Abstract
In this paper, we deduce a necessary and sufficient condition for graphs whose qlick graphs have crossing number one. We also obtain a necessary and sufficient condition for qlick graphs to have crossing number one in terms of forbidden subgraphs.
Highlights
A graph is planar if it can be drawn in the plane or on the sphere in such a way that no two of its edges intersect
It is implicit that the edges in a drawing are Jordan arcs, and it is easy to see that a drawing with the minimum number of crossing must be a good drawing, that is, each two edges have at most one point in common, which is either a common end-vertex or a crossing
The line graph of a planar graph G has crossing number one if and only if (1) or (2) holds: (1) (G) = 4 and there is unique non-cut-vertex of degree 4; (2) (G) = 5, every vertex of degree 4 is a cut-vertex, there is a unique vertex of degree 5 and it has at most 3 incident edges in any block
Summary
A graph is planar if it can be drawn in the plane or on the sphere in such a way that no two of its edges intersect. The line graph of a planar graph G has crossing number one if and only if (1) or (2) holds: (1) (G) = 4 and there is unique non-cut-vertex of degree 4; (2) (G) = 5, every vertex of degree 4 is a cut-vertex, there is a unique vertex of degree 5 and it has at most 3 incident edges in any block. Theorem 2.2 A graph G has a qlick graph with crossing number 1 if and only if G is planar and one of the following holds: (1) (G) = 3, G has exactly two adjacent non-cut-vertices of degree 3 and every other vertex of degree 3 is a cut-vertex.
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