3 research outputs found
The Detour Monophonic Convexity Number of a Graph
A set  is detour monophonic convexif  The detour monophonic convexity number is denoted by  is the cardinality of a maximum proper detour monophonic convex subset of Some general properties satisfied by this concept are studied. The detour monophonic convexity number of certain classes of graphs are determined. It is shown that for every pair of integers  and  with  there exists a connected graph  such that  and , where  is the monophonic convexity number of
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
Minimal trees and monophonic convexity
Let V be a finite set and a collection of subsets of V. Then is an alignment of V if and only if is closed under taking intersections and contains both V and the empty set. If is an alignment of V, then the elements of are called convex sets and the pair (V, ) is called an alignment or a convexity. If S ⊆ V, then the convex hull of S is the smallest convex set that contains S. Suppose X ∈ ℳ. Then x ∈ X is an extreme point for X if X∖{x} ∈ ℳ. A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. Let G = (V,E) be a connected graph and U a set of vertices of G. A subgraph T of G containing U is a minimal U-tree if T is a tree and if every vertex of V(T)∖U is a cut-vertex of the subgraph induced by V(T). The monophonic interval of U is the collection of all vertices of G that belong to some minimal U-tree. Several graph convexities are defined using minimal U-trees and structural characterizations of graph classes for which the corresponding collection of convex sets is a convex geometry are characterized