Quantifying fault recovery in multiprocessor systems

Various aspects of reliable computing are formalized and quantified with emphasis on efficient fault recovery. The mathematical model which proves to be most appropriate is provided by the theory of graphs. New measures for fault recovery are developed and the value of elements of the fault recovery vector are observed to depend not only on the computation graph H and the architecture graph G, but also on the specific location of a fault. In the examples, a hypercube is chosen as a representative of parallel computer architecture, and a pipeline as a typical configuration for program execution. Dependability qualities of such a system is defined with or without a fault. These qualities are determined by the resiliency triple defined by three parameters: multiplicity, robustness, and configurability. Parameters for measuring the recovery effectiveness are also introduced in terms of distance, time, and the number of new, used, and moved nodes and edges

On the Lengths of Symmetry Breaking-Preserving Games on Graphs

Given a graph $G$, we consider a game where two players, $A$ and $B$, alternatingly color edges of $G$ in red and in blue respectively. Let $l(G)$ be the maximum number of moves in which $B$ is able to keep the red and the blue subgraphs isomorphic, if $A$ plays optimally to destroy the isomorphism. This value is a lower bound for the duration of any avoidance game on $G$ under the assumption that $B$ plays optimally. We prove that if $G$ is a path or a cycle of odd length $n$, then $\Omega(\log n)\le l(G)\le O(\log^2 n)$. The lower bound is based on relations with Ehrenfeucht games from model theory. We also consider complete graphs and prove that $l(K_n)=O(1)$.Comment: 20 page

A structural analysis of the situation in the Middle East in 1956

We attempt to display a systematic approach for the distinction between states of equilibrium and disequilibrium in the interrelationships between nations, using as corroborative material the rapid shifts in 1956 among the various nations, brought about by the Middle Eastern situation. The psychological theory behind this approach is that of structural balance, which is pertinent in the present context to balance of power, while the logical framework involves the mathematical theory of graphs. We do not assert that this theory in its present form is predictive, but we do feel that it offers a well-organized mode of thinking which, although simple, may be fruitful. We also comment on some aspects of the Hungarian situation in terms of structural balance. It must be borne in mind throughout this article that it was written in early 1957 and that therefore the interrelationships among nations described herein reflect that time period.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/67336/2/10.1177_002200276100500204.pd

On the Group of a Graph with Respect To a Subgraph

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135216/1/jlms0457.pd

THE EXPLOSIVE GROWTH OF GRAPH THEORY

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73316/1/j.1749-6632.1979.tb17762.x.pd

Corrections: Generalized Ramsey Theory for Graphs V

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135328/1/blms0087.pd