We construct an optimal linear-time algorithm for the maximal planar subgraph problem: given a graph G, find a planar subgraph G′ of G such that adding to G′ an extra edge of G results in a non-planar graph. Our solution is based on a fast data structure for incremental planarity testing of triconnected graphs and a dynamic graph search procedure. Our algorithm can be transformed into a new opt...

We construct an optimal linear time algorithm for the maximal planar subgraph problem: given a graph G, nd a planar subgraph G 0 of G such that adding to G 0 any edge of G not present in G 0 leads to a non-planar graph. Our solution is based on a dynamic graph search procedure and a fast data structure for on-line planarity testing of triconnected graphs. Our algorithm can be transformed into a...

We prove that triangulated IC-planar and NIC-planar graphs can be recognized in cubic time. A graph is 1-planar if it can be drawn in the plane with at most one crossing per edge. A drawing is IC-planar if, in addition, each vertex is incident to at most one crossing edge and NIC-planar if two pairs of crossing edges share at most one vertex. In a triangulated drawing each face is a triangle. I...