Profile minimization on compositions of graphs

The profile minimization problem arose from the study of sparse matrix technique. In terms of graphs, the problem is to determine the profile of a graph G which is defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(G)=\min\limits_{f}\sum\limits_{v\in V(G)}\max\limits_{x\in N[v]}(f(v)-f(x)),$$\end{document} where f runs over all bijections from V(G) to {1,2,…,|V(G)|} and N[v]={v}∪{x∈V(G):xv∈E(G)}. This is equivalent to the interval graph completion problem, which is to find a super-graph of a graph G with as few number of edges as possible. The purpose of this paper is to study the profiles of compositions of two graphs.