Connectivity Labeling Scheme for Graphs on Surface
Résumé
We consider the problem of assigning each vertex of a graph $G$ a (short) label such that, for every subset $X$ of vertices of $G$ and every vertex $u$ in $G\setminus X$, the connected component of $u$ in $G\setminus X$ can be (quickly) identified from the labels of $u$ and those of all the vertices of $X$. For planar graphs with $n$ vertices we present a new labeling scheme with $O(\log n)$-bit labels and $O(\log\log n)$ query time after an almost linear time preprocessing of $X$. The scheme is significantly simpler than previous solution due to Courcelle et al.~\cite{CGKT08} and the query time is exponentially faster. Our solution generalized to Euler genus-$g$ graphs, and more generally to graphs having some constrained $k$-page embedding.
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