Title: Simplifying branched covering surface-knots by chart moves involving black vertices,
Abstract: A branched covering surface-knot is a surface-knot in the form of a branched
covering over an oriented surface-knot $F$, where we include the case when the
covering has no branch points. A branched covering surface-knot is presented by
a graph called a chart on a surface diagram of $F$. We can simplify a branched
covering surface-knot by an addition of 1-handles with chart loops to a form
such that its chart is the union of free edges and 1-handles with chart loops.
We investigate properties of such simplifications for the case when branched
covering surface-knots have a non-zero number of branch points, using chart
moves involving black vertices.