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Realizability of Graphs Maria Belk and Robert Connelly

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Motivation: the molecule problem A chemist determines the distances between atoms in a molecule: 7 7 4 4 7 4 2 5

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Motivation: the molecule problem 1.Is this a possible configuration of points in 3 dimensions? 2.What is a possible configuration satisfying these distances? 7 7 4 4 7 4 2 5

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Motivation: the molecule problem 1.Is this a possible configuration of points in 3 dimensions? 2.What is a possible configuration satisfying these distances? Unfortunately, these questions are NP-hard. We will answer a different question.

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Definitions Definition: A realization of a graph is a function which assigns to each vertex of a point in some Euclidean space. Definition: A graph is -realizable if given any realization , …, of the graph in some finite dimensional Euclidean space, there exists a realization , …, in -dimensional Euclidean space with the same edge lengths.

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Example: A path is 1-realizable.

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Examples A tree is 1 - realizable. is 2-realizable, but not 1-realizable. A cycle is also 2-realizable, but not 1-realizable. Any graph containing a cycle is not 1-realizable.

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Examples A tree is 1 - realizable. is 2-realizable, but not 1-realizable. A cycle is also 2-realizable, but not 1-realizable. Any graph containing a cycle is not 1-realizable.

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Examples A tree is 1 - realizable. is 2-realizable, but not 1-realizable. A cycle is also 2-realizable, but not 1-realizable. Any graph containing a cycle is not 1-realizable. 1 11 0 0

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Examples A tree is 1 - realizable. is 2-realizable, but not 1-realizable. A cycle is also 2-realizable, but not 1-realizable. Any graph containing a cycle is not 1-realizable.

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Theorem (Connelly) A graph is 1-realizable if and only if it is a forest (a disjoint collection of trees).

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Definition. A minor of a graph is a graph obtained by a sequence of Edge deletions and Edge contractions (identify the two vertices belonging to the edge and remove any loops or multiple edges). Theorem. (Connelly) If is - realizable then every minor of is - realizable (this means -realizability is a minor monotone graph property).

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Graph Minor Theorem Theorem (Robertson and Seymour). For a minor monotone graph property, there exists a finite list of graphs , …, such that a graph satisfies the minor monotone graph property if and only if does not have as a minor. Example: A graph is 1-realizable if and only if it does not contain as a minor.

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Graph Minor Theorem By the Graph Minor Theorem: For each , there exists a finite list of graphs , …, such that a graph is -realizable if and only if it does not have any as a minor.

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Forbidden Minors 1-realizable 2-realizable + ? 3-realizable + ?

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-sum Suppose and both contain a . The -sum of and is the graph obtained by identifying the two ’s.

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-trees -tree: -sums of ’s Example: a 1-tree is a tree

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-trees -tree: -sums of ’s Example: 2-tree

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-trees -tree: -sums of ’s Example: 3-tree

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Partial -trees Partial -tree: subgraph of a -tree Example: partial 2-tree

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Partial k-trees Theorem (Connelly). Partial -trees are -realizable.

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2-realizability Theorem (Wagner, 1937). is a partial 2-tree if and only if does not contain as a minor. Theorem (Belk, Connelly). is 2-realizable if and only if does not contain as a minor.

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Forbidden Minors 1-realizable 2-realizable 3-realizable + ?

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Theorem (Arnborg, Proskurowski, Corneil) The forbidden minors for partial 3-trees.

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Which of these graphs is 3-realizable?

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Which of these graph is 3-realizable? NO

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Which of these graph is 3-realizable? NO NO

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Which of these graph is 3-realizable? NO NO YES YES

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Are there any more forbidden minors for 3-realizability? Lemma (Belk and Connelly). If contains or as a minor then either contains or as a minor, or can be constructed by 2-sums and 1-sums of partial 3-trees, ’s, and ’s (and is thus 3-realizable). Theorem (Belk and Connelly). The forbidden minors for 3-realizability are and .

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Are there any more forbidden minors for 3-realizability? Lemma (Belk and Connelly). If contains or as a minor then either contains or as a minor, or can be constructed by 2-sums and 1-sums of partial 3-trees, ’s, and ’s (and is thus 3-realizable). Theorem (Belk and Connelly). The forbidden minors for 3-realizability are and . NO

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Forbidden Minors 1-realizable 2-realizable 3-realizable +

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is not 3-realizable

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and Theorem (Belk). and are 3-realizable. Idea of Proof: Pull 2 vertices as far apart as possible.

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Conclusion Forbidden Minors Building Blocks 1-realizable 2-realizable 3-realizable , , ,

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Open Question Which graphs are 4-realizable? – is a forbidden minor. –There is a generalization of , using a tetrahedron and a degenerate triangle. –However, there are over 75 forbidden minors for partial 4-trees.

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