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Examples of Obstructions to Apex Graphs, Edge-Apex Graphs, and Contraction-Apex Graphs International Workshop on Spatial Graphs Mike Pierce August 2016 University California, Riverside Contents Introduction Graph Minors Robertson and

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Examples of Obstructions to Apex Graphs, Edge-Apex Graphs, and Contraction-Apex Graphs

International Workshop on Spatial Graphs

Mike Pierce August 2016

University California, Riverside

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Contents

Introduction Graph Minors Robertson and Seymour’s Graph Minor Theorem mmna Graph Research What is an Apex Graph? Brute-Force Search Building Triangle-Wye Families mmne & mmnc Graph Research Edge-Apex and Contraction-Apex Contact Info & Links

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Introduction

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Graph Minors

A graph H is a minor of a graph G if performing some sequence of vertex deletions, edge deletions, or edge contractions on G results in a graph isomorphic to H. A simple minor of a graph is the result of performing any one of these actions on a graph.

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Graph Minors

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Robertson and Seymour’s Graph Minor Theorem

Graph Minor Theorem The set of finite, undirected graphs form a well-quasi-ordering under the graph minor relationship. That is, given a countable sequence of graphs {Gi} with i ∈ {1, 2, . . . } there must exists some j > i ≥ 1 such that Gi is isomorphic to a minor of Gj. Corollary: There are Finitely Many Minor-Minimal Non-P Graphs Take some graph property P such that any graph either does or does not have P, and such that P is minor-closed. Consider the set of all graphs {Gi} such that each Gi does not have P, but such that every (proper) minor of each Gi has P. This set of graphs {Gi} must be finite.

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Minor-Minimal Non-Planar

We can imagine that there are some graphs that are not planar, but that every minor of these graphs are planar. According to Robertson and Seymour’s Theorem, there must be a finite number of these graphs.

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Minor-Minimal Intrinsically Linked

A graph G is intrinsically linked if any embedding of G in S3 contains a pair of linked cycles. There are exactly seven minor-minimal intrinsically linked

  • graphs. This set of graphs is called

the Petersen Family.

Source: Wikimedia Commons

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Minor-Minimal Intrinsically Knotted

A graph G is intrinsically knotted if any embedding of G in S3 contains a knotted cycle. Classifying the entire set of minor-minimal intrinsically knotted graphs is still an open

  • problem. We know that there are

at least 263.

www.jmu.edu/_images/mathstat/sums/sumsknot15.jpg

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mmna Graph Research

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What is an Apex Graph?

A graph is apex if there is a vertex in the graph that we may remove to make the graph planar. We can think of these graphs as “one vertex away” from being planar.

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Minor-Minimal Non-Apex Graphs

This means that a minor-minimal non-apex (mmna) graph is a graph that is not apex, but such that every minor of the graph is apex.

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Examples of Minor-Minor Non-Apex Graphs

All intrinsically linked graphs are non-apex. It also turns out that all minor-minimal intrinsically linked graphs are minor-minimal non-apex.

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Examples Based on Connectivity

The first thought is to try to classify the mmna graphs based on their connectivity. We say a graph has connectivity n if the removal of n vertices is necessary to disconnect the graph into nontrivial compenents.

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Examples of Minor-Minor Non-Apex Graphs

There are exactly three disconnected mmna graphs. They are each a disjoint union of a pair of graphs from {K5, K3,3}.

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Examples of Minor-Minor Non-Apex Graphs

There are no mmna graphs of connectivity 1.

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Examples of mmna Graphs with Connectivity 2

Since apexness is based on graph planarity, many mmna graphs appear to have K5 and K3,3 as “building blocks”.

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Examples of mmna Graphs with Connectivity 2

Since apexness is based on graph planarity, many mmna graphs appear to have K5 and K3,3 as “building blocks”.

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Minor-Minimal Non-Apex Graphs

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A Few Initial Restrictions on mmna Graphs

The minimum vertex degree in any mmna graph is 3. Given a graph G with order v and minimum vertex degree δ, the size

  • f G is at least ⌈ δv

2 ⌉.

Given a graph G with order v, the maximum size of G is 1

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Minor-Minimal Non-Apex Graphs

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Minor-Minimal Non-Apex Graphs

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Brute-Force Search on 8, 9, and 10 Vertices

For doing this project, it is convenient to establish some functions to determine if a given graph G is mmna.

MMNAGraphQ[G_Graph] := Module[{}, Return[ (!ApexGraphQ[G]) && !(MemberQ[ApexGraphQ /@ SimpleMinors[G], False]) ]; ]; ApexGraphQ[G_Graph] := Module[{}, Return[ MemberQ[ PlanarGraphQ /@ Union[{G}, Table[VertexDelete[G, i], {i,VertexList[G]}]] , True] ]; ]; 21/42

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Brute-Force Search on 8, 9, and 10 Vertices

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Brute-Force Search on 8, 9, and 10 Vertices

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Brute-Force Search on 17–21 Edges

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Brute-Force Search on 17–21 Edges

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Triangle-Wye and Wye-Triangle Transforms

Let T be a 3-cycle (triangle) in a graph. We can perform a triangle-wye move on T by deleting the edges of T, and adding a new vertex to our graph adjacent to the vertices of T. Let v be a degree 3 vertex in a graph. We can perform a wye-triangle move on v by adding edges to connect the vertices adjacent to v, and then deleting v.

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Triangle-Wye Often Preserves NA

Theorem Let G be a non-apex graph with triangle T and let G′ be the result of performing triangle-wye on T where the vertex added to G′ is v. The graph G′ is non-apex if and only if G′ − v is non-planar. So unless the vertex that gets added when we perform triangle-wye causes a graph to become apex, then the graph will remain non-apex. Since performing triangle-wye on a graph preserves its size, no graph in a triangle-wye family can be a minor of another, increasing the likelyhood that some members of the triangle-wye family of an mmna graph are mmna.

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Building Triangle-Wye Families

We can write some functions to automate the process of constructing the triangle-wye family of a graph.

TriangleWyeFamily[G_Graph] := Module[{}, Return[ List[#, TriangleWyeFamily[#]] & /@ TriangleWyeCousins[G] ]; ]; TriangleWyeCousins[G_Graph] := Module[{}, Return[ DeleteGraphDuplicates[Table[TriangleWye[G, t], {t, TriangleList[G]}]] ]; ]; 28/42

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Building Triangle-Wye Families

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Building Triangle-Wye Families

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The Current mmna Total

We continued to construct mmna graphs by “gluing together” K5 and K3,3 subgraphs. We did a more liberal computer search on graphs with a minimum vertex degree of 4, 5, and 6. We performed a “smart search” for mmna graphs by looking at the simple minors of simple extensions of mmna graphs we already had. We (at least partially) created the wye-triangle-wye family for each of the new mmna graphs.

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The Current mmna Total

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The Current mmna Total

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Examples of mmna Graphs with Higher Connectivity

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mmne & mmnc Graph Research

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Edge-Apex and Contraction-Apex

A graph is edge-apex if there is some edge in the graph that we may delete to make the graph planar. A graph is contraction-apex if there is some edge in the graph that we may contract to make the graph planar.

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mmne and mmnc are not as nice as mmna

The minimum allowed vertex degree of simple mmne graphs and mmnc graphs is 2. Neither of the properties edge-apex or contraction-apex are closed under taking minors (but just barely).

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Properties of mmne and mmnc Graphs

Given a degree 2 vertex in an mmne graph, the neighbors of that vertex must themselves be neighbors. The three disconnected graphs that are the disjoint union of a pair of graphs from {K5, K3,3} are exactly the disconnected mmne and mmnc graphs. There are three mmne and mmnc graphs of connectivity 1 that are each the result of gluing together graphs in {K5, K3,3} on a single vertex.

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Examples of mmne Graphs

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The Current mmne Total

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The Current mmnc Total

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Contact Info & Links

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Contact Info

Mike Pierce math.ucr.edu/~mpierce mpierce@math.ucr.edu Supervised by Dr Thomas Mattman www.csuchico.edu/~tmattman

Current as of August 2016

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Links

Online Link to this Presentation math.ucr.edu/~mpierce/files/conferences/iwsg2016/presentation.pdf Six Variations on a Theme: Almost Planar Graphs Lipton, Mackall, Mattman, Pierce, Robertson, Thomas, Weinschelbaum CSU Chico Summer REUT; 2014 arxiv.org/abs/1608.01973 Classifying the Finite Set of Minor-Minimal Non-Apex Graphs Mike Pierce, CSU Chico Honors Thesis in Mathematics; 2014 math.ucr.edu/~mpierce/files/papers/CFSMMNAG-pierce.2014.pdf Mathematica Code Created for this Research github.com/mikepierce/MMGraphFunctions github.com/mikepierce/YTYGraphTransforms

Current as of August 2016

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