We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem—to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph—and the simultaneous planarity problem—to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required pl...

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions ...

Edge bundling is an important concept heavily used for graph visualization purposes. To enable the comparison with other established near-planarity models in graph drawing, we formulate a new edge-bundling model which is inspired by the recently introduced fan-planar graphs. In particular, we restrict the bundling to the endsegments of the edges. Similarly to 1-planarity, we call our model 1-fa...

Given an n-vertex graph and two straight-line planar drawings of the graph that have the same faces and the same outer face, we show that there is a morph (i.e., a continuous transformation) between the two drawings that preserves straight-line planarity and consists of O(n) steps, which we prove is optimal in the worst case. Each step is a unidirectional linear morph, which means that every ve...

A plane graph is a planar graph with a fixed planar embedding in the plane. In a boxrectangular drawing of a plane graph, every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal line segment or a vertical line segment, and the contour of each face is drawn as a rectangle. A planar graph is said to have a box-rectangular drawing if at least one of its plane ...

We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by a point-contact or a side-contact between the corresponding polygons. Specifically, we consider proportional contact representations, where given vertex weights are represented by the areas of the corresponding polygons. Several natural optimization goals for such repr...

This paper studies the problem of fault detection and isolation (FDI) for multi-agent systems (MAS) via complex Laplacian subject to actuator faults. A planar formation of point agents in the plane using simple and linear interaction rules related to complex Laplacian is achieved. The communication network is a directed, and yet connected graph with a fixed topology. The loss of symmetry in the...

A star coloring of a graph is a proper coloring such that no path on four vertices is 2-colored. We prove that every planar graph with girth at least 9 can be star colored using 5 colors, and that every planar graph with girth at least 14 can be star colored using 4 colors; the figure 4 is best possible. We give an example of a girth 7 planar graph that requires 5 colors to star color.

We show that obtaining the lexicographically first four coloring of a planar graph is NP–hard. This shows that planar graph four-coloring is not self-reducible, assuming P 6= NP . One consequence of our result is that the schema of [JVV 86] cannot be used for approximately counting the number of four colorings of a planar graph. These results extend to planar graph k-coloring, for k ≥ 4. Resear...

In Lombardi drawings of graphs, edges are represented as circular arcs, and the edges incident on vertices have perfect angular resolution. However, not every graph has a Lombardi drawing, and not every planar graph has a planar Lombardi drawing. We introduce k-Lombardi drawings, in which each edge may be drawn with k circular arcs, noting that every graph has a smooth 2-Lombardi drawing. We sh...