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.
45045031
Well-Covered Vector Spaces of Graphs
For any field ${\bf F}$, the set of all functions $f : V(G) \rightarrow {\bf F}$ whose sum on each maximal independent set is constant forms a vector space over ${\bf F}$. In this paper, we show that the dimension can vary depending on the characteristic of the field. We also investigate the dimensions of these vector spaces and show that while some families, such as chordal graphs, have unbounded dimension, other families, such as nonempty circulant graphs of prime order, have bounded dimension.
35054660
The Ultimate Categorical Independence Ratio of a Graph
Let $\beta(G)$ denote the independence number of a graph $G$. We introduce $A(G) = \lim_{k \rightarrow \infty}\beta(G^{k})/|V(G)|^{k}$, where the categorical graph product is used. This limit, surprisingly, lies in the range $(0,1/2]\cup\{1\}$. We can show that this limit can take any such rational number, but is there any $G$ for which $A(G)$ is irrational? A useful technique for bounding $A(G)$ is to consider special spanning subgraphs. These bounds allow us to efficiently compute $A(G)$ for many $G$. We give a condition which if true for $G$ shows that $A(G)>\beta(G)/|V(G)|$. This brings up the question: for which $G$ does $A(G)=\beta(G)/|V(G)|$? This happens if $G$ is a Cayley graph of an Abelian group or if $G$ is a connected graph that has an automorphism which has a single orbit.
5392880
On chromatic roots with negative real part
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 .
14976506
Average independence polynomials
The independence polynomial of a graph G is the function i(G, x) = Σ k ≥ 0 ik xk, where ik is the number of independent sets of vertices in G of cardinality k. We investigate here the average independence polynomial, where the average is taken over all independence polynomials of (labeled) graphs of order n. We prove that while almost every independence polynomial has a nonreal root, the average independence polynomials always have all real, simple roots.
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.
7675324
Applications of Artificial Intelligence
One of the aims of artificial intelligence has been to make machines behave intelligently as humans do. Thus the study of artificial intelligence includes the study of how humans acquire and apply knowledge, reason under uncertainty and in complex environments, and how they do planning and solve problems. Knowledge-based automation deals with the application of artificial intelligence to a production environment in order to reduce the involvement of human beings to a minimum.
27879633
To dream the invisible dream
often credited as the creator of the computer game industry, explores the “Relationships between Fun and the Computer Business.” In his article he reminisces, assesses, and projects the effects from the game industry to the computer industry as a whole. Bushnell’s article makes tangible some of the incredible distance we’ve come since the 1970s—from a world where manufacturers didn’t take cathode-ray tube use in the computer industry seriously to a time when most two-year-old children know about computers. Bushnell’s way is to mix lessons learned from the game industry with some as-yet unexplored visions of the way we will play. Henry Lieberman’s work has been that of creating programming and idea presentation environments using his favorite tools of creation objects (his work on actors), dynamic languages (his work on garbage collection), and interactive environments (his work on programming by example). Lieberman’s article, “Intelligent Graphics,” focuses on the inventive serendipity that has and can come out of the relationship between AI and graphics—an idea promoted by yet another pioneer, Muriel Cooper. He also briefly discusses programming by example, a style of interaction that he has been working to demonstrate and promote over the last couple of decades. Ken Kahn also cares deeply about programming languages, having devoted his research career to creating Prolog interpreters, parallel logic systems, and logic-oriented systems. In his article, we see him engaged in his first love, that of creating visual approaches for thinking about programs. Watching programs execute, seeing their structure, seeing their function, organizing them—all with the underlying goal of making programming as easy as child’s play. Finally, as you look to my article, you see me not working or talking about any one idea as I could, but giving a review with some context that describes many of the ideas shaping the way people are using computers today. I hope this eclectic collection of articles can serve as a celebration of our temptations to explore and design the ways computers will help us communicate, create, and record our experiences.
7478879
G-functions for the hermeneutic circle of evolution
Evolution by natural selection may include both frequency and density-dependence. Frequency-dependent selection is a kind of hermeneutic circle. As a literary term (and a school of philosophy), a hermeneutic circle describes how each word of a sentence or paragraph simultaneously draws its meaning from its context (the other words) and contributes towards the context. Frequency dependence is like this. Under frequency dependence, the fitness of an individual is not only influenced by its own heritable phenotype (in evolutionary game theory heritable phenotype = strategy) but by the frequency of strategies found among others in the population. And, the fitness accrued by individuals with a particular strategy will in time influence the frequency of strategies in the population. The challenge is to develop conceptual and modeling tools that allow us to determine the outcome of this hermeneutic circle of evolution. We need a fitness concept that defines the fitness of (or payoff to) to individuals as influenced by both their own strategy and the strategies of others. And, this fitness formulation must describe how these payoffs translate into changes in strategy frequencies. The fitness generating function, or G-function, provides this service for evolutionary game theory.
17008578
Foundations of Envisioning
This paper explores a particular kind of qualitative reasoning, called envisioning, that is capable of producing causal explanations for device behavior. It has been implemented in a computer program, ENVISION, which can analyze a wide variety of thermal, fluid, electrical, translational and rotational devices. Rather than present the technical details of the envisioning process, this paper examines the theoretical foundations upon which it is built. Many of these considerations are ones that any builder of qualitative reasoning systems must pay attention to. Two such considerations are explanation and robustness: What notion of causality is adequate for causal explanations of device behavior? How can there be any confidence in the analysis of a novel device?