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Vertex suppression in 3‐connected graphs
Author(s) -
Kriesell Matthias
Publication year - 2008
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.20277
Subject(s) - combinatorics , mathematics , vertex (graph theory) , graph , connectivity , discrete mathematics
Abstract To suppress a vertex $v$ in a finite graph G means to delete it and add an edge from a to b if a , b are distinct nonadjacent vertices which formed the neighborhood of $v$ . Let $G--x$ be the graph obtained from $G-x$ by suppressing vertices of degree at most 2 as long as it is possible; this is proven to be well defined. Our main result states that every 3‐connected graph G has a vertex x such that $G -- x$ is 3‐connected unless G is isomorphic to $K_{3,3}$ , $K_2 \times K_3$ , or to a wheel $K_{1}*C_{\ell}$ for some $\ell \geq 3$ . This leads to a generator theorem for 3‐connected graphs in terms of series parallel extensions. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 41–54, 2008