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Constructing Minimally 3-Connected Graphs

Departamento de Matemática, Universidade Federal Do Espírito, Vitória 29075-910, Brazil
Bloomberg LP, New York, NY 10165, USA
Department of Mathematics, Brooklyn College, City University of New York, Brooklyn, NY 11210, USA
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Algorithms 2021, 14(1), 9;
Received: 1 December 2020 / Revised: 22 December 2020 / Accepted: 28 December 2020 / Published: 1 January 2021
(This article belongs to the Special Issue Selected Algorithmic Papers From CSR 2020)
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 G from the cycles of G, where G is obtained from G 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 n vertices and m edges from the non-isomorphic minimally 3-connected graphs with n1 vertices and m2 edges, n1 vertices and m3 edges, and n2 vertices and m3 edges. View Full-Text
Keywords: graphs; minors; minimal 3-connected graphs; cubic graphs; algorithm graphs; minors; minimal 3-connected graphs; cubic graphs; algorithm
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MDPI and ACS Style

Costalonga, J.P.; Kingan, R.J.; Kingan, S.R. Constructing Minimally 3-Connected Graphs. Algorithms 2021, 14, 9.

AMA Style

Costalonga JP, Kingan RJ, Kingan SR. Constructing Minimally 3-Connected Graphs. Algorithms. 2021; 14(1):9.

Chicago/Turabian Style

Costalonga, João Paulo, Robert J. Kingan, and Sandra R. Kingan. 2021. "Constructing Minimally 3-Connected Graphs" Algorithms 14, no. 1: 9.

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