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Distance labeling schemes for $K_4$-free bridged graphs

$k$-Approximate distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the $k$-approximation of the distance between any two vertices $u$ and $v$ can be determined efficiently by merely inspecting the labels of $u$ and $v$, without using any other information. One of the important problems is finding natural classes of graphs admitting exact or approximate distance labeling schemes with labels of polylogarithmic size. In this paper, we describe a $4$-approximate distance labeling scheme forthe class of $K_4$-free bridged graphs. This scheme uses labels of poly-logarithmic length $O(\log n^3)$ allowing a constant decoding time. Given the labels of two vertices $u$ and $v$, the decoding function returnsa value between the exact distance $d_G(u,v)$ and its quadruple $4d_G(u,v)$.