Recognizing Polar Planar Graphs Using New Results for Monopolarity

Polar and monopolar graphs are natural generalizations of bipartite or split graphs. A graph

G

 = (

V

,

E

) is polar if its vertex set admits a partition

V

 = 

A

 ∪ 

B

such that

A

induces a complete multipartite and

B

the complement of a complete multipartite graph. If

A

is even a stable set then

G

is called monopolar.

Recognizing general polar or monopolar graphs is NP-complete and, as yet, efficient recognition is available only for very few graph classes.

This paper considers monopolar and polar graphs that are also planar. On the one hand, we show that recognizing these graphs remains NP-complete, on the other hand we identify subclasses of planar graphs on which polarity and monopolarity can be checked efficiently. The new NP-completeness results cover very restricted graph classes and are sharper than all previous known cases. On the way to the positive results, we develop new techniques for efficient recognition of subclasses of monopolar graphs. These new results extend nearly all known results for efficient monopolar recognition.