The block-cutpoint tree characterization of a covering polynomial of a graph

Robert Ellis (IIT)James Ferry, Darren Lo (Metron, Inc.)

Dhruv Mubayi (UIC)

AMS Fall Central Section MeetingNovember 6, 2010

Slide 2

Random Intersection Model B*(n,m,p)• Introduced: Karoński,

Scheinerman, Singer-Cohen `99

• Bipartite graph models collaboration– Activity nodes– Participant nodes

• Random Intersection Graph B*(n,m,p)– Bipartite edges arise

independently with constant probability

– Unipartite projection onto participant nodes

m: number of “movies”

n: number of “actors”

Unipartite projection

Collaboration graph: who’s worked with whom

Bipartite graph

Slide 3

[ ]*

1

( ( , )

( ( , , )

1 (1 )

H ER

H

mER

X N p

X N M p

p p

G

B

E

E

for

£

é ùê úë û

= - -

Subgraph H E[XH]

Erdős–Rényi RandomIntersection

60-cycle

Expected sugbraph count vs. E-R Erdős–Rényi

n = 1000 pER = 0.002

1.3

11

´ 152 10

1.2

-´ 63 10

1500

´ 54 10

´ 102 10

1700

´ 382 10

Random Intersection n = 1000 m = 100 p = 0.0045

Yields pER = 0.002

Slide 4

RC model (m = 10,000)RC model (m = 1000)RC model (m = 100)RC model (m = 10)RC model (m = 2)RC model (m = 1)

Erdős–Rényi vs. RI Model as m → ∞

Erdős–Rényi G(n,pER) model

• m = “number of movies”• pER = 0.028 (edge probability)

RI

B*(N,M,p)

Slide 5

• Theorem [Ferry, Mifflin]. For a fixed expected number of edges pER , and any graph G with n vertices, the probability of G being generated by the Random Intersection model approaches the probability of G being generated by the Erdős–Rényi model as m → ∞.

• Formula for rate of convergence:

• [(Independently) Fill, Scheinerman, Singer-Cohen `00] With m=nα, α>6, total variation distance for probability of G goes to zero as n → ∞.

( )( )

( ) ( ) ( ) ( )3 2

1/ 2 13 3

Pr 1 1 11 2 2 log

3Pr 1RC

ER ER ER ER

nGn e G c G m O m

G p p p- -

æ öæ öæö ÷ ÷ç ÷ çç ÷ ÷= + ÷- - + +ç çç ÷ ÷÷ç çç ÷ç ÷÷çç -è ø è øè ø

Erdős–Rényi vs. RI Model as m → ∞

2

n

RI

Slide 6

Idea: Let m → ∞ and fix the expected number of movies per actor at constant μ=pm.

This allows simplified asymptotic probabilities for random intersection graphs on a fixed number of nodes.

• Probability formulas are from edge clique covers• Most probable graphs have block-complete structure• Least probable graphs have connections to Turán-type

extremal graphs

RI model in the constant-μ limit

Slide 7

Edge clique covers• Unipartite projection

corresponds to an edge clique cover

• The projection-induced cover encodes collaboration structure

Hidden collaboration perspective:• Given B*(n,m,p)=G, we can infer

which clique covers are most likely

• This reveals the most likely hidden collaboration structure that produced G

Unipartite projection

B

*B

*G B

likeliest B

“size” of clique cover S “weight” of S, G

ai = #least-wt covers of size i

Covering polynomial of G

X

Xs )(

)()(wt s )(wtmin)(wt

G

i

ii xaxGs );(

wt=4, s=6

Projection

wt=4, s=7 (2 ways) wt=5 (not least-weight)

Thus wt(G) = 4 and s(G; x) = x6 + 2x7. Slide 8

Slide 9

Theorem [Lo]. Let n be fixed and let G be a fixed graph on n vertices. In the constant-m limit,

• Lower weight graphs are more likely• If G has a lower weight supergraph H, G is more likely to appear as a

subgraph of H than as an induced graph

Theorem. Let n be fixed and let G be a fixed graph on n non-isolated vertices with j cut-vertices. In the constant-m limit,

where the bi are block degrees of the cut-vertices of G, and is the bth Touchard polynomial.

0( )

b ibb i i

x xfì üï ïï ïï ïí ýï ï= ï ïï ïî þ

=å

Fixed graphs in the constant-μ limit

1)(wt

* 1);(

),,(Pr mOm

GsGpmn

G

1)(rank

1* 1),,(Pr

mO

mGpmn

G

j

i bjn

i

Slide 10

Example subgraph probabilityLet H be

rank(H) = 7 n(H)=8 2 cut-vertices; 4 blocks

31 2

1

31 3s

s s

b1= 3 b2= 2

1

Stirling numbers count partitionsof bi blocks into s “movies”

ib

s

Block-cutpointtree of H

17

26* 1

131),,8(Pr

mO

mHpm

Slide 11

Block-cutpoint tree → Least-weight supergraphs

1. Select an unvisited cut-vertex.

2. Partition incident blocks, merge, and make block-complete.

3. Update block-cutpoint tree.

4. Repeat 1 until all original cut-vertices are visited.

Slide 12

An extremal graph weight conjecture

Conjecture [Lo]. Let G have n vertices. Then

with equality iff there exists a bipartition V(G)= such that:• A=• B=• The complete (A,B)-bipartite graph is a subgraph of G• Either A or B is an independent set.

4)(wt 2nG

BA

2n

2n

Slide 13

Related simpler questions

Conjecture. Every K4-free graph G on n vertices and

edges has at least m edge-disjoint K3’s.

Theorem [Győri]. True for G with chromatic number at most 3.

Theorem. True when G is K4-free and

where k≤n2/84+O(1).

mn 42

,)3,(42 kntmn