We describe the first algorithms to compute minimum cuts in surface-embedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)-cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)-time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimum-cut algorithm computes a minimum-cost subgraph in every Z2-homology class. We also prove that finding a minimum-cost subgraph homologous to a single input cycle is NP-hard.

We describe the first algorithms to compute maximum flows in surface-embedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)-flow in O(g 7 n log 2 n log 2 C) time for integer capacities that sum to C, or in (g log n) O(g) n time for real capacities. Except for the special case of planar graphs, for which an O(n log n)-time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surface-embedded graphs follow from algorithms for general sparse graphs. Our key insight is to optimize the relative homology class of the flow, rather than directly optimizing the flow itself. A dual formulation of our algorithm computes the minimum-cost cycle or circulation in a given (real or integer) homology class.

... In passing we establish an upper bound for a related number D(G, G0), the minimum quantifier depth of a first order sentence which is true on exactly one of graphs G and G0. If G and G0 are non-isomorphic and both have n vertices, then D(G, G0) < = (n + 3)/2. This bound is tight up to an additive constant of 1. If we additionally require that a sentence distinguishing G and G0 is existential, we prove only a slightly weaker bound D(G, G0) < = (n + 5)/2.

We say that a first order formula Φ distinguishes a structure M over vocabulary L from another structure M ′ over the same vocabulary if Φ is true on M but false on M ′. A formula Φ defines an L-structure M if Φ distinguishes M from any other non-isomorphic L-structure M ′. A formula Φ identifies an n-element L-structure M if Φ distinguishes M from any other non-isomorphic n-element L-structure M ′. We prove that every n-element structure M is identifiable by a formula with quantifier rank less than (1 − 1 2k)n+k2 −k+2 and at most one quantifier alternation, where k is the maximum relation arity of M. Moreover, if the automorphism group of M contains no transposition, the same result holds for definability rather than identification. The Bernays-Schönfinkel class consists of prenex formulas in which the existential quantifiers all precede the universal quantifiers. We prove that every n-element structure M is identifiable by a formula in the Bernays-Schönfinkel class with less than (1 − 1 2k2 +2)n + k quantifiers. If in this class of identifying formulas we restrict the number of universal quantifiers to k, then less than n − √ n + k2 quantifiers suffice to identify M and, as long as we keep the number of universal quantifiers bounded by a constant, at total n − O ( √ n) quantifiers are necessary. 1

We introduce a notion of definable tree decompositions of graphs. Actually, a definable tree decomposition of a graph is not just a tree decomposition, but a more complicated structure that represents many different tree decompositions of the graph. It is definable in the graph by a tuple of formulas of some logic. In this paper, only study tree decomposition definable in fixed-point logic. We say that a definable tree decomposition is over a class of graphs if the pieces of the decomposition are in this class. We prove two general theorems lifting definability results from the pieces of a tree decomposition of a graph to the whole graph. Besides unifying earlier work on fixed-point definability and descriptive complexity theory on planar graphs and graphs of bounded tree width, these general results can be used to prove that the class of all graphs without a K5-minor is definable in fixed-point logic and that fixed-point logic with counting captures polynomial time on this class. 1

We say that a first order formula Φ defines a graph G if Φ is true on G and false on every graph G ′ non-isomorphic with G. Let D(G) be the minimal quantifier rank of a such formula. We prove that, if G is a tree of bounded degree or a Hamiltonian (equivalently, 2-connected) outerplanar graph, then D(G) = O(log n), where n denotes the order of G. This bound is optimal up to a constant factor. If h is a constant, for connected graphs with no minor Kh and degree O ( √ n/log n), we prove the bound D(G) = O ( √ n). This result applies to planar graphs and, more generally, to graphs of bounded genus. Our proof techniques are based on the characterization of the quantifier rank as the length of the Ehrenfeucht game on non-isomorphic graphs. We use the separator theorems to design a winning strategy for Spoiler in this game.

Abstract—We prove that fixed-point logic with counting captures polynomial time on all classes of graphs with excluded minors. That is, for every class C of graphs such that some graph H is not a minor of any graph in C, a property P of graphs in C is decidable in polynomial time if and only if it is definable in fixed-point logic with counting. Furthermore, we prove that for every class C of graphs with excluded minors there is a k such that the k-dimensional Weisfeiler-Lehman algorithm decides isomorphism of graphs in C in polynomial time. The Weisfeiler-Lehman algorithm is a combinatorial algorithm for testing isomorphism. It generalises the basic colour refinement algorithm and is much simpler than the known group-theoretic algorithms for deciding isomorphism of graphs with excluded minors. The main technical theorem behind these two results is a “definable structure theorem ” for classes of graphs with excluded minors. It states that graphs with excluded minors can be decomposed into pieces arranged in a treelike structure, together with a linear order of each of the pieces. Furthermore, the decomposition and the linear orders on the pieces are definable in fixed-point logic (without counting). Index Terms—descriptive complexity; graph minor theory; fixed-point logic; graph canonisation I.

Abstract. Graph isomorphism (GI) is one of the few remaining problems in NP whose complexity status couldn’t be solved by classifying it as being either NP-complete or solvable in P. Nevertheless, efficient (polynomial-time or even NC) algorithms for restricted versions of GI have been found over the last four decades. Depending on the graph class, the design and analysis of algorithms for GI use tools from various fields, such as combinatorics, algebra and logic. In this paper, we collect several complexity results on graph isomorphism testing and related algorithmic problems for restricted graph classes from the literature. Further, we provide some new complexity bounds (as well as easier proofs of some known results) and highlight some open questions. 1

Gire and Hoang introduce a xed-point logic with a `symmetric ' choice operator that makes a nondeterministic choice from a de nable set of tuples at each stage in the inductive construction of a relation, as long as the set of tuples is an automorphism class of the structure. We present a clean de nition of the syntax and semantics of this logic and investigate its expressive power. We extend the logic of Gire and Hoang with parameterized and nested xed points and rst-order combinations of xed points. We show that the ability to supply parameters to xed points strictly increases the power of the logic. Our logic can express the graph isomorphism problem and we show that, on almost all structures, it captures P , the class of problems decidable in polynomial time by a deterministic Turing machine with an oracle for graph isomorphism.

We give a deterministic algorithm to find the minimum cut in a surface-embedded graph in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm known for planar graphs, due to Ł ˛acki and Sankowski, for any constant g. Indeed, our algorithm calls Ł ˛acki and Sankowski’s recent O(n log log n) time planar algorithm as a subroutine. Previously, the best time bounds known for this problem followed from two algorithms for general sparse graphs: a randomized algorithm of Karger that runs in O(n log 3 n) time and succeeds with high probability, and a deterministic algorithm of Nagamochi and Ibaraki that runs in O(n 2 log n) time. We can also achieve a deterministic g O(g) n 2 log log n time bound by repeatedly applying the best known algorithm for minimum (s, t)-cuts in surface graphs. The bulk of our work focuses on the case where the dual of the minimum cut splits the underlying surface into multiple components with positive genus. 1