The Complexity of the Evolution of Graph Labelings

TL;DR

This paper investigates the computational complexity of transforming graph labelings through mutations, providing bounds, algorithms, and exact solutions for specific graph classes, with applications in bioinformatics, networks, and VLSI.

Contribution

It introduces new bounds, algorithms, and characterizations for the graph relabeling problem, including exact solutions for paths and stars, and addresses open problems with polynomial-time algorithms.

Findings

Vertex and edge relabelings have similar complexities.

Exact bounds for paths and stars are established.

Polynomial-time algorithms are provided for all cases.

Abstract

We study the {\sc Graph Relabeling Problem}--given an undirected, connected, simple graph $G = (V,E)$, two labelings $L$ and $L'$ of $G$, and label {\em flip} or {\em mutation} functions determine the complexity of transforming or evolving the labeling $L$ into $L'$\@. The transformation of $L$ into $L'$ can be viewed as an evolutionary process governed by the types of flips or mutations allowed. The number of applications of the function is the duration of the evolutionary period. The labels may reside on the vertices or the edges. We prove that vertex and edge relabelings have closely related computational complexities. Upper and lower bounds on the number of mutations required to evolve one labeling into another in a general graph are given. Exact bounds for the number of mutations required to evolve paths and stars are given. This corresponds to computing the exact distance between two vertices in the corresponding {\em Cayley graph}. We finally explore both vertex and edge relabeling with {\em privileged labels}, and resolve some open problems by providing precise characterizations of when these problems are solvable. Many of our results include algorithms for solving the problems, and in all cases the algorithms are polynomial-time. The problems studied have applications in areas such as bioinformatics, networks, and VLSI.