1,066 research outputs found
-Graphs of Trees
For a graph , the -graph of , denoted , is the graph whose vertex set is the collection of
minimum dominating sets, or -sets of , and two -sets are
adjacent in if they differ by a single vertex and the two different
vertices are adjacent in . In this paper, we consider -graphs of
trees. We develop an algorithm for determining the -graph of a tree,
characterize which trees are -graphs of trees, and further comment on
the structure of -graphs of trees and its connections with Cartesian
product graphs, the set of graphs which can be obtained from the Cartesian
product of graphs of order at least two.Comment: 22 pages, 3 figure
Mutually orthogonal latin squares with large holes
Two latin squares are orthogonal if, when they are superimposed, every
ordered pair of symbols appears exactly once. This definition extends naturally
to `incomplete' latin squares each having a hole on the same rows, columns, and
symbols. If an incomplete latin square of order has a hole of order ,
then it is an easy observation that . More generally, if a set of
incomplete mutually orthogonal latin squares of order have a common hole of
order , then . In this article, we prove such sets of
incomplete squares exist for all satisfying
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