On the Strong Equitable Vertex 2-Arboricity of Complete Bipartite Graphs

An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The strong equitable vertexk-arboricity of G, denoted by vak≡(G), is the smallest integer t such that G can be equitably partitioned into t′ induced forests for every t′≥t, where the maximum degree of each induced forest is at most k. In this paper, we provide a general upper bound for va2≡(Kn,n). Exact values are obtained in some special cases.

Equitable persistent coverage of non-convex environments with graph-based planning

In this article, we tackle the problem of persistently covering a complex non-convex environment with a team of robots. We consider scenarios where the coverage quality of the environment deteriorates with time, requiring every point to be constantly revisited. As a first step, our solution finds a partition of the environment where the amount of work for each robot, weighted by the importance of each point, is equal. This is achieved using a power diagram and finding an equitable partition through a provably correct distributed control law on the power weights. Compared with other existing partitioning methods, our solution considers a continuous environment formulation with non-convex obstacles. In the second step, each robot computes a graph that gathers sweep-like paths and covers its entire partition. At each planning time, the coverage error at the graph vertices is assigned as weights of the corresponding edges. Then, our solution is capable of efficiently finding the optimal open coverage path through the graph with respect to the coverage error per distance traversed. Simulation and experimental results are presented to support our proposal.

Equitable Partition of Graphs into Independent Sets and Cliques

A graph is (k, l) if its vertex set can be partitioned into k independent sets and l cliques. Deciding if a graph is (k, l) can be seen as a generalization of coloring, since deciding is a graph belongs to (k, 0) corresponds to deciding if a graph is k-colorable. A coloring is equitable if the cardinalities of the color classes differ by at most 1. In this paper, we generalize both the (k, l) and the equitable coloring problems, by showing that deciding whether a given graph can be equitably partitioned into k independent sets and l cliques is solvable in polynomial time if max(k, l) 2, and NP complete otherwise.