A study on token digraphs

TL;DR

This paper investigates properties of token digraphs, a generalization of token graphs, including connectivity, kernels, girth, and coloring, and proves NP-completeness for kernel existence in 2-token digraphs.

Contribution

It introduces and analyzes various properties of token digraphs, extending known results and establishing NP-completeness for kernel detection in 2-token digraphs.

Findings

Analyzed strong and unilateral connectivity of token digraphs.

Extended results on clique and chromatic numbers to token digraphs.

Proved NP-completeness for kernel existence in 2-token digraphs.

Abstract

For a digraph $D$ of order $n$ and an integer $1 \leq k \leq n-1$, the $k$-token digraph of $D$ is the graph whose vertices are all $k$-subsets of vertices of $D$ and, given two such $k$-subsets $A$ and $B$, $(A,B)$ is an arc in the $k$-token digraph whenever $\{a\} = A \setminus B$, $\{b\} = B \setminus A$, and there is an arc $(a,b)$ in $D$. Token digraphs are a generalization of token graphs. In this paper, we study some properties of token digraphs, including strong and unilateral connectivity, kernels, girth, circumference and Eulerianity. We also extend some known results on the clique and chromatic numbers of $k$-token graphs, addressing the bidirected clique number and dichromatic number of $k$-token digraphs. Additionally, we prove that determining whether $2$-token digraphs have a kernel is NP-complete.