Abstract
Call a connected planar graphG legal if it has at least two nodes, no parallel edges or self-loops and at most two terminals (degree 1 nodes) and all terminals and degree 2 nodes are exterior. This class of graphs arose in connection with a two-dimensional generating system for modeling growth by binary cell division. Showing that any permitted pattern can be generated properly requires a matching or pairing lemma. The vertex set of a legal graph withn nodes can be split intop adjacent pairs ands singletons withs p, resulting in a matching which includes at least\(2\left[ {\frac{n}{3}} \right]\) nodes. This bound is sharp in the sense that there are legal graphs for which this matching is maximum. The matching can be implemented by a linear time algorithm. A legal graph witht terminals and n≥4 nodes has a spanning tree with at most\(\left[ {\frac{{n - t}}{2}} \right] + t\) terminals; this bound is sharp. Such a spanning tree can be constructed by an algorithm which operates in almost linear time.
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This paper was supported in part by the National Science Foundation under Grant NSF-MCS-78-04725.
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Carlyle, J.W., Greibach, S.A. & Paz, A. Matching and spanning in certain planar graphs. Math. Systems Theory 16, 159–183 (1983). https://doi.org/10.1007/BF01744575
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DOI: https://doi.org/10.1007/BF01744575