| Summary: | 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. 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 <inline-formula><math display="inline"><semantics><msup><mi>G</mi><mo>′</mo></msup></semantics></math></inline-formula> from the cycles of <i>G</i>, where <inline-formula><math display="inline"><semantics><msup><mi>G</mi><mo>′</mo></msup></semantics></math></inline-formula> is obtained from <i>G</i> by one of the two operations above. We eliminate isomorphic duplicates using certificates generated by McKay’s isomorphism checker nauty. The algorithm consecutively constructs the non-isomorphic minimally 3-connected graphs with <i>n</i> vertices and <i>m</i> edges from the non-isomorphic minimally 3-connected graphs with <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> vertices and <inline-formula><math display="inline"><semantics><mrow><mi>m</mi><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula> edges, <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> vertices and <inline-formula><math display="inline"><semantics><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow></semantics></math></inline-formula> edges, and <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula> vertices and <inline-formula><math display="inline"><semantics><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow></semantics></math></inline-formula> edges.
|
|---|
