Enumerating the Digitally Convex Sets of Powers of Cycles and Cartesian Products of Paths and Complete Graphs

TL;DR

This paper studies digitally convex sets in specific graph classes, establishing bijections with binary strings and arrays, and deriving formulas for their enumeration, advancing understanding of digital convexity in graph products.

Contribution

It introduces new bijections and formulas for counting digitally convex sets in powers of cycles and Cartesian products of paths and complete graphs.

Findings

Bijection between cyclic binary strings and digitally convex sets in cycle powers

Closed formula for digitally convex sets in Cartesian products of complete graphs

Enumeration of digitally convex sets in Cartesian products of paths via binary arrays

Abstract

Given a finite set $V$, a convexity $\mathscr{C}$, is a collection of subsets of $V$ that contains both the empty set and the set $V$ and is closed under intersections. The elements of $\mathscr{C}$ are called convex sets. The digital convexity, originally proposed as a tool for processing digital images, is defined as follows: a subset $S\subseteq V(G)$ is digitally convex if, for every $v\in V(G)$, we have $N[v]\subseteq N[S]$ implies $v\in S$. The number of cyclic binary strings with blocks of length at least $k$ is expressed as a linear recurrence relation for $k\geq 2$. A bijection is established between these cyclic binary strings and the digitally convex sets of the $(k-1)^{th}$ 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, $P_n \square P_m$, and certain types of $n \times m$ binary arrays.