Connected Domatic Packings in Node-capacitated Graphs

Alina Ene; Nitish Korula; Ali Vakilian

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arXiv · 1305.4308

Connected Domatic Packings in Node-capacitated Graphs

Alina Ene·Nitish Korula·Ali Vakilian

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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}

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Alina Ene, Nitish Korula, Ali Vakilian. 2013-05-19. Connected Domatic Packings in Node-capacitated Graphs. https://arxiv.org/abs/1305.4308

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