We show that the reachability problem for directed graphs that are either K3,3-free or K5-free is in unambiguous log-space, UL ∩ coUL. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL ∩ coUL. Our algorithm decomposes the graphs into biconnected and triconnected components. This gives a tree structure on these components. The non-planar components are replaced by planar components that maintain the reachabilty properties. For K5-free graphs we also need a decomposition into fourconnected components. A careful analysis finally gives a polynomial size planar graph which can be computed in log-space. We show the same upper bound for computing distances in K3,3-free and K5-free directed graphs and for computing longest paths in K3,3-free and K5-free directed acyclic 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.

The Graph Isomorphism problem restricted to planar graphs has been known to be solvable in polynomial time many years ago. In terms of complexity classes however, the exact complexity of the problem has been established only very recently. It was proved in [6] that planar graph isomorphism can be computed within logarithmic space. Since there is a matching hardness result [12], this shows that the problem is complete for L. Although this could be considered as a result in algorithmics its proof relies on several important new developments in the area of logarithmic space complexity classes and reflects the close connections between algorithms and complexity theory. In this column we give an overview of this result mentioning the developments that led to it.

We discuss the definability of finite graphs in first-order logic with two relation symbols for adjacency and equality of vertices. The logical depth D(G) of a graph G is equal to the minimum quantifier depth of a sentence defining G up to isomorphism. The logical width W(G) is the minimum number of variables occurring in such a sentence. The logical length L(G) is the length of a shortest defining sentence. We survey known estimates for these graph parameters and discuss their relations to other topics (such as the efficiency of the Weisfeiler-Lehman algorithm in isomorphism testing, the evolution of a random graph, quantitative characteristics of the zero-one law, or the contribution of Frank Ramsey to the research on Hilbert’s Entscheidungsproblem). Also, we trace the behavior of the descriptive complexity of a graph as the logic becomes more restrictive (for example, only definitions with a bounded number of variables or quantifier alternations are allowed) or more expressible

We show that, for k constant, k-tree isomorphism can be decided in logarithmic space by giving an O(k log n) space canonical labeling algorithm. The algorithm computes a unique tree decomposition, uses colors to fully encode the structure of the original graph in the decomposition tree and invokes Lindell’s tree canonization algorithm. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for k-trees are all complete for deterministic logspace. Completeness for logspace holds even for simple structural properties of k-trees. We also show that a variant of our canonical labeling algorithm runs in time O((k + 1)! n), where n is the number of vertices, yielding the fastest known FPT algorithm for k-tree isomorphism.

In this paper, we propose a novel Fourier-theoretic approach for estimating the symmetry group G of a geometric object X. Our approach takes as input a geometric similarity matrix between low-order combinations of features of X and then searches within the tree of all feature permutations to detect the sparse subset that defines the symmetry group G of X. Using the Fouriertheoretic approach, we construct an efficient marginalbased search strategy, which can recover the symmetry group G effectively. The framework introduced in this paper can be used to discover symmetries of more abstract geometric spaces and is robust to deformation noise. Experimental results show that our approach can fully determine the symmetries of many geometric objects. 1

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Ever since Reingold’s deterministic logspace algorithm [66] for undirected graph reachability, logspace algorithms for various combinatorial problems have been discovered and it is now a flourishing area of research. Notable examples include special cases of directed graph reachability and planar graph isomorphism [23]. In this interesting article, Johannes Köbler, Sebastian Kuhnert and Oleg Verbitsky discuss the structural properties of interval graphs and other technical ingredients that go into their recent logspace isomorphism algorithm for interval graphs, along with some generalizations and new directions.

The symmetry of a graph is measured by its automorphism group: the set of permutations of the vertices so that all edges and non-edges are preserved. There are natural questions which arise when considering the automorphism group and there are several interesting results in this area that are not well-known. This talk presents some of these results and maybe even proves one or two. First, we will prove that almost all graphs have trivial automorphism group. Second, we will briefly discuss the relation of the graph automorphism and group intersection problems. Then, we will discuss Babai’s constrcution that any finite group with n elements can be represented by a graph on 2n vertices (other than three exceptions). Finally, we will mention there exists a subgroup of Sn that is the automorphism group of no graph of size less than 1 2 ( n 1 2 n). 1 Almost all graphs are rigid Before we can begin the proof of this fact, recall the Chernoff-Hoeffding bounds. Theorem 1.1 ([DP09]). Let X = ∑ n i=1 Xi be a sum of identically distributed independent random variables Xi where Pr(Xi = 1) = p, Pr(Xi = 0) = q = 1 − p. Then, we have the following relative Chernoff-Hoeffding bound for all ε> 0: Pr[X < (1 − ε)np] ≤ e −npε2 /2, Pr[X> (1 + ε)np] ≤ e −npε 2 /2 1.1 Properties of G(n, p) Lemma 1.2. Let ε be a function on n with ε(n)> 0. Then, the probability that G, distributed as G(n + 1, p), has all npε2 vertices of degree deg v ∈ ((1 − ε)np, (1 + ε)np) is at least 1 − 2(n + 1)e 2. Proof. Let Xi,j be the indicator variable for the edge {i, j} appearing in G(n + 1, p) (1 ≤ i < j ≤ n + 1). The expected value is p. By linearity of expectation, E[deg i] = E[Xi,j] = np. j�i By the Chernoff bound, And similarly, Pr[deg i < (1 − ε)np] ≤ e −npε2 /2 Pr[deg i> (1 + ε)np] ≤ e −npε2 /2 1 Hence, Pr[deg i � ((1 − ε)np, (1 + ε)np)] ≤ 2e −npε2 /2 Thus, the probability that all vertices have degree within the requested bounds is at least

Graph isomorphism is an important and widely studied computational problem with a yet unsettled complexity. However, the exact complexity is known for isomorphism of various classes of graphs. Recently, [8] proved that planar isomorphism is complete for log-space. We extend this result further to the classes of graphs which exclude K3,3 or K5 as a minor, and give a log-space algorithm. Our algorithm decomposes K3,3 minor-free graphs into biconnected and those further into triconnected components, which are known to be either planar or K5 components [20]. This gives a triconnected component tree similar to that for planar graphs. An extension of the log-space algorithm of [8] can then be used to decide the isomorphism problem. For K5 minor-free graphs, we consider 3-connected components. These are either planar or isomorphic to the four-rung mobius ladder on 8 vertices or, with a further decomposition, one obtains planar 4-connected components [9]. We give an algorithm to get a unique decomposition of K5 minor-free graphs into bi-, tri- and 4-connected components, and construct trees, accordingly. Since the algorithm of [8] does not deal with four-connected component trees, it needs to be modified in a quite non-trivial way.