4772181
Convex-Round and Concave-Round Graphs
We introduce two new classes of graphs which we call convex-round, respectively concave-round graphs. Convex-round (concave-round) graphs are those graphs whose vertices can be circularly enumerated so that the (closed) neighborhood of each vertex forms an interval in the enumeration. Hence the two classes transform into each other by taking complements. We show that both classes of graphs have nice structural properties. We observe that the class of concave-round graphs properly contains the class of proper circular arc graphs and, by a result of Tucker [ Pacific J. Math., 39 (1971), pp. 535--545], is properly contained in the class of general circular arc graphs. We point out that convex-round and concave-round graphs can be recognized in O(n+m) time (here n denotes the number of vertices and m the number of edges of the graph in question). We show that the chromatic number of a graph which is convex-round (concave-round) can be found in time O(n+m) (O(n2)). We describe optimal O(n+m) time algorithms for finding a maximum clique, a maximum matching, and a Hamiltonian cycle (if one exists) for the class of convex-round graphs. Finally, we pose a number of open problems and conjectures concerning the structure and algorithmic properties of the two new classes and a related third class of graphs.
4789976
In-Tournament Digraphs
Abstract In this paper we introduce a generalization of digraphs that are locally tournaments (and hence of tournaments). This is the class of in-tournament digraphs-the set of predecessors of every vertex induces a tournament. We show that many properties of local tournament digraphs can be extended even to in-tournament digraphs. For instance, any strongly connected in-tournament digraph has a directed hamiltonian cycle. We prove that the underlying graph of any in-tournament digraph is l-homotopic. We investigate the problem of which graphs are orientable as in-tournament digraphs and prove that any graph representable as an intersection subgraph of a unicyclic graph can be so oriented. It is shown that there is a polynomial algorithm for recognizing those graphs that can be oriented as in-tournament digraphs.
14351696
A sufficient condition for a semicomplete multipartite digraph to be Hamiltonian
Abstract A multipartite tournament is an orientation of a complete k -partite graph for some k ⩾ 2. A factor of a digraph D is a collection of vertex disjoint cycles covering all the vertices of D . We show that there is no degree of strong connectivity which together with the existence of a factor will guarantee that a multipartite tournament is Hamiltonian. Our main result is a sufficient condition for a multipartite tournament to be Hamiltonian. We show that this condition is general enough to provide easy proofs of many existing results on paths and cycles in multipartite tournaments. Using this condition, we obtain a best possible lower bound on the length of a longest cycle in any strongly connected multipartite tournament.
43442956
Representation characterizations of chordal bipartite graphs
A bipartite graph is chordal bipartite if it does not contain an induced cycle of length at least six. We give three representation characterizations of chordal bipartite graphs. More precisely, we show that a bipartite graph is chordal bipartite if and only if the complement is the intersection graph of a family of pairwise compatible claws in a weighted hypercircle. (A hypercircle is a graph which consists of internally vertex disjoint paths between two distinguished vertices, and a claw in a hypercircle is a connected subgraph containing exactly one of the two distinguished vertices.) We also introduce two classes of bipartite graphs, both containing interval bigraphs and interval containment bigraphs. They are compatible subtree intersection bigraphs and compatible subtree containment bigraphs. We show that these two classes are identical to the class of chordal bipartite graphs.
10979198
Weakly Hamiltonian-connected ordinary multipartite tournaments
Abstract We characterize weakly Hamiltonian-connected ordinary multipartite tournaments. Our result generalizes such a characterization for tournaments by Thomassen and implies a polynomial algorithm to decide the existence of a Hamiltonian path connecting two given vertices in an ordinary multipartite tournament and finds one, if it exists.
31025959
A Note on Spanning Local Tournaments in Locally Semicomplete Digraphs
We prove a result which implies that every k-connected locally semicomplete digraph which is not a semicomplete digraph must contain a k-connected spanning local tournament. This improves an earlier result of Guo and partially answers a question of Jackson and Thomassen.
2763612
Information Marginalization on Subgraphs
Real-world data often involves objects that exhibit multiple relationships; for example, ‘papers' and ‘authors' exhibit both paper-author interactions and paper-paper citation relationships. A typical learning problem requires one to make inferences about a subclass of objects (e.g. ‘papers'), while using the remaining objects and relations to provide relevant information. We present a simple, unified mechanism for incorporating information from multiple object types and relations when learning on a targeted subset. In this scheme, all sources of relevant information are marginalized onto the target subclass via random walks. We show that marginalized random walks can be used as a general technique for combining multiple sources of information in relational data. With this approach, we formulate new algorithms for transduction and ranking in relational data, and quantify the performance of new schemes on real world data—achieving good results in many problems.
14327978
On Exploring the Power-Law Relationship in the Itemset Support Distribution
We identify and explore in this paper an important phenomenon which points out that the power-law relationship appears in the distribution of itemset supports. Characterizing such a relationship will benefit many applications such as providing the direction of tuning the performance of the frequent-itemset mining. Nevertheless, due to the explosive number of itemsets, it will be prohibitively expensive to retrieve characteristics of the power-law relationship in the distribution of itemset supports. As such, we also propose in this paper a valid and cost-effective algorithm, called algorithm PPL, to extract characteristics of the distribution without the need of discovering all itemsets in advance. Experimental results demonstrate that algorithm PPL is able to efficiently extract the characteristics of the power-law relationship with high accuracy.
7486661
The Language of Search
This paper is concerned with a class of algorithms that perform exhaustive search on propositional knowledge bases. We show that each of these algorithms defines and generates a propositional language. Specifically, we show that the trace of a search can be interpreted as a combinational circuit, and a search algorithm then defines a propositional language consisting of circuits that are generated across all possible executions of the algorithm. In particular, we show that several versions of exhaustive DPLL search correspond to such well-known languages as FBDD, OBDD, and a precisely-defined subset of d-DNNF. By thus mapping search algorithms to propositional languages, we provide a uniform and practical framework in which successful search techniques can be harnessed for compilation of knowledge into various languages of interest, and a new methodology whereby the power and limitations of search algorithms can be understood by looking up the tractability and succinctness of the corresponding propositional languages.
16721854
Learning from labeled and unlabeled data on a directed graph
We propose a general framework for learning from labeled and unlabeled data on a directed graph in which the structure of the graph including the directionality of the edges is considered. The time complexity of the algorithm derived from this framework is nearly linear due to recently developed numerical techniques. In the absence of labeled instances, this framework can be utilized as a spectral clustering method for directed graphs, which generalizes the spectral clustering approach for undirected graphs. We have applied our framework to real-world web classification problems and obtained encouraging results.
8956273
Hierarchical Diagnosis of Multiple Faults
Due to large search spaces, diagnosis of combinational circuits is often practical for finding only single and double faults. In principle, system models can be compiled into a tractable representation (such as DNNF) on which faults of arbitrary cardinality can be found efficiently. For large circuits, however, compilation can become a bottleneck due to the large number of variables necessary to model the health of individual gates. We propose a novel method that greatly reduces this number, allowing the compilation, as well as the diagnosis, to scale to larger circuits. The basic idea is to identify regions of a circuit, called cones, that are dominated by single gates, and model the health of each cone with a single health variable. When a cone is found to be possibly faulty, we diagnose it by again identifying the cones inside it, and so on, until we reach a base case. We show that results combined from these hierarchical sessions are sound and complete with respect to minimum-cardinality diagnoses. We implement this method on top of the diagnoser developed by Huang and Darwiche in 2005, and present evidence that it significantly improves the efficiency and scalability of diagnosis on the ISCAS-85 circuits.