Connected Domatic Packings in Node-capacitated Graphs

A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. A dominating set is connected if the subgraph induced by its vertices is connected. The connected domatic partition problem asks for a partition of the nodes into connected dominating sets. The connected domatic number of a graph is the size of a largest connected domatic partition and it is a well-studied graph parameter with applications in the design of wireless networks. In this note, we consider the fractional counterpart of the connected domatic partition problem in nodecapacitated graphs. Let n be the number of nodes in the graph and let k be the minimum capacity of a node separator in G. Fractionally we can pack at most k connected dominating sets subject to the capacities on the nodes, and our algorithms construct packings whose sizes are proportional to k. Some of our main contributions are the following: • An algorithm for constructing a fractional connected domatic packing of size Ω (k) for nodecapacitated planar and minor-closed families of graphs. • An algorithm for constructing a fractional connected domatic packing of size Ω (k/ lnn) for node-capacitated general graphs. Dept. of Computer Science, University of Illinois, Urbana, IL, 61801. Supported in part by NSF grant CCF-1016684 and a Chirag Foundation graduate fellowship. This work was done while the author was visiting the IBM Almaden Research center. [email protected]. Google Research, New York, NY 10011. Part of this work was done while the author was a student at University of Illinois. [email protected]. Dept. of Computer Science, University of Illinois, Urbana, IL, 61801. Supported in part by NSF grant CCF-1016684 and a Siebel Scholar award. [email protected].