Partitioning a power set into union-free classes

Partitioning a graph into alliance free sets

A strong defensive alliance in a graph G = (V, E) is a set of vertices A ⊆ V , for which every vertex v ∈ A has at least as many neighbors in A as in V − A. We call a partition A, B of vertices to be an alliance-free partition, if neither A nor B contains a strong defensive alliance as a subset. We prove that a connected graph G has an alliance-free partition exactly when G has a block that is ...

Partitioning networks into classes of mutually isolated nodes ?

Following the work of Newman, we model complex networks with random graphs of a given degree distribution. In particular we study the case where all vertices have exactly the same degree (random regular graphs). We survey some recent results on the problem of partitioning such a graph into the smallest possible number of classes of mutually isolated vertices, known in graph theory as the colora...

Partitioning Patches into Test-equivalence Classes for Scaling Program Repair

Automated program repair is a problem of finding a transformation (called a patch) of a given incorrect program that eliminates the observable failures. It has important applications such as providing debugging aids, automatically grading assignments and patching security vulnerabilities. A common challenge faced by all existing repair techniques is scalability to large patch spaces, since ther...

Partitioning a Graph into a Dominating Set, a Total Dominating Set, and something else 1Michael

A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, Ars Comb. 89 (2008), 159–162) implies that every connected graph of minimum degree at least three has a dominating set D and a total dominating set T which are disjoint. We show that the Petersen graph is the only such graph for which D ∪ T necessarily contains all vertices of the graph.

Partitioning a Set