Given a connected graph G=(V,E)\documentclass[12pt]{minimal}
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\begin{document}$$G=(V,E)$$\end{document}. A subset C⊆V\documentclass[12pt]{minimal}
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\begin{document}$$C\subseteq V$$\end{document} is a dominating set if every vertex of V is either in C or adjacent to a vertex in C. Further, C is a connected dominating set if C is a dominating set and the induced subgraph G[C] is connected. The Minimum Connected Dominating Set (Min-CDS) problem asks to find a connected dominating set with the minimum size, which finds applications in communication networks, in particular, as a virtual backbone in wireless sensor networks. This paper focuses on a variant of the classic Min-CDS problem, called Minimum Connected Dominating Set with Labeling (Min-CDSL), in which we are given a connected graph with vertex labels, and are required to find a connected dominating set C such that the number of labels in C (instead of |C|) is minimized. Min-CDSL is apparently a generalization of Min-CDS, and is undoubtedly NP\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {NP}$$\end{document}-complete\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {complete}$$\end{document}. We give an approximation algorithm for Min-CDSL within performance ratio bounded by ln|V(G)|+span(G)+1\documentclass[12pt]{minimal}
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\begin{document}$$\ln |V(G)|+\mathrm {span}(G)+1$$\end{document}, where span(G)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {span}(G)$$\end{document} refers to the maximum span of the input labeled graph (i.e., the number of connected components of the induced subgraph by a single label). In general, span(G)≪|V(G)|\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {span}(G)\ll |V(G)|$$\end{document} and for a series of labeled graphs span(G)=O(1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {span}(G)=O(1)$$\end{document}. For a random graph G∈Gn,p\documentclass[12pt]{minimal}
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\begin{document}$$G\in {G_{n,p}}$$\end{document}, span(G)=O(ln|V(G)|)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {span}(G)=O(\ln |V(G)|)$$\end{document} almost surely, and thus our approximation ratio is O(ln|V(G)|)\documentclass[12pt]{minimal}
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\begin{document}$$O(\ln |V(G)|)$$\end{document} which is reasonable comparing with the best known approximation ratio ln|V(G)|+1\documentclass[12pt]{minimal}
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\begin{document}$$\ln |V(G)|+1$$\end{document} for Min-CDS.
