平面グラフとその周辺の話題kfujiwara/sendai/...until Graph Minors appeared. • Perhaps,...

平面グラフとその周辺の話題

河原林健一国立情報学研究所

[email protected] http://www.nii.ac.jp/~k_keniti

日常生活には，平面構造のネットワークが多い．例えば，１．鉄道網２．道路網 etc….

マイナー操作

Introduction

ネットワークの解析のためには「平面グラフ」の研究が必要丌可欠！

平面判定

これらの「平面構造」という特徴を利用することで，ネットワークの理論的な解析が可能 → VLSI等のネットワーク構築 → 迅速なカーナビゲーションシステムの情報アップデート etc…

私の研究

平面グラフの一例

平面グラフとは，「辺をそれぞれ交差しないように平面上に描くことができるグラフ」を指す

Introduction マイナー操作平面判定私の研究

The Four Color Theorem

Straight line embedding

3-polytope

Polyhedral etc

Every planar graph can be drawn

in the plane with straight edges

Fáry-Wagner

planar graph

Steinitz

Every 3-connected planar graph

is the skeleton of a polytope.

3-connected planar graph

Polyhedral version

Andre’ev

Every 3-connected planar graph

is the skeleton of a convex polytope

such that every edge

touches the unit sphere

The Four Color Theorem

Straight line embedding

3-polytope

Polyhedral etc

Planar network（平面ネットワーク）

Subway, Road etc (鉄道網、道路網等).

How can computer recognize? （コンピューターにどうやって認識さえるか？）

This needs some deep THEORY! （理論研究が必要！）

Need to convert into graphs! （グラフの言葉に変える必要がある！）

How can computer recognize planar graph?

Remember, computer can only accept 0 or 1!

How can computer recognize a planar graph??

From the “0/1” input, everybody should share the same “picture” (without seeing the actual map).

We need some deep results in graph theory.

Every planar graph is 4-colorable

Unique embedding theorem

Every planar graph (essentially) has a unique embedding (Whitney, 1935).

So your embedding and my embedding are SAME! Point: Save the following information: Face, vertices and edges. Give “orientation” to each face. The above information can give an embedding

(uniquely “extendable”)

Whitney Equivalence

Introduction

問題：不えられたネットワークが「平面グラフ」かどうかを，どのように判定するか？

問題：不えられたネットワークが「平面グラフ」ならどのように「よい」平面グラフを得るか？→この問題の解答を得るためには

「マイナー操作」を理解する必要がある．

マイナー操作

ネットワークを理論的に解析するためには，実際のネットワークが平面（またそれに近い構造の）グラフかどうかを判定する必要がある！！

また、平面に高速に埋め込む必要がある！！

平面判定私の研究

Introduction マイナー操作

どんな平面グラフでも，次の３つの操作を加えた後も平面グラフの性質は変わらない

１．辺を取り除く．

２．頂点を取り除く．

３．辺を縮約する．

マイナー操作とは？？？

この３つの操作をグラフ理論の分野では「マイナー操作」という


マイナー操作

１．辺を取り除く．

２．頂点を取り除く．




マイナー操作

マイナー操作

１．頂点を取り除く．

２．辺を取り除く．


Introduction 平面判定私の研究

この３つの操作（１．頂点を取り除く２．辺を取り除く３．辺を縮約する）をしても平面グラフがその性質を保存することを

「マイナー操作に関して閉じている」

と呼ぶ

マイナー操作

Introduction 平面判定

重要：「全ての平面グラフはマイナー操作に関して閉じている」が，「マイナー操作に関して閉じているグラフが全て平面グラフ」ではない！

私の研究

平面判定

不えられたグラフが平面グラフかどうかを判定するためには，「非平面グラフ」から，「平面グラフ」になる「最小」のグラフは何かを調べる必要がある．

マイナー操作

Introduction

非平面グラフ

大きいグラフ平面グラフマイナー操作

１．頂点を取り除く．

２．辺を取り除く．


小さいグラフ

平面グラフ

非平面グラフとは限らない．

私の研究

クラトスキーの定理その１

非平面の「最小」なグラフは，K3,3とK5である．与えられたグラフが平面グラフである必要十分条件は，「どのようなマイナー操作を

しても，K3,3とK5にならない」というこ

とである

非平面から平面になる「最小」のグラフは何か？




平面判定

K3,3とK5グラフの例

左: K5グラフ右: K3,3グラフ

私の研究

平面グラフに対しては，

「禁止」すべき「最小グラフ」は，K3,3とK5のみである．


平面判定

クラトスキーの定理その２

平面グラフかどうかを判断するための指標になる．平面グラフかどうかの判定に利用されている．

私の研究

Kuratowski定理の数学的応用

と計算機（アルゴリズム的）応用

Planar graphs are closed under taking minors(minor-closed).

There are only two “forbidden” minors, namely K5 and K3,3

There are many other “minor-closed” graphs, for example, graphs on surfaces, linklessly embeddable graphs, knotless embeddable graphs in 3-space etc.

What about other minor-closed graph property (preserved when taking minors)?

３．曲面上に埋め込めるグラフ（ドーナツ状のトーラス）


私の研究平面判定

Given a “link” in 3-space. Is it knotted??

Seminal series of ≥ 23 papers(> 500 pages) Perhaps the deepest theory in graph theory Powerful results on excluded minors:

• Every minor-closed graph property (preserved when taking minors) has a finite set of excluded minors [Wagner’s Conjecture]

• Every minor-closed graph property can be decided in polynomial time [Graph Minor Algorithm]

1. A far-generalization of Kuratowski’s theorem (Wagner’s conjecture).

2. Many new “concepts” in the proof. Tree-width, Tangle, Representativty, clique-sum etc.

3. A real “tour de force”. A seminal decomposition theorem and structure theorem.

4. A solution to one of the original Gary-Johnson’s problems (the disjoint paths problem) .

5. Many algorithmic applications.

Graph Minor Decomposition Thm Structure of H-minor-free Graphs [GM16—Robertson & Seymour 2003]

• Every H-minor-free graph can be written

as O(1)-clique sums of graphs (gluing).

(Tree-structure (Tree decomposition))

• Each summand is a (basic) graph that can be O(1)-almost-embedded into a bounded-genus surface (almost 2-dimentional)

• O(1) constants depend only on |V(H)|

Tree-Decomposition

Almost-Embeddable Graphs

Apex Surface

Vortex

A shorter proof of graph minor algorithm (Wollan and KK, STOC’10)

(< 50 pages, it does not depend on graph minor decomposition theorem) Much shorter proof of the graph minor

structure theorem (Wollan and KK, 2010) (< 50 pages) Faster graph minor testing.

Seminal series of ≥ 23 papers(> 500 pages) Perhaps the deepest theory in graph theory Powerful results on excluded minors:

• Every minor-closed graph property (preserved when taking minors) has a finite set of excluded minors [Wagner’s Conjecture]

• Every minor-closed graph property can be decided in polynomial time [Graph Minor Algorithm]

平面グラフの判定の計算量 (Hopcroft, Tarjan)

平面グラフは，線形時間で判定できる（ここでいう『線形時間』とは，入力の頂点数に関して線形関数のステップという意味である）．

この判定アルゴリズムを導き出すために，

クラトスキーの定理が利用されている．



実用の場面では，平面グラフの判定を可能な限り迅速に行う必要がある！！

クラトスキーの定理その１

非平面の「最小」なグラフは，K3,3とK5である．与えられたグラフが平面グラフである必要十分条件は，「どのようなマイナー操作を

しても，K3,3とK5にならない」というこ

とである

非平面から平面になる「最小」のグラフは何か？



私の研究の一例

実際のネットワーク網は，「完全な平面」とは限らない．(例えば，交通網における立体交差や鉄道網での地下トンネル等）

→ 不えられたグラフが，ｋ交差グラフであるかの判定を線形時間で行うことは可能か？

→ 不えられたグラフが，ｋ-平面グラフであるかの判定を線形時間で行うことは可能か？

…等を明らかにする必要がある



最近の研究で，以下の性質のグラフを

線形時間で判定することが可能であることを明らかにした（STOC’07,STOC’08,FOCS’08,

STOC’０９、FOCS‘09等）．

１．ｋ-平面グラフ．

２．ｋ交差グラフ．

３．曲面上に埋め込めるグラフ

（例えばドーナツ状のトーラス）．



３．曲面上に埋め込めるグラフ（ドーナツ状のトーラス）



• Mainly joint work with B. Reed (and his students), partially with B. Mohar.

• (half (1/3?) of) Algorithmic Graph Minor Program (10+ STOC FOCS papers).

• Problem: Either give an embedding or a certificate(NOT embeddable).

• All of these embedding problems should be done in LINEAR time !!

• We are “almost” there !

New Algorithmic Program • After Ripton-Tarjan’s algorithm, not so many known

until Graph Minors appeared.

• Perhaps, this is because we did NOT know “certificates” (Not having an embedding), until Graph Minors appeared.

• Right now, there is a “significant” generalization of Graph Minors’ algorithm.

• This gives rise to many new “LINEAR TIME ALGORITHMS” for graph embedding.

• Embedding graphs on a fixed surface, FOCS’08 .

• Crossing Number (at most k) STOC’07.

• “Densely” embeddable on a fixed surface (face-width at least k) STOC’08

• Planar graph with k apices, FOCS’09.

• Linkless embedding& Knotless embedding.

• “Almost” embeddable graphs (Graph Minors)

Surface

• By a surface, we mean a compact 2-dimensional manifold.

• We consider plane with crosscaps or handles.

• Plane + crosscap: Projective planar.

• Plane + handle : Torus

• Plane + two crosscaps : Klein bottle

Klein Bottle

Embedding into a surface

• NP-complete to determine Euler genus of a given graph. (Thomassen, 1988)

• For fixed Euler genus k, O(nk) algorithm. (Filotti, Miller and Reif, 1990)

• O(n3) algorithm. (Robertson and Seymour, 1995)

• Linear time algorithm. (Mohar, 1997)

• Mohar’s algorithm gives either an embedding in the surface of Euler genus k or an obstruction (One of minimal forbidden minors.)

Mohar’s algorithm

• First LINEAR time algorithm. • One of the most important graph

algorithms. But,,, 1. Proof > 100 pages. 2. Proof consists of 6 other papers ! 3. Hidden constant is huge…

New Algorithm(FOCS’08)

• With B. Reed and B. Mohar

• We get another linear time algorithm.

• Better in the following sense:

1. Proof < 15 pages

2. Hidden constant is better..

• Planarity Testing (Hopcroft-Tarjan etc)

• Embedding graphs on a fixed surface (Mohar + FOCS’08) .



• Planar graph with k apices FOCS’09.

• Linkless embedding& Knotless embedding.

• “Almost” embeddable graphs(Graph Minors)

What is crossing number ?

• Roughly, planar graph + k edge crossings. • So it is close to planar graphs. • Formally, a graph has crossing number at

most k if it can be embedded in the plane with at most k edge-crossings.

• NP-complete to compute the exact crossing number.

• For fixed k, however, it is more feasible.

• O(nk+1) is easy for crossing number

Crossing Number

Property of Crossing Number

• Planar graphs and Graphs on a fixed surface are closed under taking minors, but a graph with k crossings is not.

Not closed under minors

Some hard crossing number problems

• What is crossing number of the complete graph of order k ?

• Only known up to k=14! • What is crossing number of the balance

complete bipartite of order 2k ? • This is a well-known conjecture by

Turan. (Called Turan’s “Brick Factory” Problem.)

Computing Crossing Number

• NP-complete to decide the crossing number (Gary and Johnson, 1982)

• For fixed k, however, it is easy to decide whether or not a give graph has crossing number at most k in O(nk+1).

• For fixed k, Grohe (STOC’01 and JCSS 2004) gave an O(n2) algorithm (i.e, f(k) n2)

Result

• Theorem [KK and REED]

For fixed k, there is a linear time algorithm for deciding whether or not a given graph can be embedded in a plane with at most k crossings.

• Actually the algorithm does give a desired embedding if one exists.

• It also implies the planarity with at most K edges problem.

• Embedding graphs on a fixed surface, FOCS’08 .



• Planar graph with k apices, FOCS’09.

• Linkless embedding& Knotless embedding. SOCG’10

• “Almost” embeddable graphs (Graph Minors)

How can computer recognize in 3-space

Remember, computer can only accept 0 or 1!

How can computer recognize a “knot” in 3-space?

This is actually a big open question in Math!

The Unknotting Problem

Given a “link” in 3-space.

Is it knotted??

History

Haken gave an “algorithm” to decide in

1950’s. Not polynomial time. In P?? One of the most challenging problems

in Topology, Graph Theory and Theoretical Computer Science.

Open for >50 years…

Linklessly embeddable graphs

homological, homotopical,…

equivalent

embeddable in 3-space without linked cycles

Apex graph

Linkless Embedding Linkless embedding in 3-space: Embedding a

graph in 3-space so that there is no “link”.

Given a graph G, can you decide if there is a linkless embedding? (Yes, by Robertson-Seymour-Thomas, 1995)

Theorem[RST]

A graph G has a linkless embedding if and only if G does not contain Petersen family as a minor.

The Petersen family

(graphs arising from K6 by Δ-Y and Y- Δ)

K6 and Petersen Graph


平面判定

Planar graphs = No K5 nor K3,3 as a minor

左: K5グラフ右: K3,3グラフ

私の研究

Linkless Embedding Linkless embedding in 3-space: Embedding a graph

in 3-space so that there is no “link”. Given a graph G, can you decide if there is a

linkless embedding? (Yes, by Robertson-Seymour-Thomas, 1995)

Theorem[RST] A graph G has a linkless embedding if and only if G

does not contain Petersen family as a minor. Theorem[RST] Essentially unique embedding (for 4-connected

graphs)

Unique embedding theorem

Every planar graph (essentially) has a unique embedding (Whitney, 1935).

So your embedding and my embedding is SAME! Point: Save the following information: Face, vertices and edges. Give “orientation” to each face. The above information can give an embedding

(uniquely “extendable”)

Recent Progress

Flat embedding: Embedding in 3-space so that each cycle bounds a disk. So a generalization of planar embedding to 3-space! Flat embedding -> Linkless embedding But Linkless embedding does not imply flat embedding.

Flat embedding VS unknotting problem

Conjecture 1 (Lovasz, Schrijver, RST, 1995) Poly time algorithm to test if a GIVEN

embedding is FLAT. Conjecture => the unknotting problem! Converse?? Theorem(SOCG’10): Two big conjectures are equivalent!

Linkless Embedding Linkless embedding in 3-space: Embedding a graph

in 3-space so that there is no “link”. Given a graph G, can you decide if there is a

linkless embedding? (Yes, by Robertson-Seymour-Thomas, 1995)

If there is, can you construct?? (conjectured yes, by Lovasz, Schrijver, RST)

Yes! O(n2) algorithm (first P time algorithm! SOCG’10))

Theorem: O(n) time algorithm to find a flat

embedding in 3-space!

平面グラフの判定の計算量 (Hopcroft, Tarjan)

平面グラフは，線形時間で判定できる（ここでいう『線形時間』とは，入力の頂点数に関して線形関数のステップという意味である）．

この判定アルゴリズムを導き出すために，

クラトスキーの定理が利用されている．



実用の場面では，平面グラフの判定を可能な限り迅速に行う必要がある！！

Open Problem The Unknotting Problem

Given a “link” in 3-space.

Is it knotted??

まとめ

平面グラフのKuratowski定理は、グラフ理論、理論計算機分野における中心的課題。

数学においてはグラフマイナー理論を導き出す（Robertson&Seymour)。

平面性の線形時間判定（Hopcroft&Tarjan)

トポロジーの中心的課題にも関連

研究課題は多数残っている！

平面グラフとその周辺の話題kfujiwara/sendai/...until Graph Minors appeared. • Perhaps,...

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