Refining the characterization of projective graphs

Archdeacon showed that the class of graphs embeddable in the projective plane is characterized by a set of 35 excluded minors. Robertson, Seymour and Thomas in an unpublished result found the excluded minors for the class of k-connected graphs embeddable on the projective plane for k = 1,2,3. We give a short proof of that result and then determine the excluded minors for the class of internally 4-connected projective graphs. Hall showed that a 3-connected graph diff_x000B_erent from K5 is planar if and only if it has K3,3 as a minor. We provide two analogous results for projective graphs. For any minor-closed class of graphs C, we say that a set of k-connected graphs E disjoint from C is a k-connected excludable set for C if all but a _x000C_finite number of k-connected graphs not in C have a minor in E. Hall\u27s result is equivalent to saying that {K3,3} is a 3-connected excludable set for the class of planar graphs. We classify all minimal 3-connected excludable sets and fi_x000C_nd one minimal internally 4-connected excludable set for the class of projective graphs. In doing so, we also prove strong splitter theorems for 3-connected and internally 4-connected graphs that could have application to other problems of this type

Testing 4-critical plane and projective plane multiwheels using Mathematica

In this article we explore 4-critical graphs using Mathematica. We generate graph patterns according [1, D. Zeps. On building 4-critical plane and projective plane multiwheels from odd wheels, arXiv:1202.4862v1]. Using the base graph, minimal planar multiwheel and in the same time minimal according projective pattern built multiwheel, we build minimal multiwheels according [1], Weforward two conjectures according graphs augmented according considered patterns and their 4-criticallity, and argue them to be proved here if the paradigmatic examples of this article are accepted to be parts of proofs

Miscellaneous properties of embeddings of line, total and middle graphs

Chartrand et al. (J. Combin. Theory Ser. B 10 (1971) 12–41) proved that the line graph of a graph G is outerplanar if and only if the total graph of G is planar. In this paper, we prove that these two conditions are equivalent to the middle graph of G been generalized outerplanar. Also, we show that a total graph is generalized outerplanar if and only if it is outerplanar. Later on, we characterize the graphs G such that Full-size image (<1 K) is planar, where Full-size image (<1 K) is a composition of the operations line, middle and total graphs. Also, we give an algorithm which decides whether or not Full-size image (<1 K) is planar in an Full-size image (<1 K) time, where n is the number of vertices of G. Finally, we give two characterizations of graphs so that their total and middle graphs admit an embedding in the projective plane. The first characterization shows the properties that a graph must verify in order to have a projective total and middle graph. The second one is in terms of forbidden subgraphs

The reconstruction of planar graphs

The object of this thesis is to investigate the Reconstruction Problem for planar graphs. This study naturally leads to related topics concerning certain nonplanar graphs and the use of their embeddings on appropriate surfaces to reconstruct them. The principal aim of this work is to find new techniques of reconstruction and to increase the number of classes of graphs known to be reconstructible. In achieving this aim, various important properties of graphs, such as connectivity and uniqueness of embeddings, are explored, and new results on these topics are obtained. Part I, which consists of three chapters, contains a historic a l, non-technical introduction and general graph-theoretical definitions, notation and results. Some new concepts in reconstruction are also presented, notably the idea of reconstructor sets. Part II of the thesis deals with the vertex-reconstruction of maximal planar graphs: Chapter 4 is concerned with the vertex-recognition of maximal planarity, whereas Chapter 5 deals with the vertex-reconstruction. Part III deals with edge-reconstruction: planar graphs with minimum valency 5 and 4-connected planar graphs are reconstructed in Chapters 6 and 7 respectively. In Chapter 7, extensive use is made of the concept of reconstructor sets introduced in Chapter 3. This chapter also contains a brief discussion on the reconstruction of graphs from edge-contracted subgraphs, a problem which, in certain cases, can be regarded as dual to the Edge-reconstruction Problem. Part IV is concerned with extending the results and techniques of the previous chapters to nonplanar graphs. Chapter 8 discusses where the previous techniques fa il, and indicates where new methods are needed. In Chapter 9, all graphs which triangulate some surface and have connectivity 3 are edge-reconstructed. Certain graphs which triangulate the torus or the projective plane are also shown to be weakly vertex-reconstructible. Chapter 10 deals with the edge-reconstruction of all graphs which triangulate the projective plane. The Appendix proves a conjecture of Harary on the cutvertex-reconstruction of trees. One technique used here ties up with a method employed in previous chapters on edge-reconstruction

On building 4-critical plane and projective plane multiwheels from odd wheels

We build unbounded classes of plane and projective plane multiwheels that are 4-critical that are received summing odd wheels as edge sums modulo two. These classes can be considered as ascending from single common graph that can be received as edge sum modulo two of the octahedron graph O and the minimal wheel W3. All graphs of these classes belong to 2n-2-edges-class of graphs, among which are those that quadrangulate projective plane, i.e., graphs from Gr\"otzsch class, received applying Mycielski's Construction to odd cycle.Comment: 10 page

Cubic Partial Cubes from Simplicial Arrangements

We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.Comment: 11 pages, 10 figure