1 research outputs found
Enumerating the Digitally Convex Sets of Powers of Cycles and Cartesian Products of Paths and Complete Graphs
Given a finite set , a convexity , is a collection of subsets
of that contains both the empty set and the set and is closed under
intersections. The elements of are called convex sets. The
digital convexity, originally proposed as a tool for processing digital images,
is defined as follows: a subset is digitally convex if, for
every , we have implies . The number of
cyclic binary strings with blocks of length at least is expressed as a
linear recurrence relation for . A bijection is established between
these cyclic binary strings and the digitally convex sets of the
power of a cycle. A closed formula for the number of digitally convex sets of
the Cartesian product of two complete graphs is derived. A bijection is
established between the digitally convex sets of the Cartesian product of two
paths, , and certain types of binary arrays.Comment: 16 pages, 3 figures, 1 tabl