599 Dynamic Generators of Topologically Embedded Graphs David Eppstein* Abstract equator followed byj turns around a longitude. The relations We provide a data structure for maintaining an embedding of a fundamental group describe null-homotopic combina- of a graph on a surface (represented combinatorially by a tions of generators: in the toms, the relation goglgo-ig~-t = permutation of edges around each vertex) and computing 1 implies that the group is commutative. generators of the fundamental group of the surface, in amor- In addition to their topological interest, fundamental tized time O(logn + logg(loglogg) 3) per update on a sur- groups and their generators have other applications. Cutting face of genus g; we can also test orientability of the surface a surface along its generator curves produces a planar surface in the same time, and maintain the minimum and maximum with holes at the cuts, and Enckson and Har-Peled [8] have spanning tree of the graph in time O(log n + log4 g) per up- recently investigated the problem of finding short cutsets in date. Our data structure allows edge insertion and deletion nonplanar surfaces. As we will see, this cutting technique as well as the dual operations; these operations may implic- allows one to apply planar graph algorithms such as sepa- itly change the genus of the embedding surface. We apply rator construction [1] and tree decomposition [5] to nonpla- similar ideas to improve the constant factor in a separator nar graphs. We will also show how to use the generators theorem for low-genus graphs, and to find in linear time a to test orientability of surfaces and maintain minimum span- tree-decomposition of low-genus low-diameter graphs. ning trees of graphs embedded in surfaces. Subgroups of the fundamental group correspond to cov- 1 Introduction ering spaces, the most important of which is the universal cover. In the example of the toms, the universal cover is a In this paper we introduce a tool, the tree-cotree decomposi- plane, and the endpoints of curves gag1t lift to a planar lattice. tion, for constructing generators of fundamental groups of If one chooses any pair of generators of the lattice, curves surfaces, and apply it to several algorithmic problems for from the origin to these points map to a pair of generators graphs embedded on surfaces. for the fundamental group of the toms. Thus construction of The fundamental group of a surface measures the ability generators is also closely related to lattice basis reduction, to transform one curve continuously into another. Two Euclid's algorithm, and continued fractions. closed curves co and ct on a surface are homotopic if there exists a continuous map from the cylinder [0, 1] × S t to 1.1 New Results the surface such that each cylinder boundary i × S 1 is mapped homeomorphically to curve ci. The elements of We show the following results. the fundamental group are equivalence classes of curves • We provide a simple dynamic graph data structure under homotopy. The identity element is the class of curves for maintaining a dynamic graph, embedded in a 2- homotopic to the boundary of a disk, the inverse of a curve manifold, subject to updates that insert or delete edges, is formed by tracing the same curve backwards, and the changing the ordering of edges around each vertex concatenation of two curves forms their product in the group. and thus implicitly changing the underlying 2-manifold. For further exposition see any topology text, e.g. [17]. Our structure can also handle the dual operations of Fundamental groups are often infinite, but can be spec- edge contraction and expansion, and maintains a set ified finitely by a system of generators and relations. The of generators for the fundamental group of the surface, generators of a fundamental group are a system of closed as well as additional information such as the minimum curves with the property that any other curve can be gener- spanning tree of the graph or the orientability of the ated (up to homotopy) by concatenations of closed curves surface, in amortized time O(g log n) per update where in the system. For instance, on a toms, two natural closed g is the genus of the surface at the time of each update. curves to use as generators are an equatoral circle go and a longitudinal circle gl; any other closed curve c on the toms • By combining our simple structure with ideas of sepa- is homotopic to a curve g~g~ that winds i times around the rator based sparsification [6] and recent polylogarith- mic algorithms for dynamic connectivity in general *University of California, Irvine, Dept. of Information & Computer graphs [12,18] we further improve our time bounds, to Science, lrvine, CA 92697, USA, eppsl:eS.n@:i.cs, uc:J., edu O(logn + logg(loglogg) 3) per update for generators 600 and ofientability, and O (log n + (log g)4) per update for on dynamic connectivity and minimum spanning trees in minimum spanning trees. embedded planar graphs [7] which worked by decomposing the graph into two subgraphs, a minimum spanning tree and a We improve by a factor of x/~ the constant factor in maximum spanning cotree. We use a similar decomposition the best previous separator theorem for bounded genus here, however for surfaces of nonzero genus we end up with graphs [1]. leftover edges that belong neither to the tree nor to the cotree. We provide an efficient algorithm for our previous non- These extra edges are what we use to form the generators of constructive result, that genus g graphs with diameter D the surface. We also combine the dynamic graph algorithms have a tree-decomposition with treewidth O(gD) [5]. from that previous work, with a technique for maintaining small representative subgraphs [6] and more recent methods 1.2 Related Work of Holm et al. [12] to achieve time bounds that match the best of these dynamic graph algorithms. The previous work most close to ours is on construction of canonical schemata and cutsets. 2 Representing an Embedding A canonical schema for a 2-manifold is a set of gen- erators for its fundamental group, having a single prespeci- We consider maps: graphs embedded on two-dimensional fled relation (1] g2ig2i+lg2i-1 g2i+l- 1 for oriented surfaces, I]I g/2 manifolds, in such a way that the cells of the embedding are for unoriented surfaces). The generators must have a com- disks. We do not require that our graphs be simple. Such mon basepoint, but unfike the ones in our construction they an embedding on an orientable manifold can be represented are not required to follow the edges of a given graph on the by specifying a clockwise ordering of the edges around each surface: they may pass across the interior of cells. Vegter vertex [10, Section 3.2.4]. However for our purposes it is and Yap [19] showed that canonical schemata exist with total convenient to use a somewhat more general representation, complexity O(ng) where g is the surface genus; this bound is the graph-encoded map [3, 13], which also allows unofi- tight in the worst case, and they also showed that a canonical entable embeddings. We now briefly describe this represen- schema can be found in time O(ng). The generators we find tation. can have the same O(gn) total complexity, but we can find an Suppose we are given a map, such as the embedding impficit representation of the generators in finear time. How- of K3,3 shown on the left of Figure 1. Aflag of the map is ever, the generators we find are not necessarily in canonical defined to be the triple formed by a mutually adjacent vertex, form. edge, and cell of the map. For a flag F, let V(F) denote the Recently, Erickson and Har-Peled [8] studied the prob- vertex, E(F) denote the edge, and C(F) denote the cell. We lem of finding a cutset, that is, a set of edges the com- define three operations that change one flag F into another: plement of which forms a topological disk on the given • Sv(F) is a flag formed by the edge and cell of F but surfaces. Erickson and Har-Peled found algorithms for with a different vertex, found at the other endpoint of computing minimum-length cutsets in polynomial time for E(F). bounded-genus embeddings, but showed the problem to be NP-complete in the case of unbounded genus. In the ap- • SE(F) is a flag formed by the vertex and cell of F pfications to tree decomposition and separator theorems, we but with a different edge, found by tracing around the use our techniques to find cutsets with a guaranteed bound boundary of C(F) from E(F) in the direction of V(F). on the total length, but our cutsets may be far from the mini- • Sc(F) is a flag formed by the vertex and edge of F mum possible length. but with a different cell, the cell that is the next one Planar separator theorems have been long studied [14] clockwise or counterclockwise around V(F) from C(F) and there has been much work on improved constants for in the direction of E(F). such theorems [16]. More generally, for graphs embedded on genus g surfaces, a separator with O(~/~) vertices can Each of these operations produces a unique well-defined be found in O(n) time [1]. Specifically, Aleksandrov and flag, and the neighboring relation between flags is symmet- Djidjev [1] showed that, for any e > 0, one can find a set ric: Sv(Sv(F)) = Se(Se(F)) = Sc(Sc(F)) = F.