Complexity of graph evolutions

TL;DR

The paper develops a cost-based framework for graph construction sequences, defining the delay cost $\nu(e,x)$ and total cost $\nu(x)$ to quantify the complexity of building a graph via a sequence that respects endpoint containment. It yields tight extremal results for a variety of graph families: maximum costs for regular graphs follow the simple bound $\nu^*(G)=q(p+q)$, with explicit values for cycles, complete graphs, complete bipartite graphs, hypercubes, stars, paths, wheels, and double stars, and it characterizes tree extremals and disjoint unions. Minimum costs are shown to be achieved by greedy sequences, with explicit constructions and closed forms such as $\nu_*(K_n)=\frac{(n-1)n(n+1)(n+4)}{12}$ and $\nu_*(C_n)=6n-4$, along with asymptotic comparisons that reveal distinct discriminative power for min- and max-cost measures. The work lays a foundation for using construction costs as a proxy for graph-building program complexity and suggests broader applicability to operational research, network design, and analysis of graph operations.

Abstract

A permutation of the elements of a graph is a {\it construction sequence} if no edge is listed before either of its endpoints. The complexity of such a sequence is investigated by finding the delay in placing the edges, an {\it opportunity cost} for the construction sequence. Maximum and minimum cost c-sequences are provided for a variety of graphs and are used to measure the complexity of graph-building programs.

Figures (3)

• Figure 1: The double star $D_5$
• Figure 2: The suspension $\Sigma C_5$ of a 5-cycle
• Figure 3: $W_{6, 6}$

Theorems & Definitions (41)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof