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Elimination of local bridges. (English) Zbl 0958.05033
Summary: Vertices of degree different from 2 in a graph \(K\) are called main vertices of \(K\), and paths joining these vertices are branches of \(K\). Let \(K\) be a subgraph of \(G\). It is shown that if \(G\) is 3-connected (modulo \(K\)), then it is possible to replace branches of \(K\) by other branches joining the same pairs of main vertices of \(K\) such that \(G\) has no bridges with respect to the new subgraph whose vertices of attachment all lie on a single branch of \(K\). We present a linear time algorithm that either performs such a task, or finds a Kuratowski subgraph \(K_5\) or \(K_{3,3}\) in a subgraph of \(G\) formed by a branch \(e\) and those bridges of \(K\) in \(G\) that are attached only to the branch \(e\).

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C85 Graph algorithms (graph-theoretic aspects)
68R10 Graph theory (including graph drawing) in computer science
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