For a given undirected graph G, the minimum rank of G is defined to be the smallest possible rank over all real symmetric matrices A whose (i, j)th entry is non-zero whenever i / =j and {i, j} is an edge in G. The path cover number of G is the minimum number of vertexdisjoint paths occurring as induced subgraphs of G that cover all the vertices of G. For trees, the relationship between minimum ...

In the year 2000, exactly one hundred years after David Hilbert posed his now famous list of 23 open problems, The Clay Mathematics Institute (CMI) announced its seven Millennium Problems. (http://www. claymath.org/millennium). The Gazette has asked leading Australian mathematicians to put forth their own favourite ‘Millennium Problem’. Due to the Gazette’s limited budget, we are unfortunately ...

The dissociation number of a graph G is the number of vertices in a maximum size induced subgraph of G with vertex degree at most 1. A k-path vertex cover of a graph G is a subset S of vertices of G such that every path of order k in G contains at least one vertex from S. The minimum 3-path vertex cover is a dual problem to the dissociation number. For this problem we present an exact algorithm...