Fast Parallel Algorithms for the Subgraph Homeomorphism & the Subgraph Isomorphism Problems for Classes of Planar Graphs

23 pagesWe consider the problems of subgraph homeomorphism with fixed pattern graph, recognition, and subgraph isomorphism for some classes of planar graphs. Following the results of Robertson and Seymour on forbidden minor characterization, we show that the problems of fixed subgraph homeomorphism and recognition for any family of planar graphs closed under minor taking are in NC (i.e., they can be solved by an algorithm running in poly-log time using polynomial number of processors). We also show that the related subgraph isomorphism problem for biconnected outerplanar ·graphs is in NC. This is the first example of a restriction of subgraph isomorphism to a non-trivial graph family admitting an NC algorith

Subgraph detection for average detectability of LTI systems

International audienceObservation and detection of networked systems aim to reconstruct the evolution of the system based on the measurement of few nodes. In large-scale networks, reconstructing the exact state of each node becomes more complex and in practice it is often superfluous. Reconstructing an aggregated version of the system is often sufficient. In the light of this observation, we consider the notion of average detectability: A system is said to be average detectable if it is possible to reconstruct the average of the subset of its unmeasured nodes. We show here that for a particular type of system, that is negative uniform networks, the average detectability property is reached when the subgraph induced by the unmeasured nodes is regular. Thus, we study the detection of such regular induced subgraph and we propose an algorithm to complete this task. We introduce also the relaxed notion of quasi-regularity ensuring an approximate reconstruction of the average. This paper presents algorithms to detect regular induced subgraphs (RIS) and quasi-regular induced subgraph (q-RIS). We propose an extension to detect multiple quasi-regular induced subgraphs (mq-RIS) in order to reconstruct the average of several subgraphs of the system. Finally we apply our method to the evolution of an epidemic spreading over a simulated contact network over the largest cities in France based on a SIS model

Induced Subgraph Isomorphism on proper interval and bipartite permutation graphs *

Abstract Given two graphs G and H as input, the Induced Subgraph Isomorphism (ISI) problem is to decide whether G has an induced subgraph that is isomorphic to H. This problem is NP-complete already when G and H are restricted to disjoint unions of paths, and consequently also NP-complete on proper interval graphs and on bipartite permutation graphs. We show that ISI can be solved in polynomial time on proper interval graphs and on bipartite permutation graphs, provided that H is connected. As a consequence, we obtain that ISI is fixed-parameter tractable on these two graph classes, when parametrised by the number of connected components of H. Our results contrast and complement the following known results: W [1]-hardness of ISI on interval graphs when parametrised by the number of vertices of H, NP-completeness of ISI on connected interval graphs and on connected permutation graphs, and NP-completeness of Subgraph Isomorphism on connected proper interval graphs and connected bipartite permutation graphs

Tree comparison: enumeration and application to cheminformatics

Graphs are a well-known data structure used in many application domains that rely on relationships between individual entities. Examples are social networks, where the users may be in friendship with each other, road networks, where one-way or bidirectional roads connect crossings, and work package assignments, where workers are assigned to tasks. In chem- and bioinformatics, molecules are often represented as molecular graphs, where vertices represent atoms, and bonds between them are represented by edges connecting the vertices. Since there is an ever-increasing amount of data that can be treated as graphs, fast algorithms are needed to compare such graphs. A well-researched concept to compare two graphs is the maximum common subgraph. On the one hand, this allows finding substructures that are common to both input graphs. On the other hand, we can derive a similarity score from the maximum common subgraph. A practical application is rational drug design which involves molecular similarity searches. In this thesis, we study the maximum common subgraph problem, which entails finding a largest graph, which is isomorphic to subgraphs of two input graphs. We focus on restrictions that allow polynomial-time algorithms with a low exponent. An example is the maximum common subtree of two input trees. We succeed in improving the previously best-known time bound. Additionally, we provide a lower time bound under certain assumptions. We study a generalization of the maximum common subtree problem, the block-and-bridge preserving maximum common induced subgraph problem between outerplanar graphs. This problem is motivated by the application to cheminformatics. First, the vast majority of drugs modeled as molecular graphs is outerplanar, and second, the blocks correspond to the ring structures and the bridges to atom chains or linkers. If we allow disconnected common subgraphs, the problem becomes NP-hard even for trees as input. We propose a second generalization of the maximum common subtree problem, which allows skipping vertices in the input trees while maintaining polynomial running time. Since a maximum common subgraph is not unique in general, we investigate the problem to enumerate all maximum solutions. We do this for both the maximum common subtree problem and the block-and-bridge preserving maximum common induced subgraph problem between outerplanar graphs. An arising subproblem which we analyze is the enumeration of maximum weight matchings in bipartite graphs. We support a weight function between the vertices and edges for all proposed common subgraph methods in this thesis. Thus the objective is to compute a common subgraph of maximum weight. The weights may be integral or real-valued, including negative values. A special case of using such a weight function is computing common subgraph isomorphisms between labeled graphs, where labels between mapped vertices and edges must be equal. An experimental study evaluates the practical running times and the usefulness of our block-and-bridge preserving maximum common induced subgraph algorithm against state of the art algorithms

On The Complexity of Isomorphism In Finite Group Theory and Symbolic Dynamics

This thesis looks at the complexity of isomorphism in two fairly distinct areas of mathematics. First, we consider several computational problems related to sub-shifts of finite type and, in particular, conjugacy restricted to&nbsp;k-block&nbsp;codes: verifying a proposed&nbsp;k-block&nbsp;conjugacy, deciding if two shifts admit a&nbsp;k-block conjugacy, reducing the representation size of a shift via a&nbsp;k-block conjugacy, and recognizing if a sofic shift is a shift of finite type. We give a polynomial-time algorithm for verification, show GI-hardness for deciding conjugacy, show NP-hardness for reducing representation size, and give a polynomial-time algorithm for recognizing shifts of finite type. Our approach focuses on 1-block conjugacies between vertex shifts, from which we generalize to&nbsp;k-block conjugacies and to edge shifts. We also highlight several open problems. Second, we consider isomorphism between quotients of centrally indecomposible genus 2 p-groups. We show isomorphism between quotients of such groups by non-central subgroups can be determined in polynomial time. The centrally indecomposible genus 2 groups split into two cases: flat and sloped [20]. We give a polynomial-time algorithm which correctly decides isomorphism between quotients of the flat genus 2 groups H♭1 (Fq) by central subgroups whenever the algorithm succeeds; we believe the algorithm always succeeds and have tested it on tens of millions of random examples. We again highlight several open problems.</p