64 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
The generalized 3-edge-connectivity of lexicographic product graphs
The generalized -edge-connectivity of a graph is a
generalization of the concept of edge-connectivity. The lexicographic product
of two graphs and , denoted by , is an important graph
product. In this paper, we mainly study the generalized 3-edge-connectivity of
, and get upper and lower bounds of .
Moreover, all bounds are sharp.Comment: 14 page
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