a glimpse at algorithmic graph...

Algorithmic Graph Theory 1

A Glimpse at Algorithmic A Glimpse at Algorithmic Graph TheoryGraph Theory

吴耀琨吴耀琨上海交通大学数学系上海交通大学数学系

(Based on a (Based on a pptppt made by M. made by M. GolumbicGolumbic))

上海交大理科班讨论班, 2009/9/27

2Algorithmic Graph Theory

Martin Charles Golumbic, Landmarks in Algorithmic Graph Theory: Martin Charles Golumbic, Landmarks in Algorithmic Graph Theory: A personal Retrospective, LNCS 5420 (2009) 1A personal Retrospective, LNCS 5420 (2009) 1——14.14.

www.dis.uniroma1.it/~seminf/seminars/www.dis.uniroma1.it/~seminf/seminars/golumbic.pptgolumbic.ppt

Defining some termsDefining some terms

graph:graph: a collection of vertices and edgesa collection of vertices and edgescoloring a graph:coloring a graph:assigningassigning a color to every a color to every vertexvertex suchsuch that that adjacentadjacent vertices have different colorsvertices have different colors

independent set:independent set: a collection of vertices a collection of vertices NONO two of which are connectedtwo of which are connectedExampleExample: { : { d, e, f d, e, f } or the } or the greengreen setsetcliqueclique (or complete set):(or complete set):

EVERYEVERY two of which are two of which are connectedconnectedExampleExample: { : { a, b, d a, b, d } or { } or { c, e c, e } }

complement of a graph:complement of a graph:interchanging the edges and the noninterchanging the edges and the non--edgesedges

The complement G The original graph G__

directed graph:directed graph: edges have directionsedges have directions(possibly both directions)(possibly both directions)

orientation:orientation: exactly ONE direction per edgeexactly ONE direction per edge

cyclic orientation acyclic orientation

Given a family of sets Given a family of sets S(1),S(1),……,S(n),,S(n), we define its we define its intersection graph to be the graph with vertex intersection graph to be the graph with vertex set set v(1),v(1),……,v(n),v(n) and and v(i)v(j)v(i)v(j) is an edge if and only is an edge if and only if if S(i)S(i) and and S(j)S(j) have nonempty intersection.have nonempty intersection.

IntersectionIntersection Representation of a Representation of a GraphGraph

Interval GraphsInterval GraphsThe The intersection graphs of intervals on a lineintersection graphs of intervals on a line::

-- create a vertex for each intervalcreate a vertex for each interval-- connect vertices when their intervals intersectconnect vertices when their intervals intersect

Jan Feb Mar Apr May Jun July Sep Oct Nov Dec

Phase 1Phase 2

Phase 3Task 4

Task 5

4 5The interval graph G

applications in computationapplications in computationoperations researchoperations researchmolecular biologymolecular biologyschedulingschedulingdesigning circuitsdesigning circuitsrich mathematical problemsrich mathematical problems

History of Interval GraphsHistory of Interval GraphsHajosHajos 1957: 1957: Combinatorics (scheduling)Combinatorics (scheduling)BenzerBenzer 1959: 1959: Biology (genetics)Biology (genetics)Gilmore & Hoffman 1964: CharacterizationGilmore & Hoffman 1964: CharacterizationBooth & Booth & LuekerLueker 1976: First linear time 1976: First linear time

recognition recognition algorithmalgorithmMany other applications:Many other applications:

mobile radio frequency assignmentmobile radio frequency assignmentVLSI designVLSI designtemporal reasoning in AItemporal reasoning in AIcomputer storage allocationcomputer storage allocation

Scheduling ExampleScheduling Example

Lectures need to be assigned classrooms at the Lectures need to be assigned classrooms at the University.University.

Lecture #a: 9:00Lecture #a: 9:00--10:1510:15Lecture #b: 10:00Lecture #b: 10:00--12:0012:00etc.etc.

Conflicting lectures Conflicting lectures →→ Different roomsDifferent roomsHow many rooms?How many rooms?

Scheduling Example (cont.)Scheduling Example (cont.)

Scheduling Example (graphs)Scheduling Example (graphs)

(a) The interval graph (b) Its complement (disjointness)

Offline Coloring Interval GraphsOffline Coloring Interval Graphs

Interval graphs have special properties.Interval graphs have special properties.These special structural properties These special structural properties guarantees some efficient algorithms.guarantees some efficient algorithms.The greedy coloring algorithm sweeps The greedy coloring algorithm sweeps across from left to right to assign colors across from left to right to assign colors which causes no conflict and is as small which causes no conflict and is as small as possible (Firstas possible (First--Fit).Fit).

Coloring Intervals (greedy)Coloring Intervals (greedy)

Coloring Interval GraphsColoring Interval Graphs

Is the greedy coloring algorithm optimal in Is the greedy coloring algorithm optimal in minimizing the number of used colors?minimizing the number of used colors?

Coloring Intervals (greedy)Coloring Intervals (greedy)P (needs 4 colors)

Coloring Interval GraphsColoring Interval Graphs

The clique at point P

Is greedy the best we can do in Is greedy the best we can do in general?general?

Can we prove optimality? Can we prove optimality?

Is greedy the best we can do?Is greedy the best we can do?Can we prove optimality? Can we prove optimality? Yes: It uses the smallest # colors.Yes: It uses the smallest # colors.

Proof: Let k be the number of colors used.

Look at the point P, when color k was used first (This is the left endpoint of the first interval with color k).

At P all the colors 1 to k-1 were busy!

We are forced to use k colors at P and the corresponding intervals form a clique of size k in the interval graph.

Greedy the best we can do !Greedy the best we can do !

Formally,

(1) at least k colors are required

(because of the clique)

(2) greedy succeeded using k colors.

Therefore,

the solution is optimal. Q.E.D.

Perfect GraphsPerfect Graphs

AA graph is perfect if and only if the clique graph is perfect if and only if the clique number and the chromatic number of each number and the chromatic number of each induced induced subgraphsubgraph of it are equal.of it are equal.

Any induced subgraph of a perfect graph is Any induced subgraph of a perfect graph is perfect.perfect.

IntervalInterval graphs are perfect.graphs are perfect.

Perfect Perfect Graphs (Contd.)Graphs (Contd.)

AA graph is perfect if and only if the clique graph is perfect if and only if the clique number and the chromatic number of each number and the chromatic number of each induced induced subgraphsubgraph of it are equal.of it are equal.

IsIs the complement of an interval the complement of an interval graphgraph perfect?perfect?

Namely, is the smallest size of a transversal of a Namely, is the smallest size of a transversal of a family of intervals equal to the biggest size of a family of intervals equal to the biggest size of a set of pariwise disjoint intervals from the same set of pariwise disjoint intervals from the same family?family?

Perfect Graph Theorem (Perfect Graph Theorem (LovaszLovasz, born March 9, , born March 9, 19481948 ) : ) : A graph is perfect A graph is perfect iffiff its complement is its complement is perfect. perfect.

Laszlo Laszlo LovaszLovasz (1972). "Normal (1972). "Normal hypergraphshypergraphs and the perfect and the perfect graph conjecture". Discrete Mathematics 2: 253graph conjecture". Discrete Mathematics 2: 253––267.267.

Laszlo Laszlo LovaszLovasz won both the Wolf prize and the won both the Wolf prize and the Knuth prize in 1999. He is the president of the Knuth prize in 1999. He is the president of the International Mathematical Union since 2007.International Mathematical Union since 2007.

Can you think of a simple graph which is not Can you think of a simple graph which is not perfect?perfect?

AnAn induced cycle of odd length at least 5 is induced cycle of odd length at least 5 is called an odd hole. An induced called an odd hole. An induced subgraphsubgraphthat is the complement of an odd hole is that is the complement of an odd hole is called an odd called an odd antiholeantihole. .

AA graph that does not contain any odd graph that does not contain any odd holes or odd holes or odd antiholesantiholes is called a Berge is called a Berge graph. All perfect graphs are Berge graphs.graph. All perfect graphs are Berge graphs.

Strong Perfect Graph Theorem. A graph is perfect if and onlyif it is a Berge graph. Namely, the odd holes and odd antiholesare a complete list of obstructions for perfectness of graphs.

Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2006), "The strong perfect graph theorem". Annals of Mathematics 164 (1): 51--229.

Maria Maria ChudnovskyChudnovsky

Department of Industrial Department of Industrial Engineering and Operations Engineering and Operations ResearchResearch

Department of Mathematics Department of Mathematics

Columbia UniversityColumbia University

Interview with Chudnovsky

How good is the greedy algorithm How good is the greedy algorithm for online coloring interval graphs?for online coloring interval graphs?The smallest The smallest number ofnumber of colors needed to color colors needed to color an interval graph is its clique number k.an interval graph is its clique number k.For any order of the arrival of intervals, we use For any order of the arrival of intervals, we use the greedy algorithm (Firstthe greedy algorithm (First--Fit) to color them. Fit) to color them. The number of colors used in the worst case is The number of colors used in the worst case is denoted denoted FF(kFF(k) .) .FF(k) FF(k) ≤≤ 8k8k--3; 3; FF(k) FF(k) ≥≥ 4.99k when k is 4.99k when k is sufficiently large.sufficiently large.Conjecture: As Conjecture: As kk tends to infinity, the ratio tends to infinity, the ratio FFFF((kk)/)/kk tends to 5.tends to 5.

Characterizing Interval GraphsCharacterizing Interval Graphs

Properties of interval graphsProperties of interval graphsHow to recognize themHow to recognize themTheir mathematical structureTheir mathematical structure

Characterizing Interval GraphsCharacterizing Interval Graphs

Properties of interval graphsProperties of interval graphsHow to recognize themHow to recognize themTheir mathematical structureTheir mathematical structure

Two properties characterize interval graphs:

- The Chordal Graph Property

- The co-TRO Property

The coThe co--TRO PropertyTRO Property

The transitive orientation (TRO) of the The transitive orientation (TRO) of the complement complement i.e., the complement must have a TROi.e., the complement must have a TRO

Not transitive ! Transitive !

Interval Graphs are coInterval Graphs are co--TROTRO

The complement of an Interval graph has a The complement of an Interval graph has a transitive orientation!transitive orientation!

-- Why?Why?

The complement is the disjointness graph.

So, orient from the earlier interval

to the later interval.

The Chordal Graph PropertyThe Chordal Graph Propertychordal graph: chordal graph:

every cycle of length every cycle of length >> 4 has a chord4 has a chord(connecting two vertices that are not consecutive)(connecting two vertices that are not consecutive)

i.e., they may not contain chordless cycles!i.e., they may not contain chordless cycles!

Interval Graphs are ChordalInterval Graphs are Chordal

Interval graphs may not contain chordless cycles!Interval graphs may not contain chordless cycles!

-- i.e., they are chordal. Why?i.e., they are chordal. Why?

Interval graphs may not contain chordless cycles!Interval graphs may not contain chordless cycles!

-- i.e., they are chordal. Why?i.e., they are chordal. Why?

Let [v(1)v(2)Let [v(1)v(2)……v(n)] be a cycle corresponding to v(n)] be a cycle corresponding to intervals I(1),intervals I(1),……,I(n). Without loss of generality, ,I(n). Without loss of generality, suppose that the right endpoint of I(1) is the suppose that the right endpoint of I(1) is the smallest among all right endpoints of intervals smallest among all right endpoints of intervals I(1),I(1),……,I(n). Then we see that v(2) and v(n) are ,I(n). Then we see that v(2) and v(n) are joined by an edge which is a chord of the cycle if joined by an edge which is a chord of the cycle if nn≥≥4.4.

There are tons of different There are tons of different characterizations (definitions, characterizations (definitions, representations) of representations) of chordalchordal graphs graphs (interval (interval graphs). This fact is really graphs). This fact is really AMAZINGAMAZING!!ChordalChordal graphs are exactly the graphs are exactly the intersection graphs of a family of intersection graphs of a family of subtreessubtreesof a common host tree.of a common host tree.When the host tree is a path, the When the host tree is a path, the chordalchordalgraph is an interval graph.graph is an interval graph.

Gilmore and Hoffman (1964)Gilmore and Hoffman (1964)

Theorem:Theorem:A graph A graph G G is an interval graph is an interval graph ifif and only if and only if G G Is chordal and Is chordal and its complement its complement G G is transitively is transitively orientableorientable. .

This provides the basis for the first set of recognition algorithms in the early 1970’s.

A Mystery in the LibraryA Mystery in the LibraryThe Berge Mystery Story:The Berge Mystery Story:

Six professors had been to the library on the Six professors had been to the library on the day that the rare tractate was stolen. day that the rare tractate was stolen.

Each had entered once, stayed for some time Each had entered once, stayed for some time and then left. and then left.

If two were in the library at the same time, then If two were in the library at the same time, then at least one of them saw the other. at least one of them saw the other.

Detectives questioned the professors and Detectives questioned the professors and gathered the following testimony: gathered the following testimony:

Abe said that he saw Burt and Eddie Abe said that he saw Burt and Eddie Burt reported that he saw Abe and Ida Burt reported that he saw Abe and Ida Charlotte claimed to have seen Charlotte claimed to have seen DesmondDesmond and Idaand IdaDesmond said that he saw Abe and IdaDesmond said that he saw Abe and IdaEddie testified to seeing Burt and CharlotteEddie testified to seeing Burt and CharlotteIda said that she saw Charlotte and EddieIda said that she saw Charlotte and Eddie

One of the Professor LIED One of the Professor LIED by asserting to see by asserting to see somebody whom he did not see!!somebody whom he did not see!! Who was it?Who was it?

The Facts:The Facts:

Solving the MysterySolving the Mystery

The Testimony Graph

Clue #1:

Double arrows imply TRUTH

Undirected Testimony Graph

We know there is a lie, since {A, B, I, D} is a chordless 4-cycle.

Intersecting Intervals Intersecting Intervals cannotcannotform Chordless Cyclesform Chordless Cycles

Burt Desmond

No place for Ida’s interval:It must hit both B and D but cannot hit A.

Impossible!

There are three chordless 4-cycles:{A, B, I, D} {A, D, I, E} {A, E, C, D}

The liar must be a common member of all the above 4-cycles and hence is either Abe or Desmond.

WHO IS THE LIAR? Abe or Desmond ?

One professor from the chordless 4One professor from the chordless 4--cycle must be a liar.cycle must be a liar.

Solving the Mystery (cont.)Solving the Mystery (cont.)

WHO IS THE LIAR? Abe or Desmond ?

If Abe were the liar and Desmond truthful, then {A, B, I, D} would remain a chordless 4-cycle in the interval graph with the time periodof each professor staying in the library as an intersection model.

Therefore:

Desmond is the liar.

Was Desmond Stupid or Was Desmond Stupid or Just Ignorant?Just Ignorant?

If Desmond had studied algorithmic graph theory, he If Desmond had studied algorithmic graph theory, he would have known that his testimony to the police would have known that his testimony to the police would not hold up.would not hold up.

Can you formulate any general results on those kind Can you formulate any general results on those kind of lies which will be detected? Namely, how to of lies which will be detected? Namely, how to recognize the testimony graph from which the liar can recognize the testimony graph from which the liar can be detected? Can you try to develop an efficient be detected? Can you try to develop an efficient algorithm to detect such a liar provided it is possible algorithm to detect such a liar provided it is possible to be detected?to be detected?

Many other Families of Many other Families of Intersection GraphsIntersection Graphs

Victor Klee, in a paper in 1969:Victor Klee, in a paper in 1969:

``What are the intersection graphs of arcs ``What are the intersection graphs of arcs in a circle?in a circle?’’’’

``What are the intersection graphs of arcs in a ``What are the intersection graphs of arcs in a circle?circle?““

``What are the intersection graphs of arcs in a circle?``What are the intersection graphs of arcs in a circle?““

KleeKlee’’s paper was an implicit challenges paper was an implicit challenge-- consider a whole variety of problems consider a whole variety of problems -- on many kinds of intersection graphs. on many kinds of intersection graphs.

The problem should be of intrinsic interest in even a The problem should be of intrinsic interest in even a very special form, but should admit of interesting very special form, but should admit of interesting extensions. In my opinion, a good problem is extensions. In my opinion, a good problem is sufficiently specific so that even the specific form is of sufficiently specific so that even the specific form is of interest to someone, but of course itinterest to someone, but of course it’’s best if a specific s best if a specific solution inspires further questions and generalizations. I solution inspires further questions and generalizations. I deal with specific case, if a meaningful (i.e, not obvious deal with specific case, if a meaningful (i.e, not obvious but not impossible) one can be found. Then but not impossible) one can be found. Then ““brainstormbrainstorm””, looking for natural generalizations and, if , looking for natural generalizations and, if possible, applications. possible, applications. –– Victor L. Klee Victor L. Klee

Families of Intersection Families of Intersection GraphsGraphsArising from ApplicationsArising from Applications

boxes in the planeboxes in the planepaths in a treepaths in a treechords of a circlechords of a circlespheres in 3spheres in 3--spacespacetrapezoids, parallelograms, curves of functionstrapezoids, parallelograms, curves of functionsmany other geometrical and topological bodiesmany other geometrical and topological bodies

Families of Intersection Families of Intersection Graphs Graphs Arising from ApplicationsArising from Applications

boxes in the planeboxes in the planepaths in a treepaths in a treechords of a circlechords of a circlespheres in 3spheres in 3--spacespacetrapezoids, parallelograms, curves of functionstrapezoids, parallelograms, curves of functionsmany other geometrical and topological bodiesmany other geometrical and topological bodies

The Algorithmic Problems:– recognize them– color them– find maximum cliques – find maximum independent sets

The Interval Graph Sandwich The Interval Graph Sandwich ProblemProblem

BenzerBenzer’’ss original problem original problem partial intersection datapartial intersection dataIs it consistent ?Is it consistent ?

For complete data this is the For complete data this is the recognition problem for interval graphs recognition problem for interval graphs (polynomial)(polynomial)For partial data we arrive at a different For partial data we arrive at a different model model thatthat is NPis NP--complete.complete.

Interval Graph Sandwich ProblemInterval Graph Sandwich Problem

given a partially specified graph given a partially specified graph EE11 required edgesrequired edgesEE22 optional edgesoptional edgesEE33 forbidden edgesforbidden edges

Can you fillCan you fill--in some of the optional in some of the optional edges, soedges, so that the that the result will be an interval result will be an interval graph?graph?Namely, is there a set E such that Namely, is there a set E such that EE11⊆⊆EE⊆⊆ EE11 ∪∪ EE3 3

and that (V,E) is an interval graph?and that (V,E) is an interval graph?

Golumbic & Shamir (1993): Golumbic & Shamir (1993): NPNP--CompleteComplete

Interval Probe GraphsInterval Probe Graphs

A special tractable case of interval A special tractable case of interval graph graph sandwich problemsandwich problemComputational biology motivatedComputational biology motivated

Interval probe graph: vertices are partitionedInterval probe graph: vertices are partitionedP P probesprobes & N & N nonnon--probesprobes (independent set)(independent set)

can fillcan fill--in some of the N x N edges,in some of the N x N edges,

so that the result will be an interval graphso that the result will be an interval graph

Example: Interval Probe GraphsExample: Interval Probe GraphsNon-Probes are white

Probe graph NOT a Probe graph no matter how you partition vertices!

(Golumbic, Maffray, Morel, Annals of Operations Research, (Golumbic, Maffray, Morel, Annals of Operations Research, DOI10.1007/s10479DOI10.1007/s10479--009009--05840584--66 ) A bipartite graph is an ) A bipartite graph is an interval probe graph if and only if it contains none of the six interval probe graph if and only if it contains none of the six forbidden graphs in the figure below.forbidden graphs in the figure below.

A A ChordalChordal Graph Sandwich Graph Sandwich ProblemProblem

The The kthkth power of a graph G is the graph power of a graph G is the graph on the same vertex set and two different on the same vertex set and two different vertices are joined by an edge if and only if vertices are joined by an edge if and only if their distance in G is at most k.their distance in G is at most k.A Conjecture of A Conjecture of GavoilleGavoille : Suppose G is : Suppose G is not a not a chordalchordal graph and the longest graph and the longest chordlesschordless cycle in G has length k. Then, we cycle in G has length k. Then, we can find a can find a chordalchordal graph graph inbetweeninbetween G and G and the the ┌┌k/3k/3┐┐thth power of G.power of G.

Tolerance GraphsTolerance GraphsWhat if you only have 3 classrooms?What if you only have 3 classrooms?Cancel a Lecture? or show Tolerance?Cancel a Lecture? or show Tolerance?

Tolerance GraphsTolerance Graphs

Measured intersection: small, or ``tolerable’’ amount of overlap, may be ignored does NOT produce an edge

at least one of them has to be ``bothered’’

Tolerance GraphsTolerance Graphs

Assignment of positive numbers Assignment of positive numbers {{ttvv} (} (v v ∈∈ VV)) such thatsuch that

vwvw ∈∈ E if and only if E if and only if | | IIv v ∩∩ IIww | | ≥≥ min min {{ttvv ,, ttww}}

Measured intersection: small, or ``tolerable’’ amount of overlap, may be ignored does NOT produce an edge

at least one of them has to be ``bothered’’

Tolerance Graphs: ExampleTolerance Graphs: Example

c and f will no longer conflict

| Ic ∩ If | < 60 = min {tc , tf}

Rather than the arduous and systematic study of every new Rather than the arduous and systematic study of every new concept definable with a graph, the main task for the concept definable with a graph, the main task for the mathematician is to eliminate the often arbitrary and mathematician is to eliminate the often arbitrary and cubersom definitions, keeping only the cubersom definitions, keeping only the ““deepdeep””mathematical problems.mathematical problems.Of course, the deep problems may well be elusive; indeed, Of course, the deep problems may well be elusive; indeed, there have been many definitions (from Dieudonne, among there have been many definitions (from Dieudonne, among others) of what a deep problem is. In graph theory, it others) of what a deep problem is. In graph theory, it should relate to a variety of other combinatorial structures should relate to a variety of other combinatorial structures and must therefore be connected with many difficult and must therefore be connected with many difficult practical problems. Among these will be problems that practical problems. Among these will be problems that classical algebra is not able to solve completely or that the classical algebra is not able to solve completely or that the computer scientists would not attack by himself. computer scientists would not attack by himself. –– Claude Claude BergeBerge

Some BooksSome Books

Andreas Andreas BrandstBrandstäädtdt, Van Bang Le, Jeremy P. , Van Bang Le, Jeremy P. SpinradSpinrad, Graph , Graph Classes: A Survey, Classes: A Survey, SIAM,SIAM, 1999.1999.Jeremy P. Jeremy P. SpinradSpinrad, Efficient Graph Representations, AMS, 2003., Efficient Graph Representations, AMS, 2003.MartinMartin Charles Charles GolumbicGolumbic, Algorithmic Graph Theory and , Algorithmic Graph Theory and Perfect Graphs, North Holland, 2004. Perfect Graphs, North Holland, 2004. Martin Charles Martin Charles GolumbicGolumbic, Tolerance Graphs, Cambridge , Tolerance Graphs, Cambridge University Press, 2004.University Press, 2004.Terry A. McKee, F.R. Terry A. McKee, F.R. McMorrisMcMorris, Topics in Intersection Graph , Topics in Intersection Graph Theory, Theory, SIAM,SIAM, 1999. 1999. P.C. P.C. FishburnFishburn, Interval , Interval OrdersOrders and and IntervalInterval GraphsGraphs, Wiley, 1985. , Wiley, 1985. N.V.R. Mahadev, U.N. Peled, Threshold Graphs and Related N.V.R. Mahadev, U.N. Peled, Threshold Graphs and Related Topics, Elsevier, 1995. Topics, Elsevier, 1995.

Some tree-likeness parameters: Chordality,hyperbolicity and tree-length

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Shanghai Jiao Tong University

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Outline

Introduction

Hyperbolicity

Tree-length

Chordality

Hyperbolicity vs chordality

Conclusion

Tree-likeness

I Trees are graphs with some very distinctive andfundamental properties and it is legitimate to ask to whatdegree those properties can be transferred to moregeneral structures that are tree-like in some sense. – R.Diestel [32, p. 253]

I Roughly speaking, tree-likeness stands for somethingrelated to low dimensionality, low complexity, efficientinformation deduction (from local to global),information-lossless decomposition (from global intosimple pieces) and nice shape for efficient implementationof divide-and-conquer strategy.

Tree-likeness

I Trees are graphs with some very distinctive andfundamental properties and it is legitimate to ask to whatdegree those properties can be transferred to moregeneral structures that are tree-like in some sense. – R.Diestel [32, p. 253]

I Roughly speaking, tree-likeness stands for somethingrelated to low dimensionality, low complexity, efficientinformation deduction (from local to global),information-lossless decomposition (from global intosimple pieces) and nice shape for efficient implementationof divide-and-conquer strategy.

Tree-likeness, Contd.

I Researchers in different fields develop many differentmeasures of tree-likeness and it deserves to investigatethe relationship among them.

I In vast applications, one finds that the borderline betweentractable and intractable cases may be the tree-like degreeof the structure in consideration.

I Many practical structures we face with are very tree-like.

Internet and hyperbolicity

I Hyperbolicity is a measure of tree-likeness and it isreasonable to say that lower hyperbolicity stands for beingcloser to a tree.

I A network with low hyperbolicity allows many goodnetworking algorithms.

I Many experiments say that some large practicalcommunication networks, including the internet, havesurprisingly low hyperbolicity.

I It is interesting to understand why the internet has lowhyperbolicity.

Internet and hyperbolicity, Contd.

Some data from M. Soto in the CAIDA project:

Graph of the routing machines:Tree-width ≥ 234Tree-length ≤ 10Diameter = 9Gromov hyperbolicity = 3, but for for 96% of the vertices itsvalue is 1.

Autonomus system internet topology:Tree-width ≥ 82Tree-length ≤ 6Diameter = 10Gromov hyperbolicity = 2, but for 98% of the vertices its valueis 1

A problem from metric graph theory: characterizinglow hyperbolicity graphs

I Hyperbolicity takes nonnegative half-integer value.I 0-Hyperbolic graphs are just block graphs and are

well-understood.I Bandelt and Chepoi [7] obtained a characterization of all

12 -hyperbolic graphs in terms of a convexity condition. Butno corresponding finite forbidden isometric subgraphcharacterization is known till now.

I Bandelt and Chepoi [8]: “a characterization of all1-hyperbolic graphs by forbidden isometric subgraphs isnot in sight, in as much as isometric cycles of lengths up to7 may occur, thus complicating the picture”.

Finite excluded isometric subgraph characterization

Koolen and Moulton [72, p. 696]: In the proof of this propositionwe showed that G was 5-hyperbolic, although we suspect thatthe bound of 5 can be improved upon. In fact, we believe thatthe sum xy + uv in the proof of Proposition 4 can be boundedabove by 5. This would imply that only finitely many graphswould have to be excluded as isometric subgraphs – inaddition to assuming the breadth and short-cut properties– to assume that G would be 1

2 -hyperbolic. However,perhaps more importantly, this proposition indicates that theconcept of short-cuts together with the implicitly well-knownconcept of breadth could be useful for both determining thestructure and finding good bounds on the hyperbolicity ofhyperbolic graphs.

Our work

I As a small step in pursuing further understanding oftree-likeness, Chengpeng Zhang and I recently take up themodest task of comparing two parameters of tree-likeness,namely (Gromov) hyperbolicity and chordality of a graph.

I For any k ≥ 4, we show that k -chordal graphs must beb k

2 c2 -hyperbolic and there does exist a k -chordal graph with

hyperbolicity achieving this asserted bound.I We determine a complete set of 6 unavoidable isometric

subgraphs of 5-chordal graphs attaining hyperbolicity 1, asa minor attempt to respond to the general question that“what is the structure of graphs with relative smallhyperbolicity” [17, p. 62] and the even more generalquestion of “what is the structure of a very tree-like graph”.

Our work

Overview of the talk

I We only consider simple, unweighted, connected, butnot necessarily finite graphs.

I This talk will introduce three graph parameters measuringtree-likeness, namely chordality, hyperbolicity andtree-length.

I We will present some current knowledge/problems on thegeneral relationship among these parameters.

Outline

Introduction

Hyperbolicity

Tree-length

Chordality

Conclusion

Graph metric space

Any graph G together with the usual shortest-path metric on it,dG : V (G)× V (G) 7→ {0, 1, 2, . . .}, gives rise to a metric space.We often use the shorthand xy for d(x , y). Note that a pair ofvertices x and y forms an edge if and only if xy = 1.

Hyperbolicity

I For any vertices x , y , u, v of a graph G, put δG(x , y , u, v) tobe the absolute value of the difference between the largestand the second largest of the three sums

uv + xy2

,ux + vy

2, and

uy + vx2

I Clearly, δ(x , y , u, v) = 0 if x , y , u, v are not four differentvertices.

I A graph G, viewed as a metric space, is δ-hyperbolic (ortree-like with defect at most δ) provided for any verticesx , y , u, v in G it holds δ(x , y , u, v) ≤ δ and the (Gromov)hyperbolicity of G, denoted δ∗(G), is the minimum halfinteger δ such that G is δ-hyperbolic.

I Note that it may happen δ∗(G) = ∞ when G is an infinitegraph. But for a finite graph G, δ∗(G) is clearly polynomialtime computable.

Hyperbolicity

uv + xy2

,ux + vy

2, and

uy + vx2

Hyperbolicity

uv + xy2

,ux + vy

2, and

uy + vx2

Hyperbolicity

uv + xy2

,ux + vy

2, and

uy + vx2

Examples

I The hyperbolicity of a tree is 0.I The hyperbolicity of the n-cycle is bn

4c −12 if n is congruent

to 1 modulo 4 and is bn4c else.

I The hyperbolicity of a graph with diameter D is at most D2 .

Examples

Hyperbolicity is a tree-likeness measure

I The hyperbolicity of a graph is a way to measure theadditive distortion with which every four-points sub-metricof the given graph metric embeds into a tree metric [1].The condition that the hyperbolicity is zero is known as thefour-point condition (4PC) and is a characterization ofgeneral tree-like metric spaces, so-called real tree[41, 45, 67].

I Moreover, the fact that hyperbolicity is a tree-likenessparameter is reflected in the easy fact that the hyperbolicityof a graph is the maximum hyperbolicity of its 2-connectedcomponents – This observation implies the classical resultthat 0-hyperbolic graphs are exactly block graphs, namelythose graphs in which every 2-connected subgraph iscomplete, which are also known to be exactly thosediamond-free chordal graphs.

Hyperbolicity is a tree-likeness measure

I The hyperbolicity of a graph is a way to measure theadditive distortion with which every four-points sub-metricof the given graph metric embeds into a tree metric [1].The condition that the hyperbolicity is zero is known as thefour-point condition (4PC) and is a characterization ofgeneral tree-like metric spaces, so-called real tree[41, 45, 67].

I Moreover, the fact that hyperbolicity is a tree-likenessparameter is reflected in the easy fact that the hyperbolicityof a graph is the maximum hyperbolicity of its 2-connectedcomponents – This observation implies the classical resultthat 0-hyperbolic graphs are exactly block graphs, namelythose graphs in which every 2-connected subgraph iscomplete, which are also known to be exactly thosediamond-free chordal graphs.

Outline

Introduction

Hyperbolicity

Tree-length

Chordality

Conclusion

Tree-length

I A chordal graph is a graph without chordless cycle oflength greater than 3.

I The tree-length [34, 35, 78, 89] of a graph G, denotedtl(G), was introduced by Dourisboure and Gavoille in 2007and is the minimum integer k such that there is a chordalgraph G′ satisfying V (G) = V (G′), E(G) ⊆ E(G′) andmax(dG(u, v) : dG′(u, v) = 1) = k . We use the conventionthat the tree-length of a graph without any edge is 1.

I It is straightforward from the definition that chordal graphsare exactly the graphs of tree-length 1. It is also knownthat AT-free graphs, permutation graphs anddistance-hereditary graphs have tree-length at most 2 [34,p. 367].

Tree-length

Tree decomposition

The concept of tree decompositions was introduced byRobertson and Seymour in 1984 [83] and has since beenextensively studied in both mathematics and lots of appliedfields.A tree decomposition of a graph G is a tree T such that eachvertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:

I (Vertex Covering) ∪v∈V (T )Sv = V (G).I (Edge Covering) For any edge {u, w} ∈ E(G) there exists

v ∈ V (T ) such that u, w ∈ Sv .

I (Running Intersection Property) For any u ∈ V (G),{v ∈ V (T ) : u ∈ Sv} induces a subtree of T . In otherwords, for any v , w ∈ V (T ), Sv ∩ Sw can be seen in everybag along the path connecting v and w in T .

Tree decomposition

Tree-length: another definition

The length of a tree decomposition of a graph G is themaximum distance in G between two vertices in the same bagof the decomposition.

The tree-length of a graph G is just the shortest length of alltree decompositions of G.

Example (Dourisboure, Gavoille)The tree-length of an n-cycle is dn

Tree-length and hyperbolicity are comparable

The following are two results from Chepoi, Dragan, Estellon,Habib, Vaxés [25].

Theorem[25, Proposition 13] A graph G is k-hyperbolic provided itstree-length is no greater than k .

The proof is a generalization of the easy proof that every treehas hyperbolicity 0. We just take a look at a tree-decompositionwith maximum bag diameter no greater than k .

Theorem[25, Proposition 14] The inequality tl(G) ≤ 12k + 8k log2 n + 17holds for any k-hyperbolic graph G with n vertices.

Grid graph

ExampleFor any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1. Dourisboure and Gavoile showed thatthe tree-length of Gn,m is min(n, m) if n 6= m or n = m is evenand is n − 1 if n = m is odd [35, Theorem 3]. Based on thisresult it is not hard to further show thatδ∗(Gm,n) ∈ {min(m, n), min(m, n)− 1} andδ∗(Gm,n) = min(m, n)− 1 if n = m is odd. This says thatTheorem 2 is quite tight.

Is it true that the equality δ∗(Gm,n) = min(m, n)− 1 holds for allpositive integers m, n?

Grid graph

Outline

Introduction

Hyperbolicity

Tree-length

Chordality

Conclusion

Chordality

I We say that a graph is k-chordal if it does not contain anyinduced n-cycle for n > k . Clearly, trees are nothing but2-chordal graphs.

I A 4-chordal graph is also called a hole-free graph and a3-chordal graph is nothing but a chordal graph.

I The chordality of a graph G is the smallest integer k suchthat G is k -chordal [11]. Following [11], we use the notationlc(G) for this parameter as it is merely the length of thelongest chordless cycle in G when G is not a tree.

Chordality

Chordality, Contd.

The recognition of k -chordal graphs is coNP-complete fork = Θ(nε) for any constant ε > 0 [88]. Especially, to determinethe chordality of the hypercube is attracting much attentionunder the name of the snake-in-the-box problem due to itsconnection with some error-checking codes problem [71].

Nevertheless, just like many other tree-likeness parameters,quite a few natural graph classes are known to have smallchordality [15].

Chordality, Contd.

The recognition of k -chordal graphs is coNP-complete fork = Θ(nε) for any constant ε > 0 [88]. Especially, to determinethe chordality of the hypercube is attracting much attentionunder the name of the snake-in-the-box problem due to itsconnection with some error-checking codes problem [71].

Nevertheless, just like many other tree-likeness parameters,quite a few natural graph classes are known to have smallchordality [15].

Chordality and tree-length

What follows is a result of Gavoille, Katz, Katz, Paul, Peleg [51,Lemma 6].

Theorem[51, Lemma 6] [52, Theorem 3.3] If G is a k-chordal graph, thentl(G) ≤ bk

Proof.To obtain a minimal triangulation of G, it suffices to select amaximal set of pairwise parallel minimal separators of G andadd edges to make each of them a clique [81, Theorem 4.6]. Itis easy to check that each such new edge connects two pointsof distance at most b k

2c apart in G.

Outline

Introduction

Hyperbolicity

Tree-length

Chordality

Conclusion

Impossibility of bounding hyperbolicity from below interms of chordality

Firstly, we point out that a graph with low hyperbolicity mayhave arbitrarily large chordality.Indeed, take any graph G and form the new graph G′ by addingan additional vertex and connecting this new vertex with everyvertex of G. It is obvious that δ∗(G′) ≤ 1 while lc(G′) = lc(G) ifG is not a tree. Moreover, it is equally easy to see that G′ iseven 1

2 -hyperbolic if G does not have any induced 4-cycle [72,p. 695].Surely, this example does not preclude the possibility that formany important graph classes we can bound their chordality interms of their hyperbolicity.

Bound hyperbolicity from above

We now turn to show that it is possible to bound thehyperbolicity from above in terms of chordality.The following result is notified to us by Dragan [38] and ispresumably in the folklore.

TheoremEvery k-chordal graph is b k

2c-hyperbolic.

Proof.It follows directly from Theorems 2 and 5.

Bound hyperbolicity from above

We now turn to show that it is possible to bound thehyperbolicity from above in terms of chordality.The following result is notified to us by Dragan [38] and ispresumably in the folklore.

TheoremEvery k-chordal graph is b k

2c-hyperbolic.

Proof.It follows directly from Theorems 2 and 5.

A side remark

A graph is bridged [3, 75] if it does not contain any finiteisometric cycles of length at least four, or equivalently, if it iscop-win and has no chordless cycle of length 4 or 5. In contrastto Theorem 6, it is interesting to note that the hyperbolicity ofbridged graphs can be arbitrarily high [72, p. 684].

I In view of Example 4, to get better estimate than Theorem6 along the same approach one may try to beef upTheorem 5.

I We point out Dourisboure and Gavoille posed the openproblem that whether or not

tl(G) ≤ d lc(G)

3e (1)

is true [35, Question 1].I If (1) can be established, it will be the best we can expect

in the sense that for every outerplanar graph G, it holdstl(G) = d lc(G)

3 e [35, Theorem 1].

tl(G) ≤ d lc(G)

3e (1)

tl(G) ≤ d lc(G)

3e (1)

A reformulation of the open problem of Dourisboureand Gavoille

The kth-power of a graph G, denoted Gk , is the graph withV (G) as vertex set and there is an edge connecting twovertices u and v if and only if dG(u, v) ≤ k .

A Chordal Graph Sandwich Problem:For any graph G, is there always a graph inbetween G andGd lc(G)

3 e which is a chordal graph?

A reformulation of the open problem of Dourisboureand Gavoille

The kth-power of a graph G, denoted Gk , is the graph withV (G) as vertex set and there is an edge connecting twovertices u and v if and only if dG(u, v) ≤ k .

A Chordal Graph Sandwich Problem:For any graph G, is there always a graph inbetween G andGd lc(G)

3 e which is a chordal graph?

ExampleThe chordality of the n-cycle is n while the hyperbolicity of then-cycle is bn

4c −12 if n is congruent to 1 modulo 4 and is bn

4celse. It is also known that the tree-length of the n-cycle is dn

3e[35, Lemma 4]. Note that

δ∗(Cn) =

{ b n2 c2 , if n ≡ 0 (mod 4);

b n2 c2 + 1

2 , else.

ExampleFor any k ≥ 2, let Fk be the graph obtained from the 4k -cycle[v1v2 · · · v4k ] by adding the two edges {v1, v3} and{v2k+1, v2k+3}. Clearly, δ(v2, vk+2, v2k+2, v3k+2) = k − 1

2 .Furthermore, we have lc(Fk ) = 4k − 2 andδ∗(Fk ) = k − 1

2 = δ(v2, vk+2, v2k+2, v3k+2) = lc(Fk )4 .

For the graph in Fig. 1 (it is just F2 in Example 8), we haveδ(x , y , u, v) = 3

2 as it happens xy + uv = 3 + 4 = 7 andxu + yv = xv + yu = 2 + 2 = 4.

rb rvrd

Figure: A graph with hyperbolicity 32 , tree-length 2 and chordality 6.

TheoremFor each k ≥ 4, all k-chordal graphs are b k

2 c2 -hyperbolic.

It is clear that if Theorem 9 is tight for k = 4t (k = 4t + 2) then itis tight for k = 4t + 1 (k = 4t + 3). Consequently, Examples 7and 8 indeed mean that the bound reported in Theorem 9 istight for every k ≥ 4. Surely, the next natural step may be tocharacterize all those extremal k -hyperbolic graphs whose

hyperbolicity attain b k2 c2 – but there seems to be still a long haul

ahead in this direction.

Let C4, H1, H2, H3, H4 and H5 be the graphs displayed in nextslide. It is simple to check that each of them has hyperbolicity 1and is 5-chordal.

We can characterize the structure of those 5-chordal graphsachieving maximum possible hyperbolicity.

TheoremA 5-chordal graph has hyperbolicity one if and only if one ofC4, H1, H2, H3, H4, H5 appears as an isometric subgraph of it.

We can characterize the structure of those 5-chordal graphsachieving maximum possible hyperbolicity.

TheoremA 5-chordal graph has hyperbolicity one if and only if one ofC4, H1, H2, H3, H4, H5 appears as an isometric subgraph of it.

ConjectureA 6-chordal graph is 1

2 -hyperbolic if and only if it does notcontain any of a list of eleven special graphsG1, G2, G3, F2, C4, C6, Hi , i = 1, . . . , 5, as an isometricsubgraph.

Note thatlc(C4) = 4, lc(H1) = lc(H2) = 3, lc(H3) = lc(H4) = lc(H5) = 5.The next two results follow immediately from Theorem 10.

CorollaryA 4-chordal graph is 1-hyperbolic and has hyperbolicity one ifand only if it contains one of C4, H1 and H2 as an isometricsubgraph.

Corollary (Brinkmann, Koolen, Moulton)[17, Theorem 1.1] A chordal graph is 1-hyperbolic and hashyperbolicity one if and only if it contains either H1 or H2 as anisometric subgraph.

We remark that as long as every 4-chordal graph is1-hyperbolic is known, Corollary 12 also immediately followsfrom Corollary 13. We also mention that the first part ofCorollary 13, namely every chordal graph is 1-hyperbolic isimmediate from Theorem 2 as chordal graphs have tree-length1.

Corollary (Bandelt, Chepoi)[8, p. 16] A distance-hereditary graph is always 1-hyperbolicand is 1

2 -hyperbolic exactly when it is chordal, or equivalently,when it contains no induced 4-cycle.

Proof.It is easy to see that distance-hereditary graphs must be4-chordal and can contain neither H1 nor H2 as an isometricsubgraph. The result now follows from Corollary 12.

CorollaryEvery AT -free graph is 1-hyperbolic and has hyperbolicity one ifand only if it contains C4 as an isometric subgraph.

Proof.First observe that an AT -free graph must be 5-chordal. Furthernotice that the triple u, y , v is an AT in any of the graphsH1, . . . , H5. Now, an application of Theorem 10 concludes theproof.

Outline

Introduction

Hyperbolicity

Tree-length

Chordality

Conclusion

I The proof of our main results consists of a vastapplications of the triangle inequality. These triangleinequalities are obtained by examining several localcombinatorial objects, especially their behavior undercertain extremality assumptions.

I There exist several ways to connect chordality andhyperbolicity via some other tree-likeness parameters.

I There are lots of other tree-likeness parameters and theirstudy are very active in mathematics and applied fields.See the references for a small selection of some suchpapers.

I Thank you for your attention!

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