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Cobiased graphs: Single-element extensions and elementary quotients of graphic matroids

Daniel Slilaty, Thomas Zaslavsky

Abstract

Zaslavsky (1991) introduced a graphical structure called a biased graph and used it to characterize all single-element coextensions and elementary lifts of graphic matroids. We introduce a new, dual graphical structure that we call a cobiased graph and use it to characterize single-element extensions and elementary quotients of graphic matroids.

Cobiased graphs: Single-element extensions and elementary quotients of graphic matroids

Abstract

Zaslavsky (1991) introduced a graphical structure called a biased graph and used it to characterize all single-element coextensions and elementary lifts of graphic matroids. We introduce a new, dual graphical structure that we call a cobiased graph and use it to characterize single-element extensions and elementary quotients of graphic matroids.
Paper Structure (19 sections, 26 theorems, 7 equations, 4 figures)

This paper contains 19 sections, 26 theorems, 7 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Cobiased Graphs
  3. Join and Complete Join Matroids of Cobiased Graphs
  4. Cocircuits
  5. Bases and independent sets
  6. Rank
  7. Circuits
  8. Deletions and contractions
  9. Vertex union
  10. Two simple examples
  11. Gain graphs and linear classes of bonds
  12. Cobiased graphs from additive gain graphs
  13. Cobiased planar graphs using gains over arbitrary groups
  14. An example not realizable by gains
  15. Cycle shifting
  16. ...and 4 more sections

Key Result

Theorem 3.1

If $\mathcal{L}$ is a non-trivial linear class of bonds of $G$, then there is a matroid with element set $E(G)\cup e_0$ in which $e_0$ is neither a loop nor a coloop and whose cocircuits consist of the following: Conversely, if $N$ is a single-element extension of the graphic matroid $M(G)$ with new element $e_0$ which is neither a loop nor a coloop of $N$, then there is a non-trivial linear clas

Figures (4)

  • Figure 1: A tribond.
  • Figure 2: Every dibond is of exactly one of two possible types.
  • Figure 3: All edges are oriented in the downward direction. Every tribond is of one of the two types shown.
  • Figure 4: Figure for the proof of Proposition \ref{['P:GainsFromLabelings']}.

Theorems & Definitions (52)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • ...and 42 more