2 research outputs found
Simple realizability of complete abstract topological graphs simplified
An abstract topological graph (briefly an AT-graph) is a pair
where is a graph and is a set of pairs of its edges. The AT-graph is simply
realizable if can be drawn in the plane so that each pair of edges from
crosses exactly once and no other pair crosses. We show that
simply realizable complete AT-graphs are characterized by a finite set of
forbidden AT-subgraphs, each with at most six vertices. This implies a
straightforward polynomial algorithm for testing simple realizability of
complete AT-graphs, which simplifies a previous algorithm by the author. We
also show an analogous result for independent -realizability,
where only the parity of the number of crossings for each pair of independent
edges is specified.Comment: 26 pages, 17 figures; major revision; original Section 5 removed and
will be included in another pape