of a graph is bounded in terms of its rank width, we also obtain a polynomial time isomorphism test for graph classes of bounded clique width.
The computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general graph isomorphism problem. Every chordal graph can be represented as the intersection graph of some subtrees of a representing tree, and the leafage of a chordal graph is defined to be the minimum number of leaves in a representing tree for it. We prove that chordal graph isomorphism is fixed parameter tractable with leafage as parameter. In the process we intro…
A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from the first graph onto the second can be found in time poly(n).
A graph is said to be circular-arc if the vertices can be associated with arcs of a circle so that two vertices are adjacent if and only if the corresponding arcs overlap. It is proved that the isomorphism of circular-arc graphs can be tested by the Weisfeiler-Leman algorithm after individualization of two vertices.
Circular-arc graphs are intersection graphs of arcs on the circle. The aim of our work is to present a polynomial time algorithm testing whether two circular-arc graphs are isomorphic. To accomplish our task we construct decomposition trees, which are the structures representing all normalized intersection models of circular-arc graphs. Normalized models reflect the neighbourhood relation in circular-arc graphs and can be seen as their canonical representations; in particular, every intersection model can be easily transformed into a normalized one. Our work adapts and appropriately extends th…
We investigate the Sherali-Adams lift & project hierarchy applied to a graph isomorphism polytope whose integer points encode the isomorphisms between two graphs. In particular, the Sherali-Adams relaxations characterize a new vertex classification algorithm for graph isomorphism, which we call the generalized vertex classification algorithm. This algorithm generalizes the classic vertex classification algorithm and generalizes the work of Tinhofer on polyhedral methods for graph automorphism testing. We establish that the Sherali-Adams lift & project hierarchy when applied to a graph isomorph…
In this paper, we show that Graph Isomorphism (GI) is not $\textsf{AC}^{0}$-reducible to several problems, including the Latin Square Isotopy problem, isomorphism testing of several families of Steiner designs, and isomorphism testing of conference graphs. As a corollary, we obtain that GI is not $\textsf{AC}^{0}$-reducible to isomorphism testing of Latin square graphs and strongly regular graphs arising from special cases of Steiner $2$-designs. We accomplish this by showing that the generator-enumeration technique for each of these problems can be implemented in $β_{2}\textsf{FOLL}$, which c…
Given a function f: {0,1}^n \to {0,1}, the f-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to f up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions f we can test f-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing…
In this paper we show lower bounds for a certain large class of algorithms solving the Graph Isomorphism problem, even on expander graph instances. Spielman [25] shows an algorithm for isomorphism of strongly regular expander graphs that runs in time exp(O(n^(1/3)) (this bound was recently improved to expf O(n^(1/5) [5]). It has since been an open question to remove the requirement that the graph be strongly regular. Recent algorithmic results show that for many problems the Lasserre hierarchy works surprisingly well when the underlying graph has expansion properties. Moreover, recent work of …
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 suitab…
We study free scalar field theory on a graph, which gives rise to a modified version of discrete Green's function on a graph studied in \cite{CY}. We show that this gives rise to a graph invariant, which is closely related to the 2-dim Weisfeiler-Lehman algorithm for graph isomorphism testing. We complement this invariant by another type of graph invariants, coming from viewing graphs as quadratic forms over the integers. We explain that the combination of these two ideas give rise to an interesting approach to the graph isomorphism problem.
Hypergraphics matroids were studied first by Lorea and later by Frank et al. They can be seen as generalizations of graphic matroids. Here we show that several algorithms developed for the graphic case can be extended to hypergraphic matroids. We treat the following: the separation problem for the associated polytope, testing independence, separation of partition inequalities, computing the rank of a set, computing the strength, computing the arboricity and network reinforcement.
The lattice of integer flows of a graph is known to determine the graph up to 2-isomorphism (work of Su--Wagner and Caporaso--Viviani). In this paper we give an algorithmic construction of the graphic matroid $\calM(G)$ of a graph $G$, given its lattice of integer flows $\calF(G)$. The algorithm can then be applied to compute any other 2-isomorphism invariants (that is, matroid invariants) of $G$ from $\calF(G)$. Our method is based on a result of Amini which describes the relationship between the geometry of the Voronoi cell of $\calF(G)$ and the structure of $G$.
In this paper we give an explicit and algorithmic description of Graver basis for the toric ideal associated with a simple undirected graph and apply the basis for testing the beta model of random graphs by Markov chain Monte Carlo method.
We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph $H$ as a minor to graphs excluding $H$ as a topological subgraph. We prove that for a fixed $H$, every graph excluding $H$ as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, we prove that such a decomposition is computable by an algorithm that is fixed-parameter tractable with parameter $|H|$. We present two algorithmic applications of our structure …
The quantum hidden subgroup approach is an actively studied approach to solve combinatorial problems in quantum complexity theory. With the success of the Shor's algorithm, it was hoped that similar approach may be useful to solve the other combinatorial problems. One such problem is the graph isomorphism problem which has survived decades of efforts using the hidden subgroup approach. This paper provides a systematic approach to create arbitrarily large classes of classically efficiently solvable graph automorphism problems or easy graph automorphism problems for which hidden subgroup approac…
We give a new fpt algorithm testing isomorphism of $n$-vertex graphs of tree width $k$ in time $2^{k\operatorname{polylog} (k)}\operatorname{poly} (n)$, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time $2^{\mathcal{O}(k^5\log k)}\operatorname{poly} (n)$. 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 techniq…
A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of ${\rm Aut}(G)$. Regular covers have many applications in constructions and studies of big objects all over mathematics and computer science. We study computational aspects of regular covers that have not been addressed before. The decision problem RegularCover asks for two given graphs $G$ and $H$ whether $G$ regularly covers $H$. When $|H|=1$, this problem becomes Cayley graph recog…
As it follows from Gödel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually, this fundamental principle can be efficiently applicable in computational mathematics and complexity theory concerning the computational complexity of problems from the class NP, particularly and especially the NP-complete ones. While there is a wide set of algorithms for these problems that we call heuristic, the correctness or/and complexity of each concrete algorithm (or the probability of its correct and polynomial-time work) …
It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can distinguish many pairs of graphs, but the pairs of non-isomorphic graphs constructed by Cai, Furer and Immerman it cannot distinguish. If d is fixed, then the WL-algorithm runs in polynomial time. We will formulate the Graph Isomorphism Problem as an Orbit Problem: Given a representation V of an algebraic group G and two elements v_1,v_2 in V, decide whether v_1 and v_2 lie i…
We devise a unified framework for the design of canonization algorithms. Using hereditarily finite sets, we define a general notion of combinatorial objects that includes graphs, hypergraphs, relational structures, codes, permutation groups, tree decompositions, and so on. Our approach allows for a systematic transfer of the techniques that have been developed for isomorphism testing to canonization. We use it to design a canonization algorithm for combinatorial objects in general. This result gives new fastest canonization algorithms with an asymptotic running time matching the best known iso…
Unigraphs are graphs uniquely determined by their own degree sequence up to isomorphism. There are many subclasses of unigraphs such as threshold graphs, split matrogenic graphs, matroidal graphs, and matrogenic graphs. Unigraphs and these subclasses are well studied in the literature. Nevertheless, there are few results on superclasses of unigraphs. In this paper, we introduce two types of generalizations of unigraphs: $k$-unigraphs and $k$-strong unigraphs. We say that a graph $G$ is a $k$-unigraph if $G$ can be partitioned into $k$ unigraphs. $G$ is a $k$-strong unigraph if not only each su…
Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if it succeeds in distinguishing G from any non-isomorphic graph H. Tinhofer (1991) explored a linear programming approach to Graph Isomorphism and defined compact graphs: A graph is compact if its fractional automorphisms polytope is integral. Tinhofer noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of c…
