The interval graph completion problem of a graph G includes two class problems： the profile problem and the pathwidth problem, denoted as P（G） and PW（G） respectively, where the profile problem is to find an interval supergraph with the smallest possible number of edges; the pathwidth problem is to find an interval supergraph with the smallest possible cliquesize. These two class problems have important applications to numerical algebra, VLSI- layout and algorithm graph theory respectively; And they are known to be NP-complete for general graphs. Some classes of special graphs have been investigated in the literatures. In this paper the exact solutions of the profile and the pathwidth of the complete multipartite graph Kn1,n2,...nr （r≥ 2） are determined.
The extended profile problem is to find a proper interval supergraph with the smallest possible number of edges.The problem stems from the storage and elimination techniques of a sparse symmetric matrix A in 1950,s.It has important applications in numerical algebra,VLSI designs and molecular biology.A tree T is a connected acyclic graph.The complement of a tree T is called a co-tree,denoted by Tˉ.In this paper the exact extended profile value of a cotree Tˉ is given.
