Geometric Representations of Graphs Jan Kratochvíl, DIMATIA, Prague

Intersection Graphs {M u, u  V G } uv  E G  M u  M v  

String Graphs {M u, u  V G } uv  E G  M u  M v  

Personal Recollections 1982 – Czech-Slovak Graph Theory, Prague

Personal Recollections 1982 – Czech-Slovak Graph Theory, Prague 1983 – Prague 1990 – Tempe, Arizona

Personal Recollections 1982 – Czech-Slovak Graph Theory, Prague 1983 – Prague 1990 – Tempe, Arizona 1988 – Bielefeld, Germany

Intersection Graphs Every graph is an intersection graph.

Intersection Graphs Every graph is an intersection graph. M u = {e  E G | u  e}

Intersection Graphs Every graph is an intersection graph. uv  E G  M u  M v   M u = {e  E G | u  e}

Intersection Graphs Every graph is an intersection graph Restricting the sets

Intersection Graphs Every graph is an intersection graph Restricting the sets – by geometrical shape Motivation and applications in scheduling, biology, VLSI designs …

Intersection Graphs Every graph is an intersection graph Restricting the sets – by geometrical shape Motivation and applications in scheduling, biology, VLSI designs … Nice characterizations, interesting theoretical properties, challenging open problems

Few Examples

Interval graphs Interval graphs - Gilmore, Hoffman 1964 Fulkerson, Gross 1965 Booth, Lueker 1975 Trotter, Harary 1979 …

Few Examples Interval graphs Interval graphs - - neat characterization chordal + co-comparability - recognizble in linear time - most optimization problems solvable in polynomial time - perfect

Few Examples SEG graphs SEG graphs - Ehrlich, Even, Tarjan 1976 Scheinerman Erdös, Gyarfás 1987 JK, Nešetřil 1990 JK, Matoušek 1994 Thomassen 2002

Few Examples SEG graphs SEG graphs - - recognition NP-hard and in PSPACE, NP-membership open - coloring, independent set NP-hard, complexity of CLIQUE open - near-perfectness open

Near-perfect graph classes A graph class G is near-perfect if there exists a function f such that  (G)  f(  (G)) for every G  G.

Few Examples String graphs String graphs - Sinden 1966 Ehrlich, Even, Tarjan 1976 JK 1991 JK, Matoušek 1991 Pach, Tóth 2001 Štefankovič, Schaffer 2001, 2002

Few Examples CONV graphs CONV graphs - Ogden, Roberts 1970 JK, Matoušek 1994 Agarwal, Mustafa 2004 Kim, Kostochka, Nakprasit 2004

Few Examples PC graphs PC graphs - Fellows 1988 Koebe 1990 JK, Kostochka 1994 Spinrad JK, Pergel 2002 Pergel 2007

Few Examples Circle graphs Circle graphs - De Fraysseix 1984 Bouchet 1985 Gyarfas 1987 Unger 1988 Kloks 1993 Kostochka 1994

Few Examples Circle graphs Circle graphs - - recognizable in linear time - coloring NP-hard - independent set, clique solvable in polynomial time - near-perfect  log     O(2  ) - close bounds open

Few Examples Circular Arc graphs Circular Arc graphs - Tucker 1971, 1980 Gavril 1974 Gyarfás 1987 Spinrad 1988 Hell, Bang-Jensen, Huang 1990 …

Few Examples Circular Arc graphs Circular Arc graphs - Tucker 1971, 1980 Gavril 1974 Gyarfás 1987 Spinrad 1988 Hell, Bang-Jensen, Huang 1990 …

Outline  String graphs  CONV and PC graphs  Representations of planar graphs

1. String graphs Sinden 1966

1. String graphs Sinden 1966 = IG(regions)

1. String graphs Sinden 1966 = IG(regions) Graham 1974

1. String graphs Sinden 1966 JK, Goljan, Kučera 1982

1. String graphs Sinden 1966 JK, Goljan, Kučera 1982 Thomas 1988 IG(topologically con) = all graphs, String = IG(arc-connected sets)

1. String graphs Sinden 1966 JK, Goljan, Kučera 1982 Thomas 1988 JK 1991 – NP-hard

1. String graphs SEG CONV STRING

1. String graphs SEG CONV STRING

1. String graphs Sinden 1966 JK, Goljan, Kucera 1982 Thomas 1988 JK 1991 – NP-hard Recognition in NP?

1. String graphs Sinden 1966 JK, Goljan, Kucera 1982 Thomas 1988 JK 1991 – NP-hard Recognition in NP?

Abstract Topological Graphs  G = (V,E), R  { ef : e,f  E } is realizable if G has a drawing D in the plane such that for every two edges e,f  E, D e  D f    ef  R  G = (V,E), R =  is realizable iff G is planar

Worst case functions  Str(n) = min k s.t. every string graph on n vertices has a representation with at most k crossing points of the curves  At(n) = min k s.t. every AT graph with n edges has a realization with at most k crossing points of the edges  Lemma: Str(n) and At(n) are polynomially equivalent

String graphs requiring large representations  Thm (J.K., Matoušek 1991): At(n)  2 cn

1. String graphs Sinden 1966 JK, Goljan, Kucera 1982 Thomas 1988 JK 1991 – NP-hard Recognition in NP? Are they recognizable at all?

Thm (Pach, Tóth 2001): At(n)  n n Thm (Schaefer, Štefankovič 2001): At(n)  n2 n-2

1. String graphs Sinden 1966 JK, Goljan, Kučera 1982 JK 1991 – NP-hard Schaefer, Sedgwick, Štefankovič 2002 – String graph recognition is in NP (Lempel- Ziv compression)

1. Some subclasses

Complements of Comparability graphs (Golumbic 1977)

Co-comparability graphs

=  

Co-comparability graphs

1. Some subclasses “Zwischenring” graphs NP-hard (Middendorf, Pfeiffer)

1. Some subclasses Outerstring graphs NP-hard (Middendorf, Pfeiffer)

1. Some subclasses Outerstring graphs NP-hard (Middendorf, Pfeiffer)

1. Some subclasses Interval filament graphs (Gavril 2000) CLIQUE and IND SET can be solved in polynomial time

2. CONV and PC JK, Matoušek 1994 – recognition in PSPACE

Thm: Recognition of CONV graphs is in PSPACE  Reduction to solvability of polynomial inequalities in R:  x 1, x 2, x 3 … x n  R s.t. P 1 (x 1, x 2, x 3 … x n ) > 0 P 2 (x 1, x 2, x 3 … x n ) > 0 … P m (x 1, x 2, x 3 … x n ) > 0 ?

{M u, u  V G } uv  E G  M u  M v   MuMu MvMv MwMw MzMz

MuMu MvMv MwMw MzMz Choose X uv  M u  M v for every uv  E G X uw X uz X uv

C u  C v    M u  M v    uv  E G MuMu MvMv MwMw MzMz Replace M u by C u = conv(X uv : v s.t. uv  E G )  M u X uw X uz X uv

Introduce variables x uv, y uv  R s.t. X uv = [x uv, y uv ] for uv  E G

uv  E G  C u  C v   guaranteed by the choice C u = conv(X uv : v s.t. uv  E G )

Introduce variables x uv, y uv  R s.t. X uv = [x uv, y uv ] for uv  E G uv  E G  C u  C v   guaranteed by the choice C u = conv(X uv : v s.t. uv  E G ) uw  E G  C u  C w =  separating lines

Introduce variables x uv, y uv  R s.t. X uv = [x uv, y uv ] for uv  E G uv  E G  C u  C v   guaranteed by the choice C u = conv(X uv : v s.t. uv  E G ) uw  E G  C u  C w =  separating lines CuCu CwCw a uw x + b uw y + c uw = 0

Introduce variables x uv, y uv  R s.t. X uv = [x uv, y uv ] for uv  E G uv  E G  C u  C v   guaranteed by the choice C u = conv(X uv : v s.t. uv  E G ) uw  E G  C u  C w =  separating lines CuCu CwCw a uw x + b uw y + c uw = 0 Representation is described by inequalities (a uw x uv + b uw y uv + c uw ) (a uw x wz + b uw y wz + c uw ) < 0 for all u,v,w,z s.t. uv, wz  E G and uw  E G X uv X wz

2. Recognition – NP-membership “Guess and verify”

2. Recognition – NP-membership “Guess and verify” -INT, CA, CIR, PC, Co-Comparability -IFA – mixing characterization - CONV, SEG ? !! String – Lempel-Ziv compression

2. Recognition – NP-membership Thm (JK, Matoušek 1994): For every n there is a graph G n  SEG with O(n 2 ) vertices s.t. every SEG representation with integer endpoints has a coordinate of absolute value  2 2 n. Same for CONV (Pergel 2008).

2. CLIQUE in CONV graphs -CO-PLANAR  CONV (JK, Kuběna 99)

2. CLIQUE in CONV graphs -CO-PLANAR  CONV (JK, Kuběna 99) -Corollary: CLIQUE is NP-complete for CONV graphs. (Since INDEPENDENT SET is NP-complete for planar graphs.) -CLIQUE in SEG graphs still open (JK, Nešetřil 1990)

2. CLIQUE in MAX-TOL graphs

2. MAX-TOLERANCE (Golumbic, Trenk 2004)

2. MAX-TOLERANCE S R S = {I u | u  V G } intervals, t u  R tolerances uv  E G iff |I u  I v | ≥ max {t u, t v }

2. MAX-TOLERANCE Theorem (Kaufmann, JK, Lehmann, Subramarian, 2006): Max-tolerance graphs are exactly intersection graphs of homothetic triangles (semisquares)

2. MAX-TOLERANCE IuIu tutu TuTu IvIv TvTv

Lemma (folklore): Disjoint convex polygons are separated by a line parallel to a side of one of them.

A B C

Maximal cliques Q a maximal clique

Maximal cliques h highest basis of Q, v rightmost vertical side, t lowest diagonal side Q a maximal clique t h v

Maximal cliques Q(h,v,t) = all triangles that intersect h,v and t Q a maximal clique t h v

Claim: Q(h,v,t) = Q

Proof: 1) Q  Q(h,v,t) h

Claim: Q(h,v,t) = Q Proof: 1) Q  Q(h,v,t) 2) Q(h,v,t) is a clique

Claim: Q(h,v,t) = Q Proof: 1) Q  Q(h,v,t) 2) Q(h,v,t) is a clique Suppose a,b  Q(h,v,t) are disjoint, hence separated by a line parallel to one of the sides, say horizontal.

Claim: Q(h,v,t) = Q Proof: 1) Q  Q(h,v,t) 2) Q(h,v,t) is a clique a b

Claim: Q(h,v,t) = Q Proof: 1) Q  Q(h,v,t) 2) Q(h,v,t) is a clique b cannot intersect h, a contradiction a b h

Maximal cliques Q(h,v,t) = all triangles that intersect h,v and t Hence G has O(n 3 ) maximal cliques. Q a maximal clique t h v

2. Polygon-circle graphs PC graphs PC graphs - Fellows 1988 Koebe 1990 JK, Kostochka 1994 Spinrad JK, Pergel 2002 Pergel 2007

2. Polygon-circle graphs PC graphs PC graphs - Fellows 1988 Koebe 1990 JK, Kostochka 1994 Spinrad JK, Pergel 2002 Pergel 2007

2. Polygon-circle graphs PC graphs PC graphs - Fellows 1988 Koebe 1990 JK, Kostochka 1994 Spinrad JK, Pergel 2002 Pergel 2007

2. Polygon-circle graphs CIR PC IFA CA CHOR

2. Polygon-circle graphs CIR PC IFA CA CHOR

2. Polygon-circle graphs CIR PC IFA CA CHOR Pergel 2007

2. Polygon-circle graphs CIR PC IFA CA CHOR Pergel 2007

2. Short cycles Do short cycles help?

2. Short cycles Do short cycles mind? Does large girth help?

DISK UNIT-DISK

DISK UNIT-DISK PSEUDO-DISK

2. Short cycles Thm (J.K. 1996) Triangle-free intersection graphs of pseudodisks are planar.

2. Short cycles Thm (J.K. 1996) Triangle-free intersection graphs of pseudodisks are planar.

2. Short cycles Thm (J.K. 1996) Triangle-free intersection graphs of pseudodisks are planar. Corollary: Recognition of triangle-free PSEUDO-DISK and DISK graphs is polynomial.

Koebe (1936)

2. Short cycles Thm (J.K. 1996) Triangle-free STRING graphs are NP-hard to recognize.

2. Short cycles Thm (J.K. 1996) Triangle-free STRING graphs are NP-hard to recognize. Thm (JK, Pergel 2007) PC graphs of girth  5 can be recognized in polynomial time. Thm (JK, Pergel 2007) For each k, recognition of SEG graphs of girth  k is NP-hard.

2. Short cycles Problem: Is recognition of String graphs of girth  k NP-complete for every k ? Thm (JK, Pergel 2007) PC graphs of girth  5 can be recognized in polynomial time. Thm (JK, Pergel 2007) For each k, recognition of SEG graphs of girth  k is NP-hard.

3. Representations of planar graphs

- Planar graphs are exactly contact graphs of disks (Koebe 1934)

3. Representations of planar graphs -Planar graphs are exactly contact graphs of disks (Koebe 1934) -PLANAR  DISK -PLANAR  CONV -PLANAR  2-STRING

3. Representations of planar graphs -PLANAR  2-STRING -Problem (Fellows 1988): Planar  1-STRING ? -True: Chalopin, Gonçalves, and Ochem [SODA 2007]

3. Representations of planar graphs -PLANAR  2-STRING -Problem (Fellows 1988): Planar  1-STRING ? -True: Chalopin, Gonçalves, and Ochem [SODA 2007] - Problem: PLANAR  SEG? (Pollack, Scheinerman, West, …)

3. Representations of planar graphs -PLANAR  SEG (?) -3-colorable 4-connected triangulations are intersection graphs of segments (de Fraysseix, de Mendez 1997) -Planar triangle-free graphs are in SEG (Noy et al. 1999) -Planar bipartite graphs are grid intersection (Hartman et al. 91; Albertson; de Fraysseix et al.)

3. Bipartite planar graphs De Fraysseix, Ossona de Mendez, Pach d c f e b a a b c d e f

3. Bipartite planar graphs De Fraysseix, Ossona de Mendez, Pach a b c d e f

3. Bipartite planar graphs De Fraysseix, Ossona de Mendez, Pach a b c d e f

3. Representations of planar graphs -PLANAR  CONV -Planar graphs are contact graphs of triangles (de Fraysseix, Ossona de Mendez 1997)

3. Representations of planar graphs -PLANAR  CONV -Planar graphs are contact graphs of triangles (de Fraysseix, Ossona de Mendez 1997) -Are planar graphs contact graphs of homothetic triangles?

3. Representations of planar graphs -PLANAR  CONV -Planar graphs are contact graphs of triangles (de Fraysseix, Ossona de Mendez 1997) -Are planar graphs contact graphs of homothetic triangles? -No

3. Representations of planar graphs b c a

1 2 3 b c a ab c

1 2 3 b c a ab c

3. Planar – open problems -PLANAR  MAX-TOL? (Lehmann) (i.e. are planar graphs intersection graphs of homothetic triangles?)

3. Planar – open problems -PLANAR  MAX-TOL? (Lehmann) -Conjecture (Felsner, JK 2007): Planar 4-connected triangulations are contact graphs of homothetic triangles.

3. Planar – open problems -PLANAR  MAX-TOL? (Lehmann) -Conjecture (Felsner, JK 2007): Planar 4-connected triangulations are contact graphs of homothetic triangles. This would imply that planar graphs are intersection graphs of homothetic triangles.

3. Representations of planar graphs b c a ab c

1 2 3 b c a ab c

1 2 3 b c a ab c

4. Invitation  Graph Drawing, Crete, Sept 21 – 24, 2008  Prague MCW, July 28 – Aug 1, 2008