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...

We consider the problem of maintaining a dynamic planar graph subject to edge insertions and edge deletions that preserve planarity but that can change the embedding. We describe algorithms and data structures for maintaining information about 2and 3-vertex-connectivity, and 3and 4-edge-connectivity in a planar graph in O(n1/2) amortized time per insertion, deletion, or connectivity query. All ...

Planar graph canonization is known to be hard for L this directly follows from L-hardness of tree-canonization [Lin92]. We give a log-space algorithm for planar graph canonization. This gives completeness for log-space under AC many-one reductions and improves the previously known upper bound of AC [MR91]. A planar graph can be decomposed into biconnected components. We give a log-space procedu...

DP-coloring, also known as correspondence coloring, is introduced by Dvořák and Postle. It is a generalization of list coloring. In this paper, we show that every connected toroidal graph without triangles adjacent to 5-cycles has minimum degree at most three unless it is a 2-connected 4-regular graph with Euler characteristic (G) = 0. Consequently, every toroidal graph without triangles adjace...