A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge. A non-1-planar graph G is minimal if the graph G − e is 1-planar for every edge e of G. We construct two infinite families of minimal non-1-planar graphs and show that for every integer n ≥ 63, there are at least 2(n−54)/4 nonisomorphic minimal non-1-planar graphs of order n. It is a...

In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal (45◦) line-segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A k-planar graph is a planar graph in which each vertex has degree less or equal to k. In particular, we prove that every 4-planar graph admits a planar...

It is shown that every connected planar straight line graph with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar straight line graph with at most b(2n − 2)/3c new edges. It is also shown that every planar straight line tree with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar topological graph by adding at most bn/2c edge...

A graph is called 1-planar if it can be drawn in the plane so that each its edge is crossed by at most one other edge. In the paper, we study the existence of subgraphs of bounded degrees in 1-planar graphs. It is shown that each 1-planar graph contains a vertex of degree at most 7; we also prove that each 3-connected 1-planar graph contains an edge with both endvertices of degrees at most 20, ...

A straight-line grid drawing of a planar graph G is a drawing of G on an integer grid such that each vertex is drawn as a grid point and each edge is drawn as a straight-line segment without edge crossings. It is well known that a planar graph of n vertices admits a straight-line grid drawing on a grid of area O(n). A lower bound of Ω(n) on the area-requirement for straight-line grid drawings o...

In this paper, we extend the Grötzsch Theorem by proving that the clique hypergraph H(G) of every planar graph is 3-colorable. We also extend this result to list colorings by proving that H(G) is 4-choosable for every planar or projective planar graph G. Finally, 4-choosability ofH(G) is established for the class of locally planar graphs on arbitrary surfaces.