Abstract. 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. All of the above algorithms have in common that they compute a perfect elimination ordering (peo) of the graph's vertices. This ordering has the property that for every vertex, its neighborhood among its successors forms a clique. Fulkerson and Gross [FG65] show that a graph has a peo if and only if it is chordal, and the above methods determine whether a graph is an interval graph in linear time once a peo is known.Recognition of interval graphs in logspace follows from the results of Reif [Rei84] and Reingold [Rei08]. In this article, we describe a logspace algorithm that, given an interval graph G, constructs a canonical interval representation I G , i.e., G is isomorphic to the intersection graph of I G , and isomorphic graphs G 1 ∼ = G 2 are
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.
Abstract. We present logspace algorithms for the canonical labeling problem and the representation problem of Helly circular-arc (HCA) graphs. The first step is a reduction to canonical labeling and representation of interval intersection matrices. In a second step, the Δ trees employed in McConnell's linear time representation algorithm for interval matrices are adapted to the logspace setting and endowed with additional information to allow canonization. As a consequence, the isomorphism and recognition problems for HCA graphs turn out to be logspace complete.
Abstract. We study optimization versions of Graph Isomorphism. Given two graphs G1, G2, we are interested in finding a bijection π from V (G1) to V (G2) that maximizes the number of matches (edges mapped to edges or non-edges mapped to non-edges). We give an n O(log n) time approximation scheme that for any constant factor α < 1, computes an α-approximation. We prove this by combining the n O(log n) time additive error approximation algorithm of Arora et al. [Math. Program., 92, 2002] with a simple averaging algorithm. We also consider the corresponding minimization problem (of mismatches) and prove that it is NP-hard to α-approximate for any constant factor α. Further, we show that it is also NP-hard to approximate the maximum number of edges mapped to edges beyond a factor of 0.94. We also explore these optimization problems for bounded color class graphs which is a well studied tractable special case of Graph Isomorphism. Surprisingly, the bounded color class case turns out to be harder than the uncolored case in the approximate setting.
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