# Discrete Topological complexity

## TL;DR

This paper introduces a new combinatorial invariant called discrete topological complexity for simplicial complexes, linking it to existing invariants like LS-category, and explores its properties and relationships.

## Contribution

It presents the first definition of discrete topological complexity based solely on simplicial combinatorics and studies its connections with LS-category.

## Key findings

- Discrete topological complexity is defined using contiguous simplicial maps.
- The invariant is linked to classical LS-category in simplicial and topological contexts.
- Foundations for further combinatorial topological complexity studies are established.

## Abstract

We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex by means of the concept of contiguous simplicial maps. We study the links of this new invariant with those of simplicial and topological LS-category.