@MISC{GLIBPRE0.MIZ,
  AUTHOR = {Koch, Sebastian},
  NOTE1 = {The author is enrolled in the Johannes Gutenberg University in Mayence, Germany},
  TITLE = {Miscellaneous Graph Preliminaries},
  SECTION1 = {Preliminaries not directly related to graphs},
  SECTION2 = {Into GLIB_000},
  SECTION3 = {Into GLIB_002},
  SECTION4 = {Into GLIB_006},
  SECTION5 = {Into GLIB_007},
  SECTION6 = {Into GLIB_008},
  SECTION7 = {Into GLIB_009},
  SECTION8 = {Into GLIB_010},
  SECTION9 = {Into CHORD},
  SECTION10 = {Into GLIB_011},
  DAY = {30},
  MONTH = {December},
  YEAR = {2019},
  ADDRESS1 = {Germany\\\href{mailto:skoch02@students.uni-mainz.de}{skoch02@students.uni-mainz.de}},
  SUMMARY = {This article contains many auxiliary theorems which were missing in the Mizar 
  Mathematical Library \cite{BancerekJAR:2018} to the best of the author's knowledge. 
  Most of them regard graph theory as formalized in the \texttt{GLIB} series 
   (cf. \cite{GLIB_000.ABS}) and most of them are preliminaries needed in at least 
  one of \cite{GLIB_012.ABS,GLIB_013.ABS,GLIB_014.ABS,GLUNIR00.ABS}.},
  INTRODUCTION = {A generalized approach to graph theory as it was done in \cite{MULTI,AGT} 
  in contrast to \cite{SIMPLE,HOMO} is rather uncommon. To avoid duplication of the same 
  theorems in different formalization frameworks in the Mizar Mathematical Library 
  \cite{BancerekJAR:2018}, a generalized approach to formalization is preferable 
  (cf. \cite{GLIB_000.ABS,GLIB_010.ABS}). However, due to the sheer amount of ``obvious facts" 
  such an approach brings with it, it is only natural some of them not immediately needed 
  slip the initial formalization process. This article aims to fill some of the gaps that 
  emerged. Thereby, in most cases, preliminaries needed in at least one of     \cite{GLIB_012.ABS,GLIB_013.ABS,GLIB_014.ABS,GLUNIR00.ABS} are provided.\par
  Many theorems in this article regard the change of incident edge sets and degrees 
  of a vertex when going from one graph to a related one (e.g. when reversing 
  edge directions or adding an edge).}
}

@Article{BancerekJAR:2018,
  author={Bancerek, Grzegorz and Byli{\'{n}}ski, Czes{\l}aw and 
    Grabowski, Adam and Korni{\l}owicz, Artur and Matuszewski, Roman and 
  Naumowicz, Adam and P{\k{a}}k, Karol},
  title={The Role of the {M}izar {M}athematical {L}ibrary for Interactive Proof Development in {M}izar},
  journal={Journal of Automated Reasoning},
  year={2018},
  volume={61},
  number={1},
  pages={9--32},
  doi={10.1007/s10817-017-9440-6},
  url={https://doi.org/10.1007/s10817-017-9440-6}
}
@Article{FourDecades,
  author={Grabowski, Adam and Korni{\l}owicz, Artur and Naumowicz, Adam},
  title={Four Decades of {M}izar},
  journal={Journal of Automated Reasoning},
  year={2015},
  volume={55},
  number={3},
  pages={191--198},
  issn={1573-0670},
  doi={10.1007/s10817-015-9345-1}
}
@book{AGT,
  title={Algebraic graph theory: morphisms, monoids and matrices},
  author={Knauer, Ulrich},
  volume={41},
  year={2011},
  publisher={Walter de Gruyter}
}
@book{SIMPLE,
  TITLE = {Introduction to graph theory},
  AUTHOR = {Wilson, Robin James},
  ADDRESS = {Edinburgh},
  PUBLISHER = {Oliver & Boyd},
  YEAR = {1972},
  ISBN = {0-05-002534-1},
  PAGES = {VIII, 168 S.},
  URL = {http://scans.hebis.de/HEBCGI/show.pl?10670775_toc.pdf},
}
@book{MULTI,
  TITLE = {Graph theory},
  SERIES = {Graduate texts in mathematics ; 244},
  AUTHOR = {Bondy, John Adrian and Murty, U. S. R.},
  ADDRESS = {New York, NY},
  PUBLISHER = {Springer},
  YEAR = {2008},
  ISBN = {978-1-84628-969-9},
  PAGES = {XII, 652 S.},
  URL = {http://scans.hebis.de/HEBCGI/show.pl?19542627_toc.pdf ; http://scans.hebis.de/HEBCGI/show.pl?19542627_kap-2.pdf ; http://scans.hebis.de/HEBCGI/show.pl?19542627_ein-1.pdf}
}
@book{HOMO,
  title = {Graphs and homomorphisms},
  series = {Oxford lecture series in mathematics and its applications ; 28},
  author = {Hell, Pavol and Nesetril, Jaroslav},
  address = {Oxford},
  publisher = {Oxford Univ. Press},
  year = {2004},
  ISBN = {0-19-852817-5},
  pages = {IX, 244 S.},
  url = {http://scans.hebis.de/HEBCGI/show.pl?12385357_toc.pdf}
}