Ma/CS 6b Class 7: Minors

Transcription

1 Ma/CS 6b Class 7: Minors By Adam Sheffer Edge Subdivision iven a graph = V, E, and an edge e E, subdividing e is the operation of replacing e with a path consisting of new vertices. 1

2 raph Relations We saw two types of relations between graphs and : is a subgraph of. is an induced subgraph of. We say that is a subdivision of if is obtained by subdividing some (or all) of the edges of. raph Subdivision is a subdivision of. We refer to the added vertices as subdivision vertices. These vertices are of degree 2. 2

3 Topological Minors A graph is a topological minor of a graph if contains a subdivision of as a subgraph. Vertex Suppression iven a graph = V, E, and a vertex v V of degree 2, suppressing v is the operation of removing v and adding an edge between the two neighbors of V. 3

4 An Equivalent Definition A graph is a topological minor of a graph if can be obtained from by suppressing vertices (of degree 2) and by removing edges and vertices. No Order Notice that the order of the vertex and edge removal and of the vertex suppression does not matter (we will not formally prove this). 4

5 Test Your Intuition Is a topological minor of? No, since contains a vertex of degree 4 but does not. Test Your Intuition #2 Is K 5 a topological minor of the Petersen graph? The Petersen graph No. As before, we cannot create vertices of degree 4 by deletions and suppressions. 5

6 Edge Contraction iven a graph = V, E, and an edge e E, contracting e is the operation of removing e and merging its two endpoints. We merge any resulting parallel edges. a f a f b c d e b c/d e Minors A graph is a minor of a graph if can be obtained from by contracting edges and by removing edges and vertices. a b c a b c 6

7 Test Your Intuition #3 Is a minor of? Yes! Test Your Intuition #4 Is K 5 a minor of the Petersen graph? The Petersen graph Yes. Contract the five edges that connect an inner vertex to an outer vertex. 7

8 An Alternative Point of View Let be a minor of. Then every vertex of corresponds to a connected subgraph of. No Order #2 When performing deletions and contractions, the order of the deletions does not matter (we will not formally prove this). 8

9 Etymology What is the meaning of the word matrix? Wikipedia: coined by James Joseph Sylvester in 1850 who understood a matrix as an object giving rise to a number of determinants today called minors. Sylvester explains: I have in previous papers defined a "Matrix" as a rectangular array of terms, out of which different systems of determinants may be engendered as from the womb of a common parent. No C 3 Minors Problem. Characterize the family of graphs that do not have C 3 as a minor. Solution. These are exactly the graphs that do not contain any cycles. If a graph contains a cycle C k, we can remove everything except for this C k and then contract it to C 3. If a graph contains no cycles, we cannot obtain a cycle by contracting and removing edges and vertices. 9

10 No C 4 Minors Problem. Characterize the family of graphs that do not have C 4 as a minor. Solution. These are exactly the graphs that do not contain cycles of length 4. If a graph contains a cycle C k with k 4, we can remove everything except for this C k and then contract it to C 4. If a graph does not contain C k with k 4, we cannot obtain a C 4 by contracting and removing edges and vertices. Minors and Topological Minors Prove or disprove. Every topological minor of a graph is also a minor of. A minor is obtained by removing edges and vertices and contracting edges. A topological minor is obtained by removing edges and vertices and by suppressing vertices. The claim is true since vertex suppression can be seen as a special case of edge contraction. 10

11 Connections Between raph Relations Warm up. Which relation is stronger? is a subgraph of. is an induced subgraph of. An induced subgraph is also a subgraph, but the opposite is not always true. Connections Between raph Relations What are the connections between the following relations? is a subgraph of. is an induced subgraph of. is minor of. is a topological minor of. is a subdivision of. Subdiv. Topo. Minor Minor Induced Subgraph Subgraph 11

12 Connections (cont.) There is no relation between is a subgraph of. is a subdivision of. Subdivision and not a subgraph Subgraph and No subdivision Connections (cont.) Only suppressions Deletions and suppressions Induced Subgraph Subdiv. Subgraph Topo. Minor Minor Vertex deletions Only deletions Deletions and contractions 12

13 Minors and Bounded Degrees Claim. Consider = V, E, and let be a minor of with maximum degree three. Then is also a topological minor of. Proof is obtained from by removing edges and vertices and contracting edges. We need to show that can be obtained from by removing edges and vertices and by suppressing vertices. We first perform all of the deletions. We then need to handle contracting edges. After performing the deletions, we may assume that no vertex has degree more than three, also after performing any number of the contractions. 13

14 Proof: Two Cases We consider contracting an edge u, v. If either u or v is of degree 1, this is just a vertex removal. u v v If either u or v is of degree 2, this can be replaced with vertex suppression. w u v w v Proof: Another Case We consider the contraction of an edge e between vertices u, v. Consider the case where both u and v are of degree 3 and with a common neighbor w. w w u v We delete the edge v, w and then suppress v. 14

15 Proof: The Last Case We consider the contraction of an edge e between vertices u, v. Consider the case where both u and v are of degree 3 with no common neighbors. u v This contraction yields a vertex of degree 4. Contradiction! (since we performed the deletions first to avoid such a case) The End 15