Hyperpaths

TL;DR

This paper introduces hyperpaths, a new class of high-dimensional hypertrees analogous to paths in graph theory, and explores their properties, especially in dimension two, expanding the understanding of hypertree structures.

Contribution

It defines hyperpaths as a novel class of hypertrees, provides a family of such structures for all dimensions, and investigates their properties in dimension two.

Findings

Defined hyperpaths as high-dimensional path analogs

Constructed an infinite family of hyperpaths for each dimension

Analyzed properties of hyperpaths in dimension two

Abstract

Hypertrees are high-dimensional counterparts of graph theoretic trees. They have attracted a great deal of attention by various investigators. Here we introduce and study Hyperpaths -- a particular class of hypertrees which are high dimensional analogs of paths in graph theory. A $d$-dimensional hyperpath is a $d$-dimensional hypertree in which every $(d-1)$-dimensional face is contained in at most $(d+1)$ faces of dimension $d$. We introduce a possibly infinite family of hyperpaths for every dimension, and investigate its properties in greater depth for dimension $d=2$.