A note on graph compositions and their connection to minimax of set partitions

TL;DR

This paper explores the relationship between graph compositions, which partition vertices into connected subgraphs, and the concept of minimax vertices in set partitions, providing recursive formulas and theoretical connections.

Contribution

It introduces a recursive formula for the composition number of a class of graphs and links it to the minimax concept in set partitions, advancing theoretical understanding.

Findings

Derived a recursive formula for graph composition numbers.

Established a connection between graph compositions and minimax set partitions.

Provided theoretical insights into the structure of graph compositions.

Abstract

A graph composition is a partition of the vertex set such that each member of the partition induces a connected sub- graph, and the composition number of a graph is the number of possible graph compositions. A partition of a set S of consecutive labelled vertices is said to have a minimax vertex v in S if the label of v is the smallest label in the set of all maximum labels over all members of the partition. This paper exhibits a recursive formula for the composition number of a certain class of graphs and estab- lishes a connection between the composition numbers of this class of graphs and that of the minimax of partitions of a labelled set (the minimum label of the set of all maximum labels over every member of the partition).