41898252
Vertex-pancyclicity of edge-face-total graphs
The edge-face-total graph r(G) of a plane graph G is the graph defined on the vertex set E(G)∪F(G) so that two vertices in r(G) are joined by an edge if and only if they were adjacent or incident in G. In this paper we prove that (1) the edge-face-total graph of a 2-connected plane graph is vertex-pancyclic and there exists a connected plane graph G with cut vertices such that r(G) is non-Hamiltonian; (2) the line graph of a 2-connected plane graph with at most one face of degree ⩾4 is pancyclic.
1003852
Upper Bounds of Entire Chromatic Number of Plane Graphs
The entire chromatic number χvef (G) of a plane graph G is the least number of colors assigned to the vertices, edges and faces so that every two adjacent or incident pair of them receive different colors. Kronk and Mitchem (1973) conjectured that χvef (G) ≤ 1 + 4 for every plane graph G. In this paper we prove the conjecture for a plane graph G having χ ′(G) = 1 and give a upper bound χvef (G) ≤ 1+ 5 for all plane graphs, where χ ′(G) and 1 are the chromatic index and the maximum degree of G, respectively.
13354529
Edge-pancyclicity of coupled graphs
The coupled graph c(G) of a plane graph G is the graph defined on the vertex set V(G)∪F(G) so that two vertices in c(G) are joined by an edge if and only if they are adjacent or incident in G. We prove that the coupled graph of a 2-connected plane graph is edge-pancyclic. However, there exists a 2-edge-connected plane graph G such that c(G) is not Hamiltonian.
9810371
The edge-face choosability of plane graphs
A plane graph G is said to be k-edge-face choosable if, for every list L of colors satisfying |L(x)|=k for every edge and face x, there exists a coloring which assigns to each edge and each face a color from its list so that any adjacent or incident elements receive different colors. We prove that every plane graph G with maximum degree Δ(G) is (Δ(G)+3)-edge-face choosable.
6853966
Total chromatic number of planar graphs with maximum degree ten
41218825
Relaxed game chromatic number of graphs
This paper discusses a coloring game on graphs. Let k, d be non-negative integers and C a set of k colors. Two persons, Alice and Bob, alternately color the vertices of G with colors from C, with Alice having the first move. A color i is legal for an uncolored vertex x if by coloring x with color i, the subgraph of G induced by those vertices of color i has maximum degree at most d. Each move of Alice or Bob colors an uncolored vertex with a legal color. The game is over if either all vertices are colored, or no more vertices can be colored with a legal color. Alice's goal is to produce a legal coloring which colors all the vertices of G, and Bob's goal is to prevent this from happening. We shall prove that if G is a forest, then for k = 3, d ≥ 1, Alice has a winning strategy. If G is an outerplanar graph, then for k = 6 and d ≥ 1, Alice has a winning strategy.
377228
SPIN: mining maximal frequent subgraphs from graph databases
One fundamental challenge for mining recurring subgraphs from semi-structured data sets is the overwhelming abundance of such patterns. In large graph databases, the total number of frequent subgraphs can become too large to allow a full enumeration using reasonable computational resources. In this paper, we propose a new algorithm that mines only maximal frequent subgraphs, i.e. subgraphs that are not a part of any other frequent subgraphs. This may exponentially decrease the size of the output set in the best case; in our experiments on practical data sets, mining maximal frequent subgraphs reduces the total number of mined patterns by two to three orders of magnitude.Our method first mines all frequent trees from a general graph database and then reconstructs all maximal subgraphs from the mined trees. Using two chemical structure benchmarks and a set of synthetic graph data sets, we demonstrate that, in addition to decreasing the output size, our algorithm can achieve a five-fold speed up over the current state-of-the-art subgraph mining algorithms.
1857380
Mining RNA Tertiary Motifs with Structure Graphs
We present a novel application of graph database mining to identify tertiary motifs in RNA structures. In our method, we abstract an RNA molecule as a labeled graph and use a frequent subgraph mining technique to derive tertiary motifs. By applying our technique to ribosome RNA and transfer RNA, we have identified known RNA tertiary motifs such as the ribose zipper and U-turn, plus candidates for novel tertiary motifs. Finally, we suggest an iterative multiple structure alignment algorithm to classify tertiary motifs and generate consensus motifs.
17166482
Benchmarking the effectiveness of sequential pattern mining methods
Recently, there is an increasing interest in new intelligent mining methods to find more meaningful and compact results. In intelligent data mining research, accessing the quality and usefulness of the results from different mining methods is essential. However, there is no general benchmarking criteria to evaluate whether these new methods are indeed more effective compared to the traditional methods. Here we propose a novel benchmarking criteria that can systematically evaluate the effectiveness of any sequential pattern mining method under a variety of situations. The benchmark evaluates how well a mining method finds known common patterns in synthetic data. Such an evaluation provides a comprehensive understanding of the resulting patterns generated from any mining method empirically. In this paper, the criteria are applied to conduct a detailed comparison study of the support-based sequential pattern model with an approximate pattern model based on sequence alignment. The study suggests that the alignment model will give a good summary of the sequential data in the form of a set of common patterns in the data. In contrast, the support model generates massive amounts of frequent patterns with much redundancy. This suggests that the results of the support model require more post processing before it can be of actual use in real applications.
15982467
On intrinsic representations of 3D polygons for shape blending
Abstract An intrinsic representation of a geometric object comprises a set of parameters that are invariant under Euclidean transformations and determine the shape and size of the object uniquely. Unlike the case of a two-dimensional polygon which has essentially only one intrinsic representation, a three-dimensional (3D) polygon allows several variations. We study three different intrinsic representations of a 3D polygon, focusing on properties regarding their applications in shape blending. We show that only one of these representations, which is a natural extension to the intrinsic functions of a 3D differentiable curve, is suitable for shape blending, while the others lead to discontinuous shape transformation. Detailed discussions about the evaluation criteria, numerical sensitivity, and an inherent difficulty in shape blending of 3D polygons are also presented.
14471402
Computing quadric surface intersections based on an analysis of plane cubic curves
Computing the intersection curve of two quadrics is a fundamental problem in computer graphics and solid modeling. We present an algebraic method for classifying and parameterizing the intersection curve of two quadric surfaces. The method is based on the observation that the intersection curve of two quadrics is birationally related to a plane cubic curve. In the method this plane cubic curve is computed first and the intersection curve of the two quadrics is then found by transforming the cubic curve by a rational quadratic mapping. Topological classification and parameterization of the intersection curve are achieved by invoking results from algebraic geometry on plane cubic curves. 2003 Elsevier Science (USA). All rights reserved.