110 research outputs found
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
PlanE: Representation Learning over Planar Graphs
Graph neural networks are prominent models for representation learning over
graphs, where the idea is to iteratively compute representations of nodes of an
input graph through a series of transformations in such a way that the learned
graph function is isomorphism invariant on graphs, which makes the learned
representations graph invariants. On the other hand, it is well-known that
graph invariants learned by these class of models are incomplete: there are
pairs of non-isomorphic graphs which cannot be distinguished by standard graph
neural networks. This is unsurprising given the computational difficulty of
graph isomorphism testing on general graphs, but the situation begs to differ
for special graph classes, for which efficient graph isomorphism testing
algorithms are known, such as planar graphs. The goal of this work is to design
architectures for efficiently learning complete invariants of planar graphs.
Inspired by the classical planar graph isomorphism algorithm of Hopcroft and
Tarjan, we propose PlanE as a framework for planar representation learning.
PlanE includes architectures which can learn complete invariants over planar
graphs while remaining practically scalable. We empirically validate the strong
performance of the resulting model architectures on well-known planar graph
benchmarks, achieving multiple state-of-the-art results.Comment: Proceedings of the Thirty-Seventh Annual Conference on Advances in
Neural Information Processing Systems (NeurIPS 2023). Code and data available
at: https://github.com/ZZYSonny/Plan
The Degree Sequence of Random Graphs from Subcritical Classes
In this work we determine the expected number of vertices of degree k = k(n) in a graph with n vertices that is drawn uniformly at random from a subcritical graph class. Examples of such classes are outerplanar, series-parallel, cactus and clique graphs. Moreover, we provide exponentially small bounds for the probability that the quantities in question deviate from their expected value
Efficient Frequent Subtree Mining Beyond Forests
A common paradigm in distance-based learning is to embed the instance space into some appropriately chosen feature space equipped with a metric and to define the dissimilarity between instances by the distance of their images in the feature space. If the instances are graphs, then frequent connected subgraphs are a well-suited pattern language to define such feature spaces. Identifying the set of frequent connected subgraphs and subsequently computing embeddings for graph instances, however, is computationally intractable. As a result, existing frequent subgraph mining algorithms either restrict the structural complexity of the instance graphs or require exponential delay between the output of subsequent patterns. Hence distance-based learners lack an efficient way to operate on arbitrary graph data. To resolve this problem, in this thesis we present a mining system that gives up the demand on the completeness of the pattern set to instead guarantee a polynomial delay between subsequent patterns. Complementing this, we devise efficient methods to compute the embedding of arbitrary graphs into the Hamming space spanned by our pattern set. As a result, we present a system that allows to efficiently apply distance-based learning methods to arbitrary graph databases. To overcome the computational intractability of the mining step, we consider only frequent subtrees for arbitrary graph databases. This restriction alone, however, does not suffice to make the problem tractable. We reduce the mining problem from arbitrary graphs to forests by replacing each graph by a polynomially sized forest obtained from a random sample of its spanning trees. This results in an incomplete mining algorithm. However, we prove that the probability of missing a frequent subtree pattern is low. We show empirically that this is true in practice even for very small sized forests. As a result, our algorithm is able to mine frequent subtrees in a range of graph databases where state-of-the-art exact frequent subgraph mining systems fail to produce patterns in reasonable time or even at all. Furthermore, the predictive performance of our patterns is comparable to that of exact frequent connected subgraphs, where available. The above method considers polynomially many spanning trees for the forest, while many graphs have exponentially many spanning trees. The number of patterns found by our mining algorithm can be negatively influenced by this exponential gap. We hence propose a method that can (implicitly) consider forests of exponential size, while remaining computationally tractable. This results in a higher recall for our incomplete mining algorithm. Furthermore, the methods extend the known positive results on the tractability of exact frequent subtree mining to a novel class of transaction graphs. We conjecture that the next natural extension of our results to a larger transaction graph class is at least as difficult as proving whether P = NP, or not. Regarding the graph embedding step, we apply a similar strategy as in the mining step. We represent a novel graph by a forest of its spanning trees and decide whether the frequent trees from the mining step are subgraph isomorphic to this forest. As a result, the embedding computation has one-sided error with respect to the exact subgraph isomorphism test but is computationally tractable. Furthermore, we show that we can leverage a partial order on the pattern set. This structure can be used to reduce the runtime of the embedding computation dramatically. For the special case of Jaccard-similarity between graph embeddings, a further substantial reduction of runtime can be achieved using min-hashing. The Jaccard-distance can be approximated using small sketch vectors that can be computed fast, again using the partial order on the tree patterns
Structure and enumeration of K4-minor-free links and link diagrams
We study the class L of link-types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L and subclasses of it, with respect to the minimum number of crossings or edges in a projection of L' in L. Further, we obtain counting formulas and asymptotic estimates for the connected K4-minor-free link-diagrams, minimal K4-minor-free link-diagrams, and K4-minor-free diagrams of the unknot.Peer ReviewedPostprint (author's final draft
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