The Erase tool removes selected edges and any isolated vertices.  Note that it does not delete faces adjacent to the edge or vertex.  Typically you will use the Erase tool and either the Quadize or Insert Edges tool to change the flow of the edges in a particular area of the mesh.

Available for Edges.

The following sequence of edge erasing and Quadize demonstrates a typical workflow for using the Erase tool

Insert Edges

Insert Edges allows you to create new edges by clicking on existing edges, faces and vertices. It is a continuous mode allowing you to string edges together.  The first time you hit right click, it will allow you to start at a new position.  The second right click will exit the mode.  It will create vertices anywhere it crosses an existing edge and will automatically snap to vertices and edge centers. Zoom in if the snapping is to strong.  

Note that Insert Edges always puts the display into Control Polygon mode.  This enables snapping to work properly.

The Face, Edge or Vertex onto which the vertex will be inserted will be displayed in orange.

The first row shows a vertex going into a face; a vertex going into an edge and a vertex going into another vertex.

The second row shows right click and start a new edge string on the other side of the top face and connect it back to a vertex on the front face.

The third row demonstrates how Insert Edges works over the top of multiple faces even if there is a sharp corner by projecting down using the viewport orientation.

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