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

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).