On mark sequences in 2-digraphs

A k-digraph is an orientation of a multi-graph that is without loops and contains at most k edges between any pair of distinct vertices. So 1-digraph is an oriented graph, and complete 1-digraph is a tournament. We obtain necessary and sufficient conditions for a sequence of non-negative integers in non-decreasing order to be a sequence of numbers, called marks, attached to vertices of 2-digraph.