Minimally 3-connected graphs

Dawes, Robin W.

doi:10.1016/0095-8956(86)90074-2

Cited by 15 publications

(10 citation statements)

References 8 publications

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“…A set S of vertices and/or edges in a graph G is 3-compatible if it conforms to one of the following three types: For convenience in the descriptions to follow, we will use D1, D2, and D3 to refer to bridging a vertex and an edge, bridging two edges, and adding a degree 3 vertex, respectively. Dawes proved that if one of the operations D1, D2, or D3 is applied to a minimally 3-connected graph, then the result is minimally 3-connected if and only if the operation is applied to a 3-compatible set [8].…”

Section: Terminology Previous Results and Outline Of The Papermentioning

confidence: 99%

Constructing Minimally 3-Connected Graphs

Costalonga

Kingan

2021

Algorithms

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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. 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 n−1 vertices and m−2 edges, n−1 vertices and m−3 edges, and n−2 vertices and m−3 edges.

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Section: Terminology Previous Results and Outline Of The Papermentioning

confidence: 99%

Constructing Minimally 3-Connected Graphs

Costalonga

Kingan

2021

Algorithms

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show abstract

“…For convenience in the descriptions to follow, we will use D1, D2, and D3 to refer to bridging a vertex and an edge, bridging two edges, and adding a degree 3 vertex, respectively. Dawes proved that if one of the operations D1, D2, or D3 is applied to a minimally 3-connected graph, then the result is minimally 3-connected if and only if the operation is applied to a 3-compatible set [4].…”

Section: Figure 2 Tutte's Vertex Split Operationmentioning

confidence: 99%

Constructing minimally 3-connected graphs

Costalonga¹,

Kingan²,

Kingan

2020

Preprint

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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 nonisomorphic minimally 3-connected graphs with n vertices and m edges from the nonisomorphic 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.

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“…Weak n-connectivity, under the name minor-minimal n-connectivity, has been studied by Oxley [11] and Leo [9], primarily in the context of matroids. Of particular interest to this paper is the work of Dawes; the Main Theorem above can be seen as an extension of the following result in [5].…”

Section: Introductionmentioning

confidence: 98%

“…Minimal 3-connectivity has been studied by Dawes [5], Coullard, Gardner and Wagner [4], Halin [6,7,8], Mader [10], among others. Weak n-connectivity, under the name minor-minimal n-connectivity, has been studied by Oxley [11] and Leo [9], primarily in the context of matroids.…”

Section: Introductionmentioning

confidence: 99%