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A study on token digraphs

Cristina G. Fernandes, Carla N. Lintzmayer, Juan P. Peña, Giovanne Santos, Ana Trujillo-Negrete, Jose Zamora

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.

A study on token digraphs

Abstract

For a digraph of order and an integer , the -token digraph of is the graph whose vertices are all -subsets of vertices of and, given two such -subsets and , is an arc in the -token digraph whenever , , and there is an arc in . 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 -token graphs, addressing the bidirected clique number and dichromatic number of -token digraphs. Additionally, we prove that determining whether -token digraphs have a kernel is NP-complete.

Paper Structure

This paper contains 10 sections, 20 theorems, 9 equations, 9 figures.

Table of Contents

  1. Introduction
  2. General definitions and basic properties
  3. Strong connectivity aspects of token digraphs
  4. Kernels
  5. Unilateral digraphs and their token digraphs
  6. Girth and circumference
  7. Eulerian and Hamiltonian digraphs
  8. Bidirected cliques
  9. Acyclic partitions
  10. Final remarks

Key Result

lemma 3.1

Let $D$ be a strongly connected digraph, and let $A$ and $B$ be two nodes of $F_k(D)$. There is an $AB$-path in $F_k(D)$.

Figures (9)

  • Figure 1: A graph $G$ and its $2$-token graph $F_2(G)$.
  • Figure 2: A digraph $D$ and its $2$-token digraph $F_2(D)$.
  • Figure 3: A digraph $D$ with 3 strongly connected components, the $2$-token digraph $F_2(D)$ with 5 strongly connected components (for simplicity, we omit the bracets and place labels inside the vertices), and the digraph isomorphic to $\mathrm{CD}(F_2(D))$ whose vertex set is $V_2(4,2,1)$.
  • Figure 4: Two examples of a digraph $D$ and its $2$-token digraph $F_2(D)$. On the left, $D$ has a kernel, namely the set $\{a,c\}$, while $F_2(D)$ does not. On the right, the opposite occurs: the five external vertices form a kernel of $F_2(D)$.
  • Figure 5: The digraph $D$ for the 3-SAT formula $\phi = (x_1 \vee \bar{x}_2 \vee x_3) \wedge (\bar{x}_1 \vee x_3 \vee x_4) \wedge (x_2 \vee \bar{x}_3 \vee \bar{x}_4)$. The vertices in red form a kernel for the digraph $D' = D - u$.
  • ...and 4 more figures

Theorems & Definitions (35)

  • lemma 3.1
  • proof
  • lemma 3.2
  • proof
  • theorem 3.1
  • proof
  • corollary 3.1
  • corollary 3.2
  • theorem 4.1: vonNeumannM1944
  • theorem 4.2
  • ...and 25 more