Making a Graph Crossing-Critical by Multiplying its Edges

Making a Graph Crossing-Critical by Multiplying its Edges

A graph is crossing-critical if the removal of any of its edges decreases its crossing number. This work is motivated by the following question: to what extent is crossingcriticality a property that is inherent to the structure of a graph, and to what extent can it be induced on a noncritical graph by multiplying (all or some of) its edges? It is shown that if a nonplanar graph G is obtained by...

Making an Arbitrary Filled Graph Minimal by Removing Fill Edges

We consider the problem of removing ll edges from a lled graph G 0 to get a minimal chordal supergraph M of the original graph G; thus G M G 0. We show that a greedy strategy can be applied if ll edges are processed for removal in the reverse order of their introduction. For a lled graph with f ll edges and e original edges, we give a simple O(f(e + f)) algorithm which solves the problem and co...

Graph Drawings with no k Pairwise Crossing Edges

A geometric graph is a graph G = (V;E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . It is known that, for any xed k, any geometric graph G on n vertices with no k pairwise crossing edges contains at most O(n log n) edges. In this paper we give a new, simpler proof of this bound, and...

New almost-planar crossing-critical graph families

We show that, for all choices of integers k > 2 and m, there are simple 3-connected k-crossing-critical graphs containing more than m vertices of each even degree ≤ 2k − 2. This construction answers one half of a question raised by Bokal, while the other half asking analogously about vertices of odd degrees at least 5 in crossing-critical graphs remains open. Furthermore, our constructed graphs...

The Maximum Number of Edges in a Strongly Multiplicative Graph