University of Haifa Israelsummerschool2013.ccaba.upc.edu/documents/Lecture3... · University of Haifa Israel Paris, June 2013 . What do you see? Let’s look at a painting. Vasily

Recent Results on Intersection Graphs

Martin Charles Golumbic University of Haifa

Israel

Paris, June 2013

What do you see?

Let’s look at a painting.

Vasily Kandinsky 1866-1944

Several Circles (Einige Kreise)

Jan.–Feb., 1926 Oil on canvas 55 1/4 x 55 3/8 inches (140.3 x 140.7 cm)

Solomon R. Guggenheim Museum, New York

Solomon R. Guggenheim Founding Collection By gift 41.283. © 2012 Artists Rights Society (ARS), New York/ADAGP, Paris
















Marty Golumbic’s rendition of






The intersection graph

Vasily Kandinsky

• born on December 4, 1866, in Moscow

• studied law and economics, University of Moscow (1886-1892)

• studied art in Munich (1897-1900)

•. exhibited for the first time in Berlin (1902)

• traveled in Italy, Netherlands, North Africa and Russia (1903 -1904)

• first show in Paris at the Salon d’Automne (1904); returned to Munich

• lived in Russia (1914-1921)

• moved to Weimar then Dessau (1922-1928).

• first solo show in New York (1923)

• fled from the Nazis and settled in Neuilly-sur-Seine,

near Paris (1933-1944)

• died on December 13, 1944, in Neuilly

String Graphs

A simple curve (no self crossings)

A string diagram

Intersection graph of curves in the plane

Coloring Strings

A string diagram

6 strings; 5 colors Can you find the 5-clique?

Only the two RED curves are disjoint.

by Jean-Louis LASSEZ

Beauty is in

Beholding the

Curves

Structure

versus

Flowing Free Form

Beauty is in

Beholding the

Curves

COMPLEXITY versus

Simplicity

Hilbert Curve but its string graph is trivial – just one isolated vertex!

Shortest paths in a grid Super-impose on one grid Find its string graph Everyone intersects (at the corners)!

But, Grid-Edge Intersection, is NOT complete

complete Grid-Edge disjointness

The main topic of this lecture,

Intersection graphs of paths on a grid (VPG)

Theorem: They are EQUIVALENT !

VPG ≡ String graphs

Every curve on a plane can be approximated as a path on a grid.

VPG = String

• Any intersection point of several curves can be modified such that the corresponding paths will have pairwise intersection.

THE VPG “BEND NUMBER”

The VPG bend number b(G) of a string graph G

is the smallest number k such that G has a path

representation P on a grid where each path

changes direction at most k times.

We call these Bk-VPG graphs

i.e., each path in P has at most k bends (90○ turns).

B0-VPG ≡ 2-DIR (intersecting vertical and horizontal segments)

Early HISTORY of String Graphs • \

Sinden, 1966: • Every planar graph is a string graph • The complete graph of size five with every edge subdivided into two edges is not a string graph

Ehrlich, Even, Tarjan, 1976: • The minimum coloring of string graphs is NP-complete Golumbic, Rotem, Urrutia, 1983: • Incomparability graphs (of partial orders) are string graphs

Theorem [Golumbic, Rotem and Uruttia, 1983]

The following are equivalent. (1) G is an incomparability graph. (2) G is the intersection graph of a concatenation of permutation diagrams.

Moreover, the minimum number l of permutation diagrams equals the partial order dimension minus 1: dim(G) = l + 1

A permutation diagram single bend paths on a grid

Is there a relationship between the partial order dimension and the VPG bend number?

Question 1: How big is the gap? Question 2: When is there equality? (in addition to permutation graphs)

A concatenation of l of permutation diagrams Bl-VPG representation

Theorem [Cohen & Golumbic, 2012]

Let G be a cocomparability graph, then b(G) ≤ dim(G) - 1.

Elad Cohen [2012]:

The bound b(G) ≤ dim(G) - 1 is not at all tight.

Proposition. The complement of the Hiraguchi

graph H2n has bending number one although the

poset dimension is n.

B1-VPG representation

Open Problem #1: Find a better bound.

Or prove there are cocomparability graphs

with high VPG bend number.

Open Problem #2: What can we say about B0-VPG

permutation graphs?

• Permutation graphs are B1-VPG

• There are examples like C4 which are B0-VPG.

• In fact, all complete bipartite graphs Km,n are B0-VPG.

Open Problem #3: Is it computationally hard to test whether a permutation graph has VPG bend number 0 or 1?

The answer for cographs (P4-free graphs)

Cohen, Golumbic, Ries [2012]:

The VPG bend number for a cograph is 0 iff it is W4-free.

Otherwise, it is 1.

Moreover, is it “easy” to test whether a cograph is W4-free.

Open Problem #4: What can we say about the VPG bend

number of bipartite permutation graphs?

• They are equivalent to bipartite cocomparability graphs.

• Bipartite B0-VPG ≡ GIG bipartite perm bipartite B1-VPG

GIG are the grid intersection graphs of [Hartman, et al. 1991] or the Pure 2-DIR of [Kratochvil, 1994]

Kandinsky vs Lassez

Trivially, every disk intersection graph is a string graph.

Everything I learned about Intersection Graphs started with Claude Berge

misc. special terms

• 1-string: any two curves intersect at most once

• SEG: all curves are straight line segments

• k-DIR: line segments with only k possible slopes

• 2-DIR: line segments with only 2 possible slopes (horizontal and vertical)

Middle HISTORY of String Graphs: planarity

• \

• Scheinerman [1984] conjectured that planar graphs are contained in the family of segment graphs (SEG).

• West [1991] conjectured that every planar graphs is in 4-DIR.

(still open)

• Bipartite planar graphs are in 2-DIR. [Hartman et al., 1991] and [Fraysseix et al., 1994]

• Triangle-free planar graphs are in 3-DIR. [Castro, et al., 2002]

• Planar graphs are in 1-String. [Chalopin et al., 2007] • Chalopin and Goncalves [2009] FINALLY proved Scheinerman's

conjecture.

More Middle HISTORY of String Graphs: recognition

• Recognizing string graphs is

– NP-hard [Kratochvil, 1991]

– decidable [Pach&Toth, 2002][Schaefer&S, 2001]

– in NP, hence, NP-complete [SchaeferSS, 2003]

• Recognizing d-DIR and PURE-d-DIR graphs

is NP-complete, for d 2 [Kratochvil, 1994]

http://www.graphclasses.org/index.html

Vertex Intersection Graphs of

Paths on a Grid

Journal of Graph Algorithms and Applications, 2012 http://jgaa.info/getPaper?id=253 DOI: 10.7155/jgaa.00253

Recent HISTORY of String Graphs

Andrei Asinowski, Elad Cohen,

Martin Charles Golumbic, Vincent Limouzy,

Mariana Lipshteyn, Michal Stern

http://jgaa.info/getPaper?id=253

http://jgaa.info/getPaper?id=253

Why do we care?

• Integrated thin-film RC layout

• Rather relevant in the modern world

• Better layout is better for everyone

http://en.wikipedia.org/wiki/File:Palm_Pre.jpg

Why do we care?

• Started as a string graph problem

• Wires intersecting in a grid, turns are costly

• Minimum coloring in VPG is number of layers in layout

http://en.wikipedia.org/wiki/File:Silicon_chip_3d.png

Circle Graphs are included in B1-VPG

Recap

We’ve learned that the below relationships hold

Sun graphs

• The n-Sun graph Sn consists of an n-clique K, and n additional vertices in a stable set S

with edges between each vertex

si and {ki, ki+1 mod n}

S4

Sun graphs and B0-VPG

• A B0-VPG graph contains no induced Sn for n≥3

Moreover [by Farber, 1983] strongly chordal graphs are equivalent to sun-free chordal graphs,

so we have

• The family of chordal B0-VPG graphs are equivalent to the strongly chordal B0-VPG graphs

Proof: A B0-VPG graph is Sn-free, for n≥3

• Proof will be by contradiction – we start by assuming some Sn for n≥3 is also B0-VPG

• Let PK and PS be the paths corresponding to K and S.

• As we saw before, all PK must share a grid point.

Call it Q .

• If all paths in PK are on the same line, only two

points in S can be represented

Proof (continued)

• So, there must be two paths in PK , say Pi horizontal and Pi+1 vertical • But to get a sun, there is a path in Ps that

intersects both Pi and Pi+1 and thus must contain Q

• This is a contradiction, as Ps is now in K, not S.

Sun graphs and B1-VPG

Sn is contained in B1-VPG, for every n.

Characterizations of some subclasses of chordal B0-VPG graphs M.C. Golumbic and B. Ries, Graphs and Combinatorics [2012]

Theorem 1. Let G = (V,E) be a split graph with maximal clique K and stable set S. Then G is B0-VPG if and only if G is F -free (F given

in the Figure shown here.)

Characterizations of some subclasses of chordal B0-VPG graphs (continued)

Theorem 2. Let G = (V,E) be a chordal bull-free graph. Then G is B0-VPG if and only if for every vertex v in V , G[N(v)] is T2-free.

T2

Theorem 3. Let G = (V,E) be a chordal claw-free graph. Then G is is B0-VPG if and only if G is {S3 , N5}-free.

Chordal B0-VPG graphs - Steve Chaplick, Elad Cohen and Juraj Stacho, WG 2011

2-row B0-VPG graphs - Steve Chaplick, Elad Cohen and Juraj Stacho, WG 2011

Cograph B0-VPG graphs - Elad Cohen, Martin Golumbic and Bernard Ries, 2012 new

Recognizing and Characterizing other Special Classes of B0-VPG

GOING BEYOND B1 – VPG

Finding a string graph which is not B1-VPG

K n

3

The graph

What is Kn3

A split graph with a clique K of n vertices and a stable set S of “n choose 3” vertices

Why study this graph?

• We have proved that, for n≥33, this graph is not in B1-VPG

• This is our first example of a string graph with this property

• In general, we can construct string graphs which require an arbitrary number of bends.

Proof outline

• Suppose that K333 has a B1-VPG representation.

This will be a proof by contradiction

• Vertex v maps to path Pv

• Paths are of the form: since those are the four possible shapes and any straight lines can be slightly modified to have such a small bend

Facts about the K paths

• One of the shapes must contain 9 = ceil(33/4) paths in it. WLOG, it’s and we can label the paths 1, … , 9.

• They are in a clique, so we can, with possible small adjustments, make their bending points form an ascending sequence:

lowest path is 1 and the highest path is 9

(see next figure)

Clique paths

• The clique looks the figure on the right.

• Now we need to add the stable set paths

Stable set paths

• Consider P147, P258, and P369

• Place P258 first. Say it meets the horizontal on P5.

There are four ways for

2 and 8, indicated by bold lines.

• Most choices block the other two paths

Stable set paths

• Only one way to place all three paths

• But now try to add P169 – impossible!

• We can conclude that this graph isn’t in B1-VPG, for n≥33

THE COMPLETE HIERARCHY OF Bk -VPG GRAPHS

Complete – all containment relations are given

Equivalences and containments – proved earlier or trivial

Separating examples – some from other papers

Incomparables

X Y G1 in X-Y G2 in Y-X

Circle B0-VPG, GIG S4 BW3

B0-VPG Chordal, Chordal B1-VPG

K3,3 S4

Etc. …

THE BEND NUMBER OF PLANAR GRAPHS

Another French Connection

de Fraysseix, Ossona de Mendez and Rosenstiehl [1994]:

One can represent a planar graph G with a “⊺-contact system”

Every planar graph is a B3 -VPG

A transformation

from T to B3

We asked: Is there a planar graph that is not B2 -VPG ? ANSWER: No! All planar graphs are B2-VPG! [Chaplick and Ueckerdt, 2013]

An example of the transformation

THE BEND NUMBER OF INCOMPARABILITY GRAPHS

Not optimal ! What is the real bend number?

A B1-VPG representation

The Hiraguchi graph on 8 vertices and the permutation concatenation diagram of its complement: dim = 4

COMPLEXITY RESULTS FOR BK-VPG GRAPHS

Generally Known Results

• All the classes VPG, B0-VPG, bipartite B0-VPG are NP-Complete to recognize

• Max Indep Set on B0-VPG is NP-C

• Hamiltonian circuit/path on B0-VPG is NP-C

• Max clique on B0-VPG easily polynomial, but

Max clique on Bk-VPG, k>0, is NP-Complete

Coloring B0-VPG - Preliminaries

• Important to circuit layout

• Min-color is NP-C for STRING (and so VPG), for CIRCLE graphs (and so B1-VPG)

• We have proven: NP-C for B0-VPG, i.e., to decide if χ(G) ≤ m, m ≥ 3.

Overview of the reduction

• Start with G, B1-VPG

• Split each path with a bend into horizontal and vertical subpaths, disconnected

• Link h and v with an m-1 clique at the bend

• Result is a G′, B0-VPG, in polynomial time and size

Overview of the reduction

B1-VPG on the left becomes B0-VPG on the right

Coloring B0-VPG – reduction proof

χ(G)≤m ↔ χ(G′)≤m

• Let φ be an m-coloring for G

• Color h, v in G′ with same color as path in G

• Color m-1 cliques with remaining colors

χ(G′)≤m → χ(G)≤m

• Each m-1 clique in G′ requires m-1 colors

• Requires h, v to have the same color, as there is only one color left

• Same coloring is valid assignment to G by dissolving the cliques

Coloring B0-VPG - Approximation

• ω(G) ≤ χ(G) ≤ 2 ω(G)

• Polynomial 2-approximation scheme exists for coloring G

• There is a triangle-free B0-VPG graph H, with χ(H)=4, so the bound is tight: χ(H) = 2 ω(H)

Some other open questions

• Relation between boxicity and bending number

• B1-VPG with a subset of

• A better bound on b(G) for other families of incomparability graphs?

Mathematicians of the 20th

Century

This 20th century French mathematician wrote

the zeroth book on graph theory in 1926.

Who was André Sainte-Laguë?

"Les réseaux (ou graphes)", Paris (1926)

Charles-Edouard Jeanneret aka Le Corbusier 1925 : "Plan Voisin", Paris

Thank you

Paris

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University of Haifa Israelsummerschool2013.ccaba.upc.edu/documents/Lecture3... · University of Haifa Israel Paris, June 2013 . What do you see? Let’s look at a painting. Vasily