Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Polynomial-delay generation of functional digraphs up to isomorphism

Oscar Defrain, Antonio E. Porreca, Ekaterina Timofeeva

TL;DR

The paper addresses the enumeration of functional digraphs up to isomorphism (mapping patterns) and develops a reverse-search-based generator with $O(n^2)$ delay and linear space. A canonical isomorphism code for connected digraphs is built from unordered rooted-tree codes and a minimal-rotation criterion, enabling duplicates-free generation. The method extends to arbitrary digraphs via partition-based composition of connected components, preserving the delay and memory bounds. This work establishes the first polynomial-delay generator for functional digraphs and lays groundwork for generating restricted classes or broader graph families without sacrificing efficiency.

Abstract

We describe a procedure for the generation of functional digraphs up to isomorphism; these are digraphs with uniform outdegree 1, also called mapping patterns, finite endofunctions, or finite discrete-time dynamical systems. This procedure is based on a reverse search algorithm for the generation of connected functional digraphs, which is then applied as a subroutine for the generation of arbitrary ones. Both algorithms output solutions with $O(n^2)$ delay and require linear space with respect to the number $n$ of vertices.

Polynomial-delay generation of functional digraphs up to isomorphism

TL;DR

The paper addresses the enumeration of functional digraphs up to isomorphism (mapping patterns) and develops a reverse-search-based generator with delay and linear space. A canonical isomorphism code for connected digraphs is built from unordered rooted-tree codes and a minimal-rotation criterion, enabling duplicates-free generation. The method extends to arbitrary digraphs via partition-based composition of connected components, preserving the delay and memory bounds. This work establishes the first polynomial-delay generator for functional digraphs and lays groundwork for generating restricted classes or broader graph families without sacrificing efficiency.

Abstract

We describe a procedure for the generation of functional digraphs up to isomorphism; these are digraphs with uniform outdegree 1, also called mapping patterns, finite endofunctions, or finite discrete-time dynamical systems. This procedure is based on a reverse search algorithm for the generation of connected functional digraphs, which is then applied as a subroutine for the generation of arbitrary ones. Both algorithms output solutions with delay and require linear space with respect to the number of vertices.
Paper Structure (5 sections, 7 theorems, 14 equations, 3 figures)

This paper contains 5 sections, 7 theorems, 14 equations, 3 figures.

Table of Contents

  1. Introduction and motivation
  2. Isomorphism codes for connected functional digraphs
  3. Generation of connected functional digraphs
  4. Generation of arbitrary functional digraphs
  5. Conclusions

Key Result

lemma 1

Let $C = \langle T_1, \ldots, T_k\rangle$ be a component in canonical form with at least one nontrivial tree, and let $T_h$ be the leftmost such nontrivial tree. Furthermore, let $(U_1, U_2) = \mathop{\mathrm{unmerge}}\nolimits(T_h)$ with $V_1 = \min \{U_1, U_2\}$ and $V_2 = \max \{U_1, U_2\}$. Then is also in canonical form; we call $C^*$ the parent of $C$ and denote it by $\mathop{\mathrm{Parent

Figures (3)

  • Figure 1: Isomorphism codes for a tree (Definition \ref{['def:tree-code']}), a connected (Definition \ref{['def:component-code']}) and a disconnected functional digraph (Definition \ref{['def:funcdigraph-code']}). Notice the different nesting depths for the angled brackets of the three codes.
  • Figure 2: Reverse search tree for the generation of components of $4$ vertices, represented both in graphical form (top) and as isomorphism codes (bottom); the order of generation by the algorithm of Theorem \ref{['thm:generates-all-connected']} is displayed on the top left. Note that the actual ordering of children of each node depends on the (arbitrary) order we chose for the generation of the candidate merges; in the picture we first compute $L_i(C^*)$ for $i$ from $1$ to $k-1$, then the same for the $R_i(C^*)$: this is the ordering that has been adopted in our implementation and that will be considered in the remaining of the paper.
  • Figure 3: Generation (top to bottom, left to right) of the set of functional digraphs $\mathcal{G}_p$ over $8$ vertices with partition $p = (0, 1, 2, 0, 0, 0, 0, 0)$, corresponding to components of $2$, $3$, $3$ vertices respectively. Each component is labelled by its generation order according to the algorithm of Theorem \ref{['thm:generates-all-connected']}. As an example, the functional digraph in grey (the 7th generated one in $\mathcal{G}_p$) has isomorphism code $\langle\langle\langle1\rangle,\langle1\rangle\rangle, \langle\langle3,2,1\rangle\rangle, \langle\langle1\rangle,\langle2,1\rangle\rangle\rangle$.

Theorems & Definitions (23)

  • definition 1: code of a tree
  • definition 2: code of a connected functional digraph
  • definition 3: tree merging
  • remark 1
  • definition 4: tree unmerging
  • lemma 1
  • proof
  • proposition 1
  • proof
  • definition 5
  • ...and 13 more