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Expansion of a digraph with a doubly bidirectionally connected pair is not cactus

Hiroki Kodama

Abstract

Azuma et al. showed that strongly connected digraph without doubly bidirectionally connected pair is cactus-expandable. We show the converse; namely, If a digraph has a doubly bidirectionally connected pair, then its expansion cannot be cactus.

Expansion of a digraph with a doubly bidirectionally connected pair is not cactus

Abstract

Azuma et al. showed that strongly connected digraph without doubly bidirectionally connected pair is cactus-expandable. We show the converse; namely, If a digraph has a doubly bidirectionally connected pair, then its expansion cannot be cactus.
Paper Structure (1 section, 3 theorems, 1 equation)

This paper contains 1 section, 3 theorems, 1 equation.

Table of Contents

  1. Acknowledgement

Key Result

Theorem 12

If a digraph $G$ is strongly connected and does not have a dbcp, then there exists an expansion $\varphi \colon G' \to G$ such that $G'$ is a cactus bigraph.

Theorems & Definitions (18)

  • Definition 1: morphism
  • Definition 2: expansion
  • Definition 3: cactus digraph
  • Claim 4
  • proof
  • Definition 5: connecting point
  • Definition 6: preorder
  • Definition 7: preorder for the rooted cactus digraph
  • Claim 8
  • proof
  • ...and 8 more