Minimal non-1-planar graphs

AbstractA graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge. A non-1-planar graph G is minimal if the graph G-e is 1-planar for every edge e of G. We prove that there are infinitely many minimal non-1-planar graphs (MN-graphs). It is known that every 6-vertex graph is 1-planar. We show that the graph K7-K3 is the unique 7-vertex MN-graph

Characterizing 2-crossing-critical graphs

It is very well-known that there are precisely two minimal non-planar graphs: $K_5$ and $K_{3,3}$ (degree 2 vertices being irrelevant in this context). In the language of crossing numbers, these are the only 1-crossing-critical graphs: they each have crossing number at least one, and every proper subgraph has crossing number less than one. In 1987, Kochol exhibited an infinite family of 3-connected, simple 2-crossing-critical graphs. In this work, we: (i) determine all the 3-connected 2-crossing-critical graphs that contain a subdivision of the M\"obius Ladder $V_{10}$; (ii) show how to obtain all the not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii) show that there are only finitely many 3-connected 2-crossing-critical graphs not containing a subdivision of $V_{10}$; and (iv) determine all the 3-connected 2-crossing-critical graphs that do not contain a subdivision of $V_{8}$.Comment: 176 pages, 28 figure

Konstruksi Famili Graf Hampir Planar dengan Angka Perpotongan Tertentu

KONSTRUKSI FAMILI GRAF HAMPIR PLANAR DENGAN ANGKA PERPOTONGAN TERTENTU Benny Pinontoan1) 1) Program Studi Matematika FMIPA Universitas Sam Ratulangi Manado, 95115ABSTRAK Sebuah graf adalah pasangan himpunan tak kosong simpul dan himpunan sisi. Graf dapat digambar pada bidang dengan atau tanpa perpotongan. Angka perpotongan adalah jumlah perpotongan terkecil di antara semua gambar graf pada bidang. Graf dengan angka perpotongan nol disebut planar. Graf memiliki penerapan penting pada desain Very Large Scale of Integration (VLSI). Sebuah graf dinamakan perpotongan kritis jika penghapusan sebuah sisi manapun menurunkan angka perpotongannya, sedangkan sebuah graf dinamakan hampir planar jika menghapus salah satu sisinya membuat graf yang sisa menjadi planar. Banyak famili graf perpotongan kritis yang dapat dibentuk dari bagian-bagian kecil yang disebut ubin yang diperkenalkan oleh Pinontoan dan Richter (2003). Pada tahun 2010, Bokal memperkenalkan operasi perkalian zip untuk graf. Dalam artikel ini ditunjukkan sebuah konstruksi dengan menggunakan ubin dan perkalian zip yang jika diberikan bilangan bulat k ³ 1, dapat menghasilkan famili tak hingga graf hampir planar dengan angka perpotongan k. Kata kunci: angka perpotongan, ubin graf, graf hampir planar. CONSTRUCTION OF INFINITE FAMILIES OF ALMOST PLANAR GRAPH WITH GIVEN CROSSING NUMBER ABSTRACT A graph is a pair of a non-empty set of vertices and a set of edges. Graphs can be drawn on the plane with or without crossing of its edges. Crossing number of a graph is the minimal number of crossings among all drawings of the graph on the plane. Graphs with crossing number zero are called planar. Crossing number problems find important applications in the design of layout of Very Large Scale of Integration (VLSI). A graph is crossing-critical if deleting of any of its edge decreases its crossing number. A graph is called almost planar if deleting one edge makes the graph planar. Many infinite sequences of crossing-critical graphs can be made up by gluing small pieces, called tiles introduced by Pinontoan and Richter (2003). In 2010, Bokal introduced the operation zip product of graphs. This paper shows a construction by using tiles and zip product, given an integer k ³ 1, to build an infinite family of almost planar graphs having crossing number k

Characterizing 2-crossing-critical graphs

It is very well-known that there are precisely two minimal non-planar graphs: K5 and K3,3 (degree 2 vertices being irrelevant in this context). In the language of crossing numbers, these are the only 1-crossing-critical graphs: They each have crossing number at least one, and every proper subgraph has crossing number less than one. In 1987, Kochol exhibited an infinite family of 3-connected, simple, 2-crossing-critical graphs. In this work, we: (i) determine all the 3-connected 2-crossing-critical graphs that contain a subdivision of the Möbius Ladder V10; (ii) show how to obtain all the not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii) show that there are only finitely many 3-connected 2-crossing-critical graphs not containing a subdivision of V10; and (iv) determine all the 3-connected 2-crossing-critical graphs that do not contain a subdivision of V8