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    This is the author's version of a work that was accepted for publication in Journal of Graph Theory. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms, may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Graph Theory, VOLUME 9, ISSUE 1, 129-146, 1985, 10.1002/jgt.3190090111

    Miscellaneous Graph Preliminaries

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    This article contains many auxiliary theorems which were missing in the Mizar Mathematical Library [2] to the best of the author’s knowledge. Most of them regard graph theory as formalized in the GLIB series (cf. [8]) and most of them are preliminaries needed in [7] or other forthcoming [email protected] Gutenberg University, Mainz, GermanyGrzegorz Bancerek, CzesƂaw Bylinski, Adam Grabowski, Artur KorniƂowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, CzesƂaw Bylinski, Adam Grabowski, Artur KorniƂowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.John Adrian Bondy and U. S. R. Murty. Graph Theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. ISBN 978-1-84628-969-9.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. About graph mappings. Formalized Mathematics, 27(3):261–301, 2019. doi:10.2478/forma-2019-0024.Sebastian Koch. About graph complements. Formalized Mathematics, 28(1):41–63, 2020. doi:10.2478/forma-2020-0004.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.233

    About Graph Mappings

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    In this articles adjacency-preserving mappings from a graph to another are formalized in the Mizar system [7], [2]. The generality of the approach seems to be largely unpreceeded in the literature to the best of the author’s knowledge. However, the most important property defined in the article is that of two graphs being isomorphic, which has been extensively studied. Another graph decorator is introduced as [email protected] Gutenberg University, Mainz, GermanyGrzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421–427, 1990.Grzegorz Bancerek, CzesƂaw Bylinski, Adam Grabowski, Artur KorniƂowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.John Adrian Bondy and U. S. R. Murty. Graph Theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. ISBN 978-1-84628-969-9.CzesƂaw Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.CzesƂaw Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1): 153–164, 1990.Christopher David Godsil and Gordon Royle. Algebraic graph theory. Graduate Texts in Mathematics; 207. Springer, New York, 2001. ISBN 0-387-95220-9; 0-387-95241-1.Adam Grabowski, Artur KorniƂowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Huckenbeck. Extremal paths in graphs: foundations, search strategies, and related topics, volume 10 of Mathematical Topics. Akademie Verlag, Berlin, 1. edition, 1997. ISBN 3-05-501658-0; 978-3-05-501658-5.Tommy R. Jensen and Bjarne Toft. Graph coloring problems. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York, 1995. ISBN 0-471-02865-7.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. About supergraphs. Part I. Formalized Mathematics, 26(2):101–124, 2018. doi:10.2478/forma-2018-0009.Sebastian Koch. About supergraphs. Part II. Formalized Mathematics, 26(2):125–140, 2018. doi:10.2478/forma-2018-0010.Mike Krebs and Anthony Shaheen. Expander families and Cayley graphs: a beginners guide. Oxford University Press, Oxford, 2011. ISBN 0-19-976711-4; 978-0-19-976711-3.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.27326130

    About Vertex Mappings

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    In [6] partial graph mappings were formalized in the Mizar system [3]. Such mappings map some vertices and edges of a graph to another while preserving adjacency. While this general approach is appropriate for the general form of (multidi)graphs as introduced in [7], a more specialized version for graphs without parallel edges seems convenient. As such, partial vertex mappings preserving adjacency between the mapped verticed are formalized [email protected] Gutenberg University, Mainz, GermanyIan Anderson. A first course in discrete mathematics. Springer Undergraduate Mathematics Series. Springer, London, 2001. ISBN 1-85233-236-0.Grzegorz Bancerek, CzesƂaw Bylinski, Adam Grabowski, Artur KorniƂowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Adam Grabowski, Artur KorniƂowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. About graph mappings. Formalized Mathematics, 27(3):261–301, 2019. doi:10.2478/forma-2019-0024.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.27330331

    Extended Natural Numbers and Counters

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    This article introduces extended natural numbers, i.e. the set ℕ âˆȘ {+∞}, in Mizar [4], [3] and formalizes a way to list a cardinal numbers of cardinals. Both concepts have applications in graph [email protected] Gutenberg University, Mainz, GermanyGrzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek, CzesƂaw ByliƄski, Adam Grabowski, Artur KorniƂowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, CzesƂaw ByliƄski, Adam Grabowski, Artur KorniƂowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.John Adrian Bondy and U. S. R. Murty. Graph Theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. ISBN 978-1-84628-969-9.Adam Grabowski, Artur KorniƂowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Library Committee of the Association of Mizar Users. Number-valued functions. Mizar Mathematical Library, 2007.Library Committee of the Association of Mizar Users. Introduction to arithmetic of extended real numbers. Mizar Mathematical Library, 2006.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341–347, 2003.Andrzej Trybulec. Subsets of complex numbers. Mizar Mathematical Library, 2003.Andrzej Trybulec. Basic operations on extended real numbers. Mizar Mathematical Library, 2008.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825–829, 2001.28323924

    About Supergraphs. Part II

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    In the previous article [5] supergraphs and several specializations to formalize the process of drawing graphs were introduced. In this paper another such operation is formalized in Mizar [1], [2]: drawing a vertex and then immediately drawing edges connecting this vertex with a subset of the other vertices of the graph. In case the new vertex is joined with all vertices of a given graph G, this is known as the join of G and the trivial loopless graph K1. While the join of two graphs is known and found in standard literature (like [9], [4], [8] and [3]), the operation discribed in this article is not.Alongside the new operation a mode to reverse the directions of a subset of the edges of a graph is introduced. When all edge directions of a graph are reversed, this is commonly known as the converse of a (directed) graph.Johannes Gutenberg University, Mainz, GermanyGrzegorz Bancerek, CzesƂaw ByliƄski, Adam Grabowski, Artur KorniƂowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, CzesƂaw ByliƄski, Adam Grabowski, Artur KorniƂowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Lowell W. Beineke and Robin J. Wilson, editors. Selected Topics in Graph Theory. Academic Press, London, 1978. ISBN 0-12-086250-6.John Adrian Bondy and U. S. R. Murty. Graph Theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. ISBN 978-1-84628-969-9.Sebastian Koch. About supergraphs. Part I. Formalized Mathematics, 26(2):101–124, 2018. doi:10.2478/forma-2018-0009.Gilbert Lee. Walks in graphs. Formalized Mathematics, 13(2):253–269, 2005.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics,6(3):335–338, 1997.Klaus Wagner. Graphentheorie. B.I-HochschultaschenbĂŒcher; 248. Bibliograph. Inst., Mannheim, 1970. ISBN 3-411-00248-4.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.26212514

    Unification of Graphs and Relations in Mizar

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    A (di)graph without parallel edges can simply be represented by a binary relation of the vertices and on the other hand, any binary relation can be expressed as such a graph. In this article, this correspondence is formalized in the Mizar system [2], based on the formalization of graphs in [6] and relations in [11], [12]. Notably, a new definition of createGraph will be given, taking only a non empty set V and a binary relation E ⊆ V × V to create a (di)graph without parallel edges, which will provide to be very useful in future [email protected] Gutenberg University, Mainz, GermanyGrzegorz Bancerek, CzesƂaw ByliƄski, Adam Grabowski, Artur KorniƂowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Adam Grabowski, Artur KorniƂowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. Underlying simple graphs. Formalized Mathematics, 27(3):237–259, 2019. doi:10.2478/forma-2019-0023.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Karol Pąk. The friendship theorem. Formalized Mathematics, 20(3):235–237, 2012. doi:10.2478/v10037-012-0028-7.Gunther Schmidt and Thomas Ströhlein. Relations and graphs: discrete mathematics for computer scientists. Springer Science & Business Media, 2012.Yozo Toda. The formalization of simple graphs. Formalized Mathematics, 5(1):137–144, 1996.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.28217318

    A graph-based mathematical morphology reader

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    This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research
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