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Construction numbers: How to build a graph?

Paul C. Kainen

TL;DR

The paper studies the construction number $c(G)$, counting constructions sequences where each edge appears after its endpoints, and ties this quantity to linear extensions of the graph's element-poset. It develops a general framework with vertex-edge bounds and uses it to derive exact formulas and recurrences for several graph families, including $c(K_{1,n})=2^n(n!)^2$ and $c(P_p)=T_p$, the Tangent numbers, with $c(C_p)=p\,c(P_p)$ and $c(nK_2)=2^n\binom{3n}{3,\ldots,3}$; an Appendix gives $c(K_n)=\binom{n+1}{2}!/(2n-1)!!$. By connecting construction counts to classical sequences and poset theory, the work illuminates how topology influences construction complexity and identifies avenues for further study in maximal planar/outerplanar graphs and potential applications to self-assembly and networks.

Abstract

A construction sequence for a graph is a listing of the elements of the graph (the set of vertices and edges) such that each edge follows both its endpoints. The construction number of the graph is the number of such sequences. We determine this number for various graph families.

Construction numbers: How to build a graph?

TL;DR

The paper studies the construction number , counting constructions sequences where each edge appears after its endpoints, and ties this quantity to linear extensions of the graph's element-poset. It develops a general framework with vertex-edge bounds and uses it to derive exact formulas and recurrences for several graph families, including and , the Tangent numbers, with and ; an Appendix gives . By connecting construction counts to classical sequences and poset theory, the work illuminates how topology influences construction complexity and identifies avenues for further study in maximal planar/outerplanar graphs and potential applications to self-assembly and networks.

Abstract

A construction sequence for a graph is a listing of the elements of the graph (the set of vertices and edges) such that each edge follows both its endpoints. The construction number of the graph is the number of such sequences. We determine this number for various graph families.
Paper Structure (4 sections, 16 theorems, 20 equations)

This paper contains 4 sections, 16 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and lemmas
  3. Construction numbers for some graph families
  4. Discussion

Key Result

Lemma 1

If $G$ is connected, then the last $\delta(G)$ entries in any construction sequence $x \in {\cal C}(G)$ are edges. Moreover, if $x_i \in V(G)$, then as each vertex must be followed by the edges with which it is incident, $\deg(v_i) \leq \ell -i$.

Theorems & Definitions (21)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Proposition 1
  • Lemma 5
  • proof
  • Proposition 2
  • Theorem 1: c-number of a star
  • proof
  • ...and 11 more