Biased graphs whose matroids are special binary matroids
type: Item
instance of: scholarly article
title: Biased graphs whose matroids are special binary matroids
zbMATH Open document ID: 0786.05020
DOI: 10.1007/BF01787483
author: Thomas Zaslavsky
published in: Graphs and Combinatorics
publication date: 1992-06-25T00:00:00Z
review text: A biased graph \(\Omega\) is a graph \(\Gamma\) together with a class \({\mathcal B}\) of polygons of \(\Gamma\) such that no theta-subgraph of \(\Gamma\) contains exactly two members of \({\mathcal B}\). (Examples arise form signed graphs by letting \({\mathcal B}\) consist of the polygons with an even number of minus-signs.) A subgraph \(S\) is balanced if each polygon of \(S\) is in \({\mathcal B}\), contrabalanced if none does. The associated bias matroid \(G(\Omega)\) is defined on the edge set of \(\Gamma\) and has as circuits the balanced polygons and the minimal contrabalanced connected edge sets with cyclomatic number two. The lift \(L(\Omega)\) and the complete lift \(L_ 0(\Omega)\) are two more matroids closely related to \(G(\Omega)\). The article determines the possible structure of \(\Omega\) in the cases where one of the three assocaited matroids is known to be one of the matroids \(F_ 7\), \(R_{10}\), \(G(K_{3,3})\), \(G(K_ 4)\), \(G(K_ 5)\), and their duals. Sample result: \(L_ 0(\Omega) \cong F^*_ 7\) if and only if \(\Omega\) arises from \(K_ 4\) with all edges negatively signed. Moreover, \(L_ 0(\Omega)\) is regular if and only if \(\Omega\) is sign- biased and has no subgraph that is a subdivision of \([-K_ 4]\) or a cycle of tree unbalanced polygons.
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 4890
zbMATH Keywords: binary matroids
zbMATH Keywords: biased graph
zbMATH Keywords: polygons
zbMATH Keywords: bias matroid
zbMATH Keywords: cyclomatic number
zbMATH Keywords: lifts
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4890