Abstract
Cunningham and Edmonds [4[ have proved that a 2-connected graphG has a unique minimal decomposition into graphs, each of which is either 3-connected, a bond or a polygon. They define the notion of a good split, and first prove thatG has a unique minimal decomposition into graphs, none of which has a good split, and second prove that the graphs that do not have a good split are precisely 3-connected graphs, bonds and polygons. This paper provides an analogue of the first result above for 3-connected graphs, and an analogue of the second for minimally 3-connected graphs. Following the basic strategy of Cunningham and Edmonds, an appropriate notion of good split is defined. The first main result is that ifG is a 3-connected graph, thenG has a unique minimal decomposition into graphs, none of which has a good split. The second main result is that the minimally 3-connected graphs that do not have a good split are precisely cyclically 4-connected graphs, twirls (K 3,n for somen≥3) and wheels. From this it is shown that ifG is a minimally 3-connected graph, thenG has a unique minimal decomposition into graphs, each of which is either cyclically 4-connected, a twirl or a wheel.
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Research partially supported by Office of Naval Research Grant N00014-86-K-0689 at Purdue University.
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Coullard, C.R., Gardner, L.L. & Wagner, D.K. Decomposition of 3-connected graphs. Combinatorica 13, 7–30 (1993). https://doi.org/10.1007/BF01202787
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DOI: https://doi.org/10.1007/BF01202787