Planar graph

Abstract: In this paper, we give for constant $k$ a linear-time algorithm that, given a graph $G=(V,E)$, determines whether the treewidth of $G$ is at most $k$ and, if so, finds a tree-decomposition of $G$ with treewidth at most $k$. A consequence is that every minor-closed class of graphs that does not contain all planar graphs has a linear-time recognition algorithm. Another consequence is that a similar result holds when we look instead for path-decompositions with pathwidth at most some constant $k$.

Abstract: Graph embedding algorithms embed a graph into a vector space where the structure and the inherent properties of the graph are preserved The existing graph embedding methods cannot preserve the asymmetric transitivity well, which is a critical property of directed graphs Asymmetric transitivity depicts the correlation among directed edges, that is, if there is a directed path from u to v, then there is likely a directed edge from u to v Asymmetric transitivity can help in capturing structures of graphs and recovering from partially observed graphs To tackle this challenge, we propose the idea of preserving asymmetric transitivity by approximating high-order proximity which are based on asymmetric transitivity In particular, we develop a novel graph embedding algorithm, High-Order Proximity preserved Embedding (HOPE for short), which is scalable to preserve high-order proximities of large scale graphs and capable of capturing the asymmetric transitivity More specifically, we first derive a general formulation that cover multiple popular high-order proximity measurements, then propose a scalable embedding algorithm to approximate the high-order proximity measurements based on their general formulation Moreover, we provide a theoretical upper bound on the RMSE (Root Mean Squared Error) of the approximation Our empirical experiments on a synthetic dataset and three real-world datasets demonstrate that HOPE can approximate the high-order proximities significantly better than the state-of-art algorithms and outperform the state-of-art algorithms in tasks of reconstruction, link prediction and vertex recommendation

Abstract: INTRODUCTION. ELEMENTS OF GRAPH THEORY. The Definition of a Graph. Isomorphic Graphs and Graph Automorphism. Walks, Trails, Paths, Distances and Valencies in Graphs. Subgraphs. Regular Graphs. Trees. Planar Graphs. The Story of the Koenigsberg Bridge Problem and Eulerian Graphs. Hamiltonian Graphs. Line Graphs. Vertex Coloring of a Graph. CHEMICAL GRAPHS. The Concept of a Chemical Graph. Molecular Topology. Huckel Graphs. Polyhexes and Benzenoid Graphs. Weighted Graphs. GRAPH-THEORETICAL MATRICES. The Adjacency Matrix. The Distance Matrix. THE CHARACTERISTIC POLYNOMIAL OF A GRAPH. The Definition of the Characteristic Polynomial. The Method of Sachs for Computing the Characteristic Polynomial. The Characteristic Polynomials of Some Classes of Simple Graphs. The Le Verrier-Faddeev-Frame Method for Computing the Characteristic Polynomial. TOPOLOGICAL ASPECTS OF HUECKEL THEORY. Elements of Huckel Theory. Isomorphism of Huckel Theory and Graph Spectral Theory. The Huckel Spectrum. Charge Densities and Bond Orders in Conjugated Systems. The Two-Color Problem in Huckel Theory. Eigenvalues of Linear Polyenes. Eigenvalues of Annulenes. Eigenvalues of Moebius Annulenes. A Classification Scheme for Monocyclic Systems. Total p-Electron Energy. TOPOLOGICAL RESONANCE ENERGY. Huckel Resonance Energy. Dewar Resonance Energy. The Concept of Topological Resonance Energy. Computation of the Acyclic Polynomial. Applications of the TRE Model. ENUMERATION OF KEKULE VALENCE STRUCTURES. The Role of Kekule Valence Structures in Chemistry. The Identification of Kekule Systems. Methods for the Enumeration of Kekule Structures. The Concept of Parity of Kekule Structures. THE CONJUGATED-CIRCUIT MODEL. The Concept of Conjugated Circuits. The p-Resonance Energy Expression. Selection of the Parameters. Computational Procedure. Applications of the Conjugated-Circuit Model. Parity of Conjugated Circuits. TOPOLOGICAL INDICES. Definitions of Topological Indices. The Three-Dimensional Wiener Number. ISOMER ENUMERATION. The Cayley Generation Functions. The Henze-Blair Approach. The Polya Enumeration Method. The Enumeration Method Based on the N-Tuple Code.