Connected Domatic Packings in Node-capacitated Graphs

TL;DR

This paper studies the fractional connected domatic packing problem in node-capacitated graphs, providing algorithms that achieve packings proportional to the minimum node separator capacity, with specific results for planar, minor-closed, and general graphs.

Contribution

It introduces algorithms for fractional connected domatic packings in node-capacitated graphs, achieving sizes proportional to the minimum node separator capacity, including for planar and general graphs.

Findings

Algorithms for planar and minor-closed graphs achieve size Ω(k).

Algorithms for general graphs achieve size Ω(k / log n).

Fractional packings are proportional to the minimum node separator capacity.

Abstract

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 \emph{node-capacitated} 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: \begin{itemize} \item An algorithm for constructing a fractional connected domatic packing of size $\Omega(k)$ for node-capacitated planar and minor-closed families of graphs. \item An algorithm for constructing a fractional connected domatic packing of size $\Omega(k / \ln{n})$ for node-capacitated general graphs. \end{itemize}