15 research outputs found
A Planarity Test via Construction Sequences
Optimal linear-time algorithms for testing the planarity of a graph are
well-known for over 35 years. However, these algorithms are quite involved and
recent publications still try to give simpler linear-time tests. We give a
simple reduction from planarity testing to the problem of computing a certain
construction of a 3-connected graph. The approach is different from previous
planarity tests; as key concept, we maintain a planar embedding that is
3-connected at each point in time. The algorithm runs in linear time and
computes a planar embedding if the input graph is planar and a
Kuratowski-subdivision otherwise
The Laplacian permanental polynomial for trees
AbstractThe polynomial we consider here is the characteristic polynomial of a certain (not adjacency) matrix associated with a graph. This polynomial was introduced in connection with the problem of counting spanning trees in graphs [8]. In the present paper the properties of this polynomial are used to construct some classes of graphs with an extremal numbers of spanning trees