5845717
Graph coloring satisfying restraints
Abstract For an integer k ⩾2, a proper k-restraint on a graph G is a function from the vertex set of G to the set of k -colors. A graph G is amenably k-colorable if, for each nonconstant proper k -restraint r on G , there is a k -coloring c of G with c ( v )≠ r ( v ) for each vertex v of G . A graph G is amenable if it is amenably k -colorable and k is the chromatic number of G . For any k ≠3, there are infinitely many amenable k -critical graphs. For k ⩾ 3, we use a construction of B. Toft and amenable graphs to associate a k -colorable graph to any k -colorable finite hypergraph. Some constructions for amenable graphs are given. We also consider a related property—being strongly critical —that is satisfied by many critical graphs, including complete graphs. A strongly critical graph is critical and amenable, but the converse is not always true. The Dirac join operation preserves both amenability and the strongly critical property. In addition, the Hajos construction applied to a single edge in each of two strongly k -critical graphs yields an amenable graph. However, for any k ⩾5, there are amenable k -critical graphs for which the Hajos construction on two copies is not amenable.
2327812
The existence of uniquely -G colourable graphs
Abstract Given graphs F and G and a nonnegative integer k , a function π : V ( F ) → 1, …, k is a − G k-colouring of F if no induced copy of is monochromatic; F is − G k-chromatic if F has a − G k -colouring but no − G ( k − 1)-colouring. Further, we say F is uniquely − G k -colourable if F is − G k -chromatic and, up to a permutation of colours, it has only one − G k -colouring. Such notions are extensions of the well-known corresponding definitions from chromatic theory. It was conjectured that for all graphs G of order at least two and all positive integers k there exist uniquely − G k -colourable graphs. We prove the conjecture and show that, in fact, in all cases infinitely many such graphs exist.
19128559
Bounding the Roots of Independence Polynomials
28649866
A ramsey type problem concerning vertex colourings
Abstract Let H → k v G denote the fact that for every function π: V ( H ) → {1, …, k } there is an induced subgraph G ′ of H with G ′ ≅ G and V ( G ′) ⊆ π −1 ( i ) for some i . Folkman has shown that for all graphs G and for all positive integers k such a graph H exists. We examine here f ( G , k ), the minimum order of a graph H for which H → k v G . We show that for any fixed integer k ≥ 2 there are positive constants C 1 and C 2 such that C 1 n 2 ≤ max{ f ( G , k ): | V ( G )| = n } ≤ C 2 n 2 log 2 n .
40600226
The Complexity of Generalized Graph Colorings
Given a graph property P and positive integer k, a P k-coloring of a graph G is an assignment of one of k colors to each vertex of the graph so that the subgraphs induced by each color class have property P. This notion generalizes the standard definition of graph coloring, and has been investigated for many properties. We consider here the complexity of the decision problem. In particular, for the property —G, of not containing an induced subgraph isomorphic to G, we conjecture (and provide strong evidence) that —G k-colorability is NP-complete whenever G has order at least 3 and k ⩾ 2. The techniques rely on new NP-completeness results for hypergraph colorings.
44284809
The strongly connected reliability of complete digraphs
Given a digraph D, consider the model where each vertex is always operational, but the edges are independently operational with probability p. The strongly connected reliability of D, scRel(D, p), is the probability that the spanning subgraph of D consisting of the operational edges is strongly connected. One can view strongly connected reliability as the probability that any vertex can send information to any other vertex, given that edges fail independently. ∗Partially supported by a grant from NSERC. Communicating author.
10597458
Computational and neural mechanisms of task switching
Switching between tasks that overlap in perceptual and response characteristics is assumed to rely upon the maintenance of task representations in prefrontal cortex (PFC). However, task-switching studies demonstrate ''switch costs,'' even when there is sufficient time to prepare for a new task. These costs suggest that task-switching performance reflects a complex interplay between priming and the updating and maintenance of task representations. We describe a computational model in which this interaction is made explicit and linked to the dynamics of PFC. Simulation results account for a variety of empirical phenomena and predict a double dissociation in lateral PFC that was subsequently identified.
205225115
The ribonuclease P database
Ribonuclease P is responsible for the 5'-maturation of tRNA precursors. Ribonuclease P is a ribonucleoprotein, and in bacteria (and some Archaea) the RNA subunit alone is catalytically active in vitro, i.e. it is a ribozyme. The Ribonuclease P Database is a compilation of ribonuclease P sequences, sequence alignments, secondary structures, three-dimensional models and accessory information, available via the World Wide Web at the following URL: http://www.mbio.ncsu.edu/RNaseP/home .html
5243593
Steps Toward Automatic Theory Formation
This paper describes a theory formation system which can discover a partial axiomization of a data base represented as extensionally defined binary relations.- The system first discovers all possible intensional definitions of each binary relation in terms of the others. It then determines a minimal set of these relations from which the others can be defined. It then attempts to discover all the ways the relations of this minimal set can interact with each other, thus generating a set of inference rules. Although the system was originally designed to explore automatic techniques for theory construction for question-answering systems, it is currently being expanded to function as a symbiotic system to help social scientists explore certain kinds of data bases.
38979633
A Qualitative Physics Based on Confluences
Abstract A qualitative physics predicts and explains the behavior of mechanisms in qualitative terms. The goals for the qualitative physics are (1) to be far simpler than the classical physics and yet retain all the important distinctions (e.g., state, oscillation, gain, momentum) without invoking the mathematics of continuously varying quantities and differential equations, (2) to produce causal accounts of physical mechanisms that are easy to understand, and (3) to provide the foundations for commonsense models for the next generation of expert systems. This paper presents a fairly encompassing account of qualitative physics. First, we discuss the general subject of naive physics and some of its methodological considerations. Second, we present a framework for modeling the generic behavior of individual components of a device based on the notions of qualitative differential equations (confluences) and qualitative state. This requires developing a qualitative version of the calculus. The modeling primitives induce two kinds of behavior, intrastate and interstate, which are governed by different laws. Third, we present algorithms for determining the behavior of a composite device from the generic behavior of its components. Fourth, we examine a theory of explanation for these predictions based on logical proof. Fifth, we introduce causality as an ontological commitment for explaining how devices behave.
15658893
Theories of Causal Ordering
Abstract This paper is a response to Iwasaki and Simon [14] which criticizes de Kleer and Brown [8]. We argue that many of their criticisms, particularly concerning causality, modeling and stability, originate from the difference of concerns between engineering and economics. Our notion of causality arises from considering the interconnections of components, not equations. When no feedback is present, the ordering produced by our qualitative physics is similar to theirs. However, when feedback is present, our qualitative physics determines a causal ordering around feedback loops as well. Causal ordering is a general technique not only applicable to qualitative reasoning. Therefore we also explore the relationship between causal ordering and propagation of constraints upon which the methods of qualitative physics are based.