Orderly spanning trees with applications to graph encoding and graph drawing

The <i>canonical ordering</i> for triconnected planar graphs is a powerful method for designing graph algorithms. This paper introduces the <i>orderly pair</i> of connected planar graphs, which extends the concept of canonical ordering to planar graphs not required to be triconnected. Let <i>G</i> be a connected planar graph. We give a linear-time algorithm that obtains an orderly pair (<i>H</i>,<i>T</i>) of <i>G</i>, where <i>H</i> is a planar embedding of <i>G</i>, and <i>T</i> is an <i>orderly spanning tree</i> of <i>H</i>. As applications, we show that the technique of orderly spanning trees yields (i) the best known encoding of <i>G</i> with query support, and (ii) the first area-optimal 2-visibility drawing of <i>G</i>.