Constructing minimally 3-connected graphs

12/22/2020

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by   J. P. Costalonga, et al.

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A 3-connected graph is minimally 3-connected if removal of any edge destroys 3-connectivity. We present an algorithm for constructing minimally 3-connected graphs based on the results in (Dawes, JCTB 40, 159-168, 1986) using two operations: adding an edge between non-adjacent vertices and splitting a vertex. In order to test sets of vertices and edges for 3-compatibility, which depends on the cycles of the graph, we develop a method for obtaining the cycles of G' from the cycles of G, where G' is obtained from G by one of the two operations above. We eliminate isomorphs using certificates generated by McKay's isomorphism checker nauty. The algorithm consecutively constructs the non-isomorphic minimally 3-connected graphs with n vertices and m edges from the non-isomorphic minimally 3-connected graphs with n-1 vertices and m-2 edges, n-1 vertices and m-3 edges, and n-2 vertices and m-3 edges.