Connected realizations of joint-degree matrices

We study a restriction of the classic degree sequence graphic realization problem studied by Erdős, Gallai, Havel, and Hakimi, namely the joint-degree matrix graphic realization problem. Here, in addition to the degree sequence, a joint degree matrix is given, the (i,j)th element of which specifies the exact number of edges between vertices of degree di and vertices of degree dj. The decision and construction versions of the problem have a relatively straightforward solution. In this work, however, we focus on the corresponding connected graphic realization version of the problem. We give a necessary and sufficient condition for a connected graphic realization to exist, as well as a polynomial time construction algorithm that involves a novel recursive search of suitable local graph modifications. As a byproduct, we also suggest an alternative polynomial time algorithm for the joint-degree matrix graphic realization problem that never increases the number of connected components of the graph constructed.