Abstract
The minor crossing number of a graph $G$, $rmmcr(G)$, is defined as the minimum crossing number of all graphs that contain $G$ as a minor. We present some basic properties of this new minor-monotone graph invariant. We give estimates on mmcr for some important graph families using the topological structure of graphs satisfying \$mcr(G) ≤k$.
Highlights
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We present some basic properties of this new minor-monotone graph invariant
We suggest a possible approach to this problem, how to relate crossing number to graph minors and graph embeddings
Summary
Let G = (V, E) be a graph and let S0 denote the 2-sphere. Let D be a drawing of a graph G in S0. Let G be a graph with mcr(G) = k. If G is a minor of Gand cr(G) = k we say that Gis a realizing graph of G, and a drawing Dof Gwith exactly k crossings is a realizing drawing of G. Replace each vertex v ∈ V (G) with a vertex tree Tv and make sure that trees Tv and Tw are connected with an edge if uv ∈ E(G), and let us call the newly obtained graph G. Optimizing over all possible choices for vertex trees and their connections and over all possible drawings of these graphs yields a realizing drawing of G
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