1-planar graph

Coloring

1-planar graphs were first studied by , who showed that they can be colored with at most seven colors.{{citation

Ringel's motivation was in trying to solve a variation of total coloring for planar graphs, in which one simultaneously colors the vertices and faces of a planar graph in such a way that no two adjacent vertices have the same color, no two adjacent faces have the same color, and no vertex and face that are adjacent to each other have the same color. This can obviously be done using eight colors by applying the four color theorem to the given graph and its dual graph separately, using two disjoint sets of four colors. However, fewer colors may be obtained by forming an auxiliary graph that has a vertex for each vertex or face of the given planar graph, and in which two auxiliary graph vertices are adjacent whenever they correspond to adjacent features of the given planar graph. A vertex coloring of the auxiliary graph corresponds to a vertex-face coloring of the original planar graph. This auxiliary graph is 1-planar, from which it follows that Ringel's vertex-face coloring problem may also be solved with six colors. The graph K6 cannot be formed as an auxiliary graph in this way, but nevertheless the vertex-face coloring problem also sometimes requires six colors; for instance, if the planar graph to be colored is a triangular prism, then its eleven vertices and faces require six colors, because no three of them may be given a single color.{{citation

Edge density

Every 1-planar graph with n vertices has at most 4n − 8 edges.{{citation | editor1-last = Didimo | editor1-first = Walter | editor2-last = Patrignani | editor2-first = Maurizio

A 1-planar graph is said to be an optimal 1-planar graph if it has exactly 4n − 8 edges, the maximum possible. In a 1-planar embedding of an optimal 1-planar graph, the uncrossed edges necessarily form a quadrangulation (a polyhedral graph in which every face is a quadrilateral). Every quadrangulation gives rise to an optimal 1-planar graph in this way, by adding the two diagonals to each of its quadrilateral faces. It follows that every optimal 1-planar graph is Eulerian (all of its vertices have even degree), that the minimum degree in such a graph is six, and that every optimal 1-planar graph has at least eight vertices of degree exactly six. Additionally, every optimal 1-planar graph is 4-vertex-connected, and every 4-vertex cut in such a graph is a separating cycle in the underlying quadrangulation.{{citation

The graphs that have straight 1-planar drawings (that is, drawings in which each edge is represented by a line segment, and in which each line segment is crossed by at most one other edge) have a slightly tighter bound of 4n − 9 on the maximum number of edges, achieved by infinitely many graphs.{{citation

Complete multipartite graphs

A complete classification of the 1-planar complete graphs, complete bipartite graphs, and more generally complete multipartite graphs is known. Every complete bipartite graph of the form K2,n is 1-planar (even planar), as is every complete tripartite graph of the form K1,1,n. Other than these infinite sets of examples, the only complete multipartite 1-planar graphs are K6, K1,1,1,6, K1,1,2,3, K2,2,2,2, K1,1,1,2,2, and their subgraphs. The minimal non-1-planar complete multipartite graphs are K3,7, K4,5, K1,3,4, K2,3,3, and K1,1,1,1,3. For instance, the complete bipartite graph K3,6 is 1-planar because it is a subgraph of K1,1,1,6, but K3,7 is not 1-planar.{{citation

Computational complexity

It is NP-complete to test whether a given graph is 1-planar,{{citation | hdl-access = free | editor1-last = Tollis | editor1-first = Ioannis G. | editor2-last = Patrignani | editor2-first = Maurizio

In contrast to Fáry's theorem for planar graphs, not every 1-planar graph may be drawn 1-planarly with straight line segments for its edges.{{citation | editor1-last = Gudmundsson | editor1-first = Joachim | editor2-last = Mestre | editor2-first = Julián | editor3-last = Viglas | editor3-first = Taso

1-planar graphs have bounded local treewidth, meaning that there is a (linear) function f such that the 1-planar graphs of diameter d have treewidth at most f(d); the same property holds more generally for the graphs that can be embedded onto a surface of bounded genus with a bounded number of crossings per edge. They also have separators, small sets of vertices the removal of which decomposes the graph into connected components whose size is a constant fraction of the size of the whole graph. Based on these properties, numerous algorithms for planar graphs, such as Baker's technique for designing approximation algorithms, can be extended to 1-planar graphs. For instance, this method leads to a polynomial-time approximation scheme for the maximum independent set of a 1-planar graph.

Generalizations and related concepts

The class of graphs analogous to outerplanar graphs for 1-planarity are called the outer-1-planar graphs. These are graphs that can be drawn in a disk, with the vertices on the boundary of the disk, and with at most one crossing per edge. These graphs can always be drawn (in an outer-1-planar way) with straight edges and right angle crossings.{{citation | editor1-last = Wismath | editor1-first = Stephen | editor2-last = Wolff | editor2-first = Alexander | editor1-last = Wismath | editor1-first = Stephen | editor2-last = Wolff | editor2-first = Alexander

The 1-planar graphs include the 4-map graphs, graphs formed from the adjacencies of regions in the plane with at most four regions meeting in any point. Conversely, every optimal 1-planar graph is a 4-map graph. However, 1-planar graphs that are not optimal 1-planar may not be map graphs.{{citation

1-planar graphs have been generalized to k-planar graphs, graphs for which each edge is crossed at most k times (0-planar graphs are exactly the planar graphs). Ringel defined the local crossing number of G to be the least non-negative integer k such that G has a k-planar drawing. Because the local crossing number is the maximum degree of the intersection graph of the edges of an optimal drawing, and the thickness (minimum number of planar graphs into which the edges can be partitioned) can be seen as the chromatic number of an intersection graph of an appropriate drawing, it follows from Brooks' theorem that the thickness is at most one plus the local crossing number.{{citation

Nonplanar graphs may also be parameterized by their crossing number, the minimum number of pairs of edges that cross in any drawing of the graph. A graph with crossing number k is necessarily k-planar, but not necessarily vice versa. For instance, the Heawood graph has crossing number 3, but it is not necessary for its three crossings to all occur on the same edge of the graph, so it is 1-planar, and can in fact be drawn in a way that simultaneously optimizes the total number of crossings and the crossings per edge.

Another related concept for nonplanar graphs is graph skewness, the minimal number of edges that must be removed to make a graph planar.

References

References

1. (2013). "On drawings and decompositions of 1-planar graphs". [[Electronic Journal of Combinatorics]].
2. (2012). "Adding one edge to planar graphs makes crossing number and 1-planarity hard".
3. (2013). "[[SWAT and WADS conferences".
4. (2015). "Proc. 23rd International Symposium on Graph Drawing and Network Visualization (GD 2015)".
5. {{harvtxt. Grigoriev. Bodlaender. 2007. Grigoriev and Bodlaender state their results only for graphs with a known 1-planar embedding, and use a tree decomposition of a planarization of the embedding with crossings replaced by degree-four vertices; however, their methods straightforwardly imply bounded local treewidth of the original 1-planar graph, allowing Baker's method to be applied directly to it without knowing the embedding.
6. (2015). "Proc. 23rd International Symposium on Graph Drawing and Network Visualization (GD 2015)".