12133267
A note on graph reconstruction
Suppose G and G′ are graphs on the same vertex set V such that for each x ∈ V there is an isomorphism θx of G−x to G′−x. We prove in this paper that if there is a vertex x ∈ V and an automorphism σ of G − x such that θx agrees with σ on all except for at most three vertices of V −x, then G is isomorphic to G′. As a corollary we prove that if a graph G has a vertex which is contained in at most three bad pairs, then G is reconstructible. Here a pair of vertices x, y of a graph G is called a bad pair if there exist u, v ∈ V (G) such that {u, v} 6 = {x, y} and G− {x, y} is isomorphic to G− {u, v}. All graphs discussed here are finite simple graphs. The vertex set and edge set of a graph G are denoted by V (G) and E(G) respectively. If A is a subset of V (G), we use G|A and G−A to denote the subgraphs of G induced by A and V (G)−A respectively. When A = {x} is a singleton, we use G−x instead of G−{x}. For two subsets X, Y of V (G), we use eG(X, Y ) to denote the number of edges joining a vertex of X to a vertex of Y . For brevity, we write eG(x, X) for eG({x}, X) and eG(X) for eG(X, X). The degree of x in G is denoted by dG(x). We shall use some notations defined in [4]. Two graphs G and H are hypomorphic if there exists a bijection f : V (G) → V (H) such that G− x is ∗Current address: SFB 343, Universität Bielefeld, D-33501 Bielefeld, FRG
7791840
Pseudo-Hamiltonian-connected graphs
Abstract Given a graph G and a positive integer k , denote by G[k] the graph obtained from G by replacing each vertex of G with an independent set of size k . A graph G is called pseudo- k Hamiltonian-connected if G[k] is Hamiltonian-connected, i.e., every two distinct vertices of G[k] are connected by a Hamiltonian path. A graph G is called pseudo Hamiltonian-connected if it is pseudo- k Hamiltonian-connected for some positive integer k . This paper proves that a graph G is pseudo-Hamiltonian-connected if and only if for every non-empty proper subset X of V(G) , |N(X)|>|X| . The proof of the characterization also provides a polynomial-time algorithm that decides whether or not a given graph is pseudo-Hamiltonian-connected. The characterization of pseudo-Hamiltonian-connected graphs also answers a question of Richard Nowakowski, which motivated this paper.
206792834
Star chromatic numbers of graphs
We investigate the relation between the star-chromatic number χ(G) and the chromatic number χ(G) of a graphG. First we give a sufficient condition for graphs under which their starchromatic numbers are equal to their ordinary chromatic numbers. As a corollary we show that for any two positive integersk, g, there exists ak-chromatic graph of girth at leastg whose star-chromatic number is alsok. The special case of this corollary withg=4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose star-chromatic number equals their chromatic number. We then study the star-chromatic number of An infinite family of graphs is constructed to show that for each ε>0 and eachm≥2 there is anm-connected (m+1)-critical graph with star chromatic number at mostm+ε. This answers another question asked by Abbott and Zhou.
42953989
The game coloring number of pseudo partial k-trees
Abstract This paper introduces a new class of graphs: (a,b)-pseudo partial k-trees. In some sense, the parameters (a,b) measure a graph's deviation from being a partial k-tree. In particular, a (0,0)-pseudo partial k-tree is just a partial k-tree. We discuss the game coloring number (as well as the game chromatic number) of (a,b)-pseudo partial k-trees, and prove that the game coloring number of an (a,b)-pseudo partial k-tree is at most 3k+2a+b+2. In particular, the game coloring number of a partial k-tree is at most 3k+2. This reduces considerably the previous known upper bound for the game chromatic number of a partial k-tree. Then we describe another strategy for Alice for playing the game, which gives a better upper bound for the game coloring number of (a,b)-pseudo partial 2-trees. Namely, we prove that the game coloring number of an (a,b)-pseudo partial 2-tree is at most a+b+8. By using this result, we prove that the game coloring number of a graph embeddable on an orientable surface of genus g⩾1 is at most ⌊ 1 2 (3 1+48g +23)⌋. This is the first upper bound for the game coloring number of such graphs, and it also improves considerably the previous known upper bound for the game chromatic number of such graphs. Moreover, the strategy for Alice introduced in this paper for playing the game can be considered as a common generalization of the previous known strategies for playing the game on forests, interval graphs, outerplanar graphs and planar graphs.
17776907
Circular perfect graphs
For 1 ≤ d ≤ k, let Kk/d be the graph with vertices 0, 1, · · · , k − 1, in which i ∼ j if d ≤ |i − j| ≤ k − d. The circular chromatic number χc(G) of a graph G is the minimum of those k/d for which G admits a homomorphism to Kk/d. The circular clique number ωc(G) of G is the maximum of those k/d for which Kk/d admits a homomorphism to G. A graph G is circular perfect if for every induced subgraph H of G we have χc(H) = ωc(H). This paper surveys results on circular perfect graphs.
5956515
Homomorphisms to oriented paths
Abstract Given an oriented path P , we characterize those digraphs G which are homomorphic to P . The characterization equates the nonexistence of a homomorphism G → P with the existence of a homomorphism W → G , for some oriented path W which is not homomorphic to P . This result complements the recent polynomial time algorithm of Welzl et al. to find such a homomorphism (if one exists).
5621999
Accelerated Kernel Feature Analysis
A fast algorithm, Accelerated Kernel Feature Analysis (AKFA), that discovers salient features evidenced in a sample of n unclassified patterns, is presented. Like earlier kernel-based feature selection algorithms, AKFA implicitly embeds each pattern into a Hilbert space, H, induced by a Mercer kernel. An \ell-dimensional linear subspace of H is iteratively constructed by maximizing a variance condition for the nonlinearly transformed sample. This linear subspace can then be used to define more efficient data representations and pattern classifiers. AKFA requires O(\elln2) operations, as compared to 0(n^3) for Sch¨olkof, Smola, and M¨uller’s Kernel Principal Component Analysis (KPCA), and O(\ell^2 n^2) for Smola, Mangasarian, and Sch¨olkopf’s Sparse Kernel Feature Analysis (SKFA). Numerical experiments show that AKFA can generate more concise feature representations than both KPCA and SKFA, and demonstrate that AKFA obtains similar classification performance as KPCA for a face recognition problem.
16957901
Bridging Local and Global Data Cleansing: Identifying Class Noise in Large, Distributed Data Datasets
To cleanse mislabeled examples from a training dataset for efficient and effective induction, most existing approaches adopt a major set oriented scheme: the training dataset is separated into two parts (a major set and a minor set). The classifiers learned from the major set are used to identify noise in the minor set. The obvious drawbacks of such a scheme are twofold: (1) when the underlying data volume keeps growing, it would be either physically impossible or time consuming to load the major set into the memory for inductive learning; and (2) for multiple or distributed datasets, it can be either technically infeasible or factitiously forbidden to download data from other sites (for security or privacy reasons). Therefore, these approaches have severe limitations in conducting effective global data cleansing from large, distributed datasets.In this paper, we propose a solution to bridge the local and global analysis for noise cleansing. More specifically, the proposed effort tries to identify and eliminate mislabeled data items from large or distributed datasets through local analysis and global incorporation. For this purpose, we make use of distributed datasets or partition a large dataset into subsets, each of which is regarded as a local subset and is small enough to be processed by an induction algorithm at one time to construct a local model for noise identification. We construct good rules from each subset, and use the good rules to evaluate the whole dataset. For a given instance Ik, two error count variables are used to count the number of times it has been identified as noise by all data subsets. The instance with higher error values will have a higher probability of being a mislabeled example. Two threshold schemes, majority and non-objection, are used to identify and eliminate the noisy examples. Experimental results and comparative studies on both real-world and synthetic datasets are reported to evaluate the effectiveness and efficiency of the proposed approach.
5944615
Eliminating Class Noise in Large Datasets
This paper presents a new approach for identifying and eliminating mislabeled instances in large or distributed datasets. We first partition a dataset into subsets, each of which is small enough to be processed by an induction algorithm at one time. We construct good rules from each subset, and use the good rules to evaluate the whole dataset. For a given instance Ik, two error count variables are used to count the number of times it has been identified as noise by all subsets. The instance with higher error values will have a higher probability of being a mislabeled example. Two threshold schemes, majority and non-objection, are used to identify the noise. Experimental results and comparative studies from real-world datasets are reported to evaluate the effectiveness and efficiency of the proposed approach.
11534248
Mining in Anticipation for Concept Change: Proactive-Reactive Prediction in Data Streams
Prediction in streaming data is an important activity in the modern society. Two major challenges posed by data streams are (1) the data may grow without limit so that it is difficult to retain a long history of raw data; and (2) the underlying concept of the data may change over time. The novelties of this paper are in four folds. First, it uses a measure of conceptual equivalence to organize the data history into a history of concepts. This contrasts to the common practice that only keeps recent raw data. The concept history is compact while still retains essential information for learning. Second, it learns concept-transition patterns from the concept history and anticipates what the concept will be in the case of a concept change. It then proactively prepares a prediction model for the future change. This contrasts to the conventional methodology that passively waits until the change happens. Third, it incorporates proactive and reactive predictions. If the anticipation turns out to be correct, a proper prediction model can be launched instantly upon the concept change. If not, it promptly resorts to a reactive mode: adapting a prediction model to the new data. Finally, an efficient and effective system RePro is proposed to implement these new ideas. It carries out prediction at two levels, a general level of predicting each oncoming concept and a specific level of predicting each instance's class. Experiments are conducted to compare RePro with representative existing prediction methods on various benchmark data sets that represent diversified scenarios of concept change. Empirical evidence offers inspiring insights and demonstrates the proposed methodology is an advisable solution to prediction in data streams.
9199797
Error Detection and Impact-Sensitive Instance Ranking in Noisy Datasets
Given a noisy dataset, how to locate erroneous instances and attributes and rank suspicious instances based on their impacts on the system performance is an interesting and important research issue. We provide in this paper an Error Detection and Impact-sensitive instance Ranking (EDIR) mechanism to address this problem. Given a noisy dataset D, we first train a benchmark classifier T from D. The instances, that cannot be effectively classified by T are treated as suspicious and forwarded to a subset S. For each attribute Ai, we switch Ai and the class label C to train a classifier APi for Ai. Given an instance Ik in S, we use APi and the benchmark classifier T to locate the erroneous value of each attribute Ai. To quantitatively rank instances in S, we define an impact measure based on the Information-gain Ratio (IR). We calculate IRi between attribute Ai and C, and use IRi as the impact-sensitive weight of Ai. The sum of impact-sensitive weights from all located erroneous attributes of Ik indicates its total impact value. The experimental results demonstrate the effectiveness of our strategies.