Unavoidable Cycle-Contraction Minors of Large 2-Connected Graphs

Jørgensen conjectured that every 6-connected graph with no K6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs.

K6 minors in 6-connected graphs of bounded tree-width

In this work, we present results on the unavoidable structures in large connected and large 2-connected graphs. For the relation of induced subgraphs, Ramsey proved that for every positive integer r, every sufficiently large graph contains as an induced subgraph either Kr or Kr. It is well known that, for every positive integer r, every sufficiently large connected graph contains an induced subgraph isomorphic to one of Kr, K1,r, and Pr. We prove an analogous result for 2-connected graphs. Similarly, for infinite graphs, every infinite connected graph contains an induced subgraph isomorphic to one of the following: an infinite complete graph, an infinite star, and a ray. Using some techniques from the finite result, we give the unavoidable induced subgraphs of infinite 2-connected graphs. We then shift our attention to the relation of bipartite minors defined in 2016 by Chudnovsky, Kalai, Nevo, Novik, and Seymour. For the relation of bipartite minors, we present the unavoidable substructures of both large connected and large 2-connected bipartite graphs.

Linear connectivity forces large complete bipartite minors

It is shown that, for each fixed $k$, the problem of finding $k$ pairwise vertex-disjoint directed paths between given pairs of terminals in a directed planar graph is solvable in polynomial time.

Given a large graph, like who-calls-whom, or who-likes-whom, what behavior is normal and what should be surprising, possibly due to fraudulent activity? How do graphs evolve over time? How does influence/news/viruses propagate, over time? We focus on three topics: (a) anomaly detection in large static graphs (b) patterns and anomalies in large time-evolving graphs and (c) cascades and immunization. For the first, we present a list of static and temporal laws, including advances patterns like 'eigenspokes'; we show how to use them to spot suspicious activities, in on-line buyer-and-seller settings, in FaceBook, in twitter-like networks. For the second, we show how to handle time-evolving graphs as tensors, how to handle large tensors in map-reduce environments, as well as some discoveries such settings. For the third, we show that for virus propagation, a single number is enough to characterize the connectivity of graph, and thus we show how to do efficient immunization for almost any type of virus (SIS - no immunity; SIR - lifetime immunity; etc) We conclude with some open research questions for graph mining.

Contractible edges in 3-connected graphs that preserve a minor

It is well known that every sufficiently large connected graph G has either a vertex of high degree or a long path. If we require G to be more highly connected, then we ensure the presence of more highly structured minors. In particular, for all positive integers k, every 2-connected graph G has a series minor isomorphic to a k-edge cycle or K_{2,k}. In 1993, Oxley, Oporowski, and Thomas extended this result to 3- and internally 4-connected graphs identifying all unavoidable series minors of these classes. Loosely speaking, a series minor allows for arbitrary edge deletions but only allows edges to be contracted when they meet a degree-2 vertex. Dually, a parallel minor allows for any edge contractions but restricts the deletion of edges to those that lie in 2-edge cycles. This dissertation begins by proving the dual results to those noted above. These identify all unavoidable parallel minors for finite graphs of low connectivity. Following this, corresponding results on unavoidable minors for infinite graphs are proved. The dissertation concludes by finding the unavoidable parallel minors for 3-connected regular matroids, which combines the results for unavoidable series and parallel minors for graphs with Seymour's decomposition theorem for regular matroids.

This dissertation establishes a number of theorems related to the structure of graphs and, more generally, matroids. In Chapter 2, we prove that a 3-connected graph G that has a triangle in which every single-edge contraction is 3-connected has a minor that uses the triangle and is isomorphic to K5 or the octahedron. We subsequently extend this result to the more general context of matroids. In Chapter 3, we specifically consider the triangle-rounded property that emerges in the results of Chapter 2. In particular, Asano, Nishizeki, and Seymour showed that whenever a 3-connected matroid M has a four-point-line-minor, and T is a triangle of M, there is a four-point-line-minor of M using T. We will prove that the four-point line is the only such matroid on at least four elements. In Chapter 4, we extend a result of Dirac which states that any set of n vertices of an n-connected graph lies in a cycle. We prove that if V' is a set of at most 2n vertices in an n-connected graph G, then G has, as a minor, a cycle using all of the vertices of V'. In Chapter 5, we prove that, for any vertex v of an n-connected simple graph G, there is a n-spoked-wheel-minor of G using v and any n edges incident with v. We strengthen this result in the context of 4-connected graphs by proving that, for any vertex v of a 4-connected simple graph G, there is a K 5- or octahedron-minor of G using v and any four edges incident with v. Motivated by the results of Chapters 4 and 5, in Chapter 6, we introduce the concept of vertex-roundedness. Specifically, we provide a finite list of conditions under which one can determine which collections of graphs have the property that whenever a sufficiently highly connected graph has a minor in the collection, it has such a minor using any set of vertices of some fixed size.

Removable edges in a 5-connected graph and a construction method of 5-connected graphs

It is known that any 3-connected matroid that is large enough is certain to contain a minor of a given size belonging to one of a few special classes of matroids. This paper proves a similar unavoidable minor result for large 4-connected bicircular matroids. The main result follows from establishing the list of unavoidable minors of large 4-biconnected graphs, which are the graphs representing the 4-connected bicircular matroids. This paper also gives similar results for internally 4-connected and vertically 4-connected bicircular matroids.

A graph G has a K t minor if a graph isomorphic to K t , the complete graph on t vertices, can be obtained from a subgraph of G by contracting edges. A long-standing conjecture of Hadwiger states that every graph with no K t minor is (t − 1)-colorable. Hadwiger’s conjecture is known for t ≤ 6, and open for all t > 7.A deep theorem of Robertson and Seymour describes the structure of graphs with no K t minor. The theorem is very powerful, but it is fairly complicated to state, and the condition it gives is necessary, but not sufficient, for the exclusion of a K t minor.We prove a necessary and sufficient condition under additional restrictions on the graph G. We prove that for every integer t there exists an integer N such that every t-connected graph on at least N vertices with no K t minor has a set of at most t − 5 vertices whose deletion makes the graph planar. This is best possible in the sense that neither t-connectivity nor the size of the deleted set can be lowered, and for t > 7 some lower bound on the number of vertices is needed.

Small minors in dense graphs

Graphs are characterized by whether or not they have orientations to avoid one or more of the digraphs K&ar;3 , S&ar;3 , and P&ar;3 . K&ar;3 , S&ar;3 and P&ar;3 are created by starting with a triangle, a three point star, or a path of length three respectively, and replacing each edge with a pair of arcs in opposite directions. Conditions are described when all orientations of 3-connected and 4-connected graphs must have one or more of the above digraphs as a minor. It is shown that double wheels, and double wheels without an axle, are the only 4-connected graphs with an orientation not having a K&ar;3 -minor. For S&ar;3 , it is shown that the only 4-connected graphs which may be oriented without the minor are K5 and C26 . It is also shown that all 3-connected graphs which do not have a W5-minor have an orientation without-an S&ar;3 -minor, while every orientation of a graph with a W 6-minor has an S&ar;3 -minor. It is demonstrated that K5, C26 , and C26 plus an edge are the only 4-connected graphs with an orientation without a P&ar;3 -minor. Additionally, some restrictions on large 3-connected graphs without a P&ar;3 -minor are given, and it is shown that if a 3-connected graph has a large wheel as a minor and has an orientation without a P&ar;3 -minor, then the graph must be a wheel. Certain smaller digraphs P&ar;1 , P&ar;2 , and M = K&ar;3 \a are also considered as possible minors of orientations of graphs. It is shown that a graph has an orientation without a P&ar;1 -minor if and only if it is a forest. It is shown that every orientation of a graph has a P&ar;2 -minor if and only if the graph has T2 or K+4 as a minor. To describe graphs with an orientation without an M-minor, a similar small list of graphs is given, and it is shown that if none of the given graphs is a minor of a graph, then that graph has an orientation without an M-minor.

K6 and icosahedron minors in 5-connected projective planar graphs