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Growing Trees and Amoebas' Replications

Vladimir Gurvich, Matjaž Krnc, Mikhail Vyalyi

TL;DR

The paper develops a formal framework for amoebas—trees equipped with root multiplicities—growing via ell-extensions and studies when such growth sequences terminate (mortality) or persist (immortality). It connects mortality to confinement and completion, proving that for ell in {1,2} mortality is equivalent to completion, and deriving degree-based and structural criteria for the 1-extension case, including a complete characterization for caterpillars. Key contributions include a digraph-based bound on degrees, a complete caterpillar mortality criterion via the completion's slow/non-slow status, and a set of conjectures about equivalences and undecidability, along with proposed generalizations such as amoeba colonies and confinement graphs. The results illuminate deterministic growth mechanisms on graphs, with potential implications for graph-avoidability concepts and algorithmic classification of growth processes.

Abstract

An amoeba is a tree together with instructions how to iteratively grow trees by adding paths of a fixed length $\ell$. This paper analyses such a growth process. An amoeba is mortal if all versions of the process are finite, and it is immortal if they are all infinite. We obtain some necessary and some sufficient conditions for mortality. In particular, for growing caterpillars in the case $\ell=1$ we characterize mortal amoebas. We discuss variations of the mortality concept, conjecture that some of them are equivalent, and support this conjecture for $\ell\in\{1,2\}$.

Growing Trees and Amoebas' Replications

TL;DR

The paper develops a formal framework for amoebas—trees equipped with root multiplicities—growing via ell-extensions and studies when such growth sequences terminate (mortality) or persist (immortality). It connects mortality to confinement and completion, proving that for ell in {1,2} mortality is equivalent to completion, and deriving degree-based and structural criteria for the 1-extension case, including a complete characterization for caterpillars. Key contributions include a digraph-based bound on degrees, a complete caterpillar mortality criterion via the completion's slow/non-slow status, and a set of conjectures about equivalences and undecidability, along with proposed generalizations such as amoeba colonies and confinement graphs. The results illuminate deterministic growth mechanisms on graphs, with potential implications for graph-avoidability concepts and algorithmic classification of growth processes.

Abstract

An amoeba is a tree together with instructions how to iteratively grow trees by adding paths of a fixed length . This paper analyses such a growth process. An amoeba is mortal if all versions of the process are finite, and it is immortal if they are all infinite. We obtain some necessary and some sufficient conditions for mortality. In particular, for growing caterpillars in the case we characterize mortal amoebas. We discuss variations of the mortality concept, conjecture that some of them are equivalent, and support this conjecture for .
Paper Structure (9 sections, 12 theorems, 4 equations, 6 figures)

This paper contains 9 sections, 12 theorems, 4 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Amoebas
  3. Initial observations
  4. Orbit lemma
  5. Case of $1$-extensions
  6. Degree conditions
  7. Mortality criteria for caterpillars
  8. Directions for future work
  9. Acknowledgements.

Key Result

Proposition 1

The intersection of an area of an extension path and the original graph is a tree of depth at most $\ell-1$.

Figures (6)

  • Figure 1: Non-isomorhic 3-growths of the same copy of an amoeba. The copy is shadowed in grey, the new edge is dashed
  • Figure 2: A generation sequence for an amoeba $H$ with $\ell=2$. The $\ell$-growth of $T_4$ is $T_4$
  • Figure 3: Four ways of extension
  • Figure 4: An amoeba admitting ambiguous $3$-extensions in a generation sequence
  • Figure 5: A not mortal amoeba with one root with respect to $2$-extensions. The next step uses the left lower $P_5$ for extensions. Repetitions of the process continue forever
  • ...and 1 more figures

Theorems & Definitions (32)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 3
  • Theorem 4
  • proof
  • ...and 22 more