Clin D'oeil on L1-embeddable Planar Graphs

In this note we present some properties of LI-embeddable planar graphs. We present a characterization of graphs isometrically embeddable into half-cubes. This result implies that every planar Li-graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of a planar Li-graph G the subgraph H of G bounded by C is also Li-embeddable. In many important cases, the length of C is the dimension of the smallest cube in which H has a scale embedding. Using these facts we establish the LI-embeddability of a list of planar graphs.

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