Abstract We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of the adjacency matrix. We provide a purely combinatorial characterization of this hierarchy in terms of the walk counts. This allows us to give a complete answer to Fürer's question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison with the 2-dimensional Weisfeiler-Leman algorithm, extending the recent work of Rattan and Seppelt (SODA 2023). As another application of the characterization, we prove that almost all graphs are determined up to isomorphism in terms of the spectrum and the angles, which is of interest in view of the long-standing open problem whether almost all graphs are determined by their eigenvalues alone. Finally, we describe the exact relationship between the hierarchy and the Weisfeiler-Leman algorithms for small dimensions, as also some other important spectral characteristics of a graph such as the generalized and the main spectra.

In this note we explore the color refinement procedure — also known as the 1-dimensional Weisfeiler-Leman procedure, well-studied in connection with the Graph Isomorphism problem — in the context of the Graph Reconstruction conjecture of Ulam. A basic fact about the Ulam reconstruction conjecture is that the connectedness of a graph is determined by the deck of its vertex-deleted subgraphs, which are considered up to isomorphism. We strengthen this result by proving that connectedness of a graph can even be determined from the deck of its vertex-deleted subgraphs given only by their stable colorings (i.e., up to equivalence under color refinement). It follows as a consequence that connectedness is recognizable by Reconstruction Graph Neural Networks, which is a recently introduced GNN architecture inspired by the reconstruction conjecture (Cotta, Morris, Ribeiro 2021).

We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of the adjacency matrix. We provide a purely combinatorial characterization of this hierarchy in terms of the walk counts. This allows us to give a complete answer to Fürer's question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison to the 2-dimensional Weisfeiler-Leman algorithm, extending the recent work of Rattan and Seppelt (SODA 2023). As another application of the characterization, we prove that almost all graphs are determined up to isomorphism in terms of the spectrum and the angles, which is of interest in view of the long-standing open problem whether almost all graphs are determined by their eigenvalues alone. Finally, we describe the exact relationship between the hierarchy and the Weisfeiler-Leman algorithms for small dimensions, as also some other important spectral characteristics of a graph such as the generalized and the main spectra.

As is well known, the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical two-dimensional Weisfeiler--Leman algorithm (2-WL). On the other hand, the prominent Cai--Fürer--Immerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph $G$ of color multiplicity 4, recognizes whether or not $G$ is identifiable by 2-WL, that is, whether or not 2-WL distinguishes $G$ from every nonisomorphic graph. In fact, we solve the much more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to graphs of color multiplicity 4 with directed and colored edges. Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2-WL. The Cai--Fürer--Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai--Fürer--Immerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2-WL. One of them arises from patterns known as $(n_3)$-configurations in incidence geometry.

Lubiw showed that several variants of Graph Isomorphism are NP-complete, where the solutions are required to satisfy certain additional constraints [SICOMP 10, 1981]. One of these, called Isomorphism With Restrictions, is to decide for two given graphs X_1=(V,E_1) and X_2=(V,E_2) and a subset R\subseteq V\times V of forbidden pairs whether there is an isomorphism \pi from X_1 to X_2 such that i^\pi\ne j for all (i,j)\in R. We prove that this problem and several of its generalizations are in fact in \FPT: - The problem of deciding whether there is an isomorphism between two graphs that moves k vertices and satisfies Lubiw-style constraints is in FPT, with k and the size of R as parameters. The problem remains in FPT even if a conjunction of disjunctions of such constraints is allowed. As a consequence of the main result it follows that the problem to decide whether there is an isomorphism that moves exactly k vertices is in FPT. This solves a question left open in our article on exact weight automorphisms [STACS 2017]. - When the number of moved vertices is unrestricted, finding isomorphisms that satisfy a CNF of Lubiw-style constraints can be solved in FPT with access to a GI oracle. - Checking if there is an isomorphism π between two graphs with complexity t is also in FPT with t as parameter, where the complexity of a permutation is the Cayley measure defined as the minimum number t such that \pi can be expressed as a product of t transpositions. - We consider a more general problem in which the vertex set of a graph X is partitioned into Red and Blue, and we are interested in an automorphism that stabilizes Red and Blue and moves exactly k vertices in Blue, where k is the parameter. This problem was introduced by [Downey and Fellows 1999], and we showed [STACS 2017] that it is W[1]-hard even with color classes of size 4 inside Red. Now, for color classes of size at most 3 inside Red, we show the problem is in FPT. In the non-parameterized setting, all these problems are NP-complete. Also, they all generalize in several ways the problem to decide whether there is an isomorphism between two graphs that moves at most k vertices, shown to be in FPT by Schweitzer [ESA 2011].

We show that checking if a given hypergraph has an automorphism that moves exactly k vertices is fixed parameter tractable, using k and additionally either the maximum hyperedge size or the maximum color class size as parameters. In particular, it suffices to use k as parameter if the hyperedge size is at most polylogarithmic in the size of the given hypergraph. As a building block for our algorithms, we generalize Schweitzer's FPT algorithm [ESA 2011] that, given two graphs on the same vertex set and a parameter k, decides whether there is an isomorphism between the two graphs that moves at most k vertices. We extend this result to hypergraphs, using the maximum hyperedge size as a second parameter. Another key component of our algorithm is an orbit-shrinking technique that preserves permutations that move few points and that may be of independent interest. Applying it to a suitable subgroup of the automorphism group allows us to switch from bounded hyperedge size to bounded color classes in the exactly-k case.

In this paper we study the complexity of the following problems: Given a colored graph X=(V,E,c), compute a minimum cardinality set S of vertices such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G on [n] given by generators, i.e., a minimum cardinality subset S of [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k=n-|S| is the parameter, we give FPT algorithms. A notion closely related to fixing is called individualization. Individualization combined with the Weisfeiler-Leman procedure is a fundamental technique in algorithms for Graph Isomorphism. Motivated by the power of individualization, in the present paper we explore the complexity of individualization: what is the minimum number of vertices we need to individualize in a given graph such that color refinement "succeeds" on it. Here "succeeds" could have different interpretations, and we consider the following: It could mean the individualized graph becomes: (a) discrete, (b) amenable, (c) compact, or (d) refinable. In particular, we study the parameterized versions of these problems where the parameter is the number of vertices individualized. We show a dichotomy: For graphs with color classes of size at most 3 these problems can be solved in polynomial time (even in logspace), while starting from color class size 4 they become W[P]-hard.

We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism, denoted CHI, which has running time (2 b N) O(1), where the parameter b is the maximum size of the color classes of the given hypergraphs and N is the input size. We also describe an fpt algorithm for a parameterized coset intersection problem that is used as a subroutine in our algorithm for CHI.

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where canonical means that isomorphic graphs receive identical models. This implies that the recognition and the isomorphism problems for these graphs are solvable in logspace. For the broader class of concave-round graphs, which still possess (not necessarily proper) circular-arc models, we show that a canonical circular-arc model can also be constructed in logspace. As a building block for these results, we design a logspace algorithm for computing canonical circular-arc models of circular-arc hypergraphs; this important class of hypergraphs corresponds to matrices with the circular ones property. Furthermore, we consider the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We show that this problem is solvable in logarithmic space for the classes of proper circular-arc, concave-round, and co-convex graphs.

We present a logspace algorithm for computing a canonical labeling, in fact, a canonical interval representation, for interval graphs. To achieve this, we compute canonical interval representations of interval hypergraphs. This approach also yields a canonical labeling of convex graphs. As a consequence, the isomorphism and automorphism problems for these graph classes are solvable in logspace. For proper interval graphs we also design logspace algorithms computing their canonical representations by proper and by unit interval systems.

We show that the class of sets having polynomial size circuits, P/poly, has EXP/sup NP/-measure zero under each of the following two assumptions: EXP/sup NP//spl ne/ZPP(/spl Sigma//sub 2//sup p/)(which holds if the polynomial time hierarchy does not collapse to ZPP(/spl Sigma//sub 2//sup p/)), or NP is not small (does not have EXP-measure zero).

We present a uniform approach to investigate the relationship between the existence of complete sets for promise classes and the existence of (p-)optimal proof systems for certain languages. Central to our approach is the notion of a test set which can be used to verify that a given nondeterministic polynomial-time machine obeys the promise on a given input. Basically, we show that a promise class C has a many-one complete language if and only if there is a test set for C which has a p-optimal proof system. As an application we are able to improve earlier results. For example, we show that NP/spl cap/co-NP has a many-one complete language, provided that the set TAUT of all valid boolean formulas as well as the set SAT of all satisfiable boolean formulas have p-optimal proof systems. We also apply, the result to other classes and show, for example, that the probabilistic complexity classes BPP, RP, and ZPP have many-one complete languages, provided that the set TAUT/sub 2/ of all valid /spl Pi//sub 2/-formulas in quantified propositional logic has a p-optimal proof system. Finally it is shown that already a collapse of tally sets at the double exponential time level implies the existence of a (p-)optimal proof system for TAUT.

We show that if a self-reducible set has polynomial-size circuits, then it is low for the probabilistic class ZPP (NP). As a consequence we get a deeper collapse of the polynomial-time hierarchy PH to ZPP(NP) under the assumption that NP has polynomial-size circuits. This improves on the well-known result in Karp and Lipton [ Proceedings of the 12th ACM Symposium on Theory of Computing, ACM Press, New York, 1980, pp. 302--309] stating a collapse of PH to its second level $\Sigmap_2$ under the same assumption. Furthermore, we derive new collapse consequences under the assumption that complexity classes like UP, FewP, and C=P have polynomial-size circuits. Finally, we investigate the circuit-size complexity of several language classes. In particular, we show that for every fixed polynomial s, there is a set in ZPP(NP) which does not have O(s(n))-size circuits.

The authors consider uniform subclasses of the nonuniform complexity classes defined by Karp and Lipton [L’Enseign. Math., 28 (1982)) via the notion of advice functions. These subclasses are obtained by restricting the complexity of computing correct advice. Also, the effect of allowing advice functions of limited complexity to depend on the input rather than on the input’s length is investigated. Among other results, using the notions described above, new characterizations of (a) ${\text{NP}}^{{\text{NP}} \cap {\text{SPARSE}}} $, (b) ${\text{NP}}$ with a restricted access to an ${\text{NP}}$ oracle, and (c) the odd levels of the boolean hierarchy are given. As a consequence, it is shown that every set that is nondeterministically truth-table reducible to SAT in the sense of Rich [J. Comput. System Sci., 38 (1989), pp. 511–523) is already deterministically truth-table reducible to SAT. Furthermore, it turns out that the NP reduction classes of bounded versions of this reducibility coincide with the odd levels of the boolean hierarchy.