An algebraic theory of graph reduction - LaBRI

An Algebraic

Theory

STEFAN

ARNBORG

BRUNO

COURCELLE

Bordc’aLwl

Unllerslil’,

ANDRZEJ

Talctzce,

of Graph

Reduction

France

PROSKUROWSKI

AND

DETLEF

SEESE

Abstract, Wc show how membership in classes of graphs definable m monwhc second-order ]oglc and of bounded treewldth can be decided by finite sets of terminating reduction rules. The method is constructive in the sense that wc describe an algorlthm that wdl produce, from J formula in monxhc second-order Ioglc and an mleger k such that the class dcfmed by the formul~ IS of treewidth s k, a set of rewrite rules that rcducxs any member of the elms to one of’ firrltely many graphs, in a number of steps bounded by the size c~f the graph. This rcductmn syjtem ymlds an algorlthm that runs m time linear m the size of the graph. We illustrate our results with rcductlon systems that recognux some families of outerplanar and planar graphs. Categories .ind Subject Descriptors F.2 2 [Analysis of Algorithms and Problem Complexity], Nonnurnerical Algorithms and Pr[>blems—cotyz/] ~trczt~o/zs O;Zdzscrctc strut tztrcs; F.-1 3 [Mathematical G 2.2 Logic and Formal Languages]; Formal Languages-ckmscs defu~ed by ~runzn~ars c~r uutomafa. alcqortrhms, trees [Discrete Mathematics]; Graph Thco~–grup/~ General

Terms Algorithm\,

Languages, Perform~nce. Theory

Additlonai Key Words and Phrases Graph algebra, graph rcwritlng, m~}nadic scccrnd-order Ioglc, regular set of graphs, trecwldth,

S. Arnborg was supported by the Swed]sh Research Swechsh Board for Technical

Councd

B Courcelle was supported by the “Programmc de Recherchcs [rrformatlque” and the ESPRIT-BRA project 3299 “Computing The research

for Enginecrmg

Sciences and the

Development.

of A. Prcsskurowsk] was supported

Coordonn&es: Moth6matiques by (Graph Transform titlons”.

in part by National

Science Foundation

ct

(NSF)

gnmts (7CR 92-13439 and INT 92-141{)8. Authors” addresses: S. Arnborg, The R(,yal Instltutc

of’ Technology. NADA, KTH, S- 100-44, Stockholm, Sweden; B. Courcclle, Bordeaux-1 Umverslty, Laboratolre d’Informatiquc (associ~ au University of CNRS), 351 Cours de la Lib k, for all graphs G and G’, G *s G’ if and only if G ~(,S,F,,j G’. PROOF.

not required we ‘The

must

Recall that the contexts involved in the definition of -(~,.~) are to be generated by F. So the necessity is clear. For the sufficiency,

prove

first version

that

whenever

of the proof

G = f [ H ] for

of this theorem

appeared

a context in 1986.

f [ ], we

also

have

S. ARNBORG

1144

ET

AL.

G = r~(l’,(11, K)) for some K. We prove this by induction, for all values of H and i over the depth of the argument place in ~[ ]. The base case where ~[ ] is the context

that

removes

all sources

is easy:

G =rl*(H) The inductive an argument (i) (ii) (iii) (iv)

case has four

=r~(P,

subcases

(H,

i)).

according

to the operator

of which

H is

in the expression:

G =~[Hl G =jlH]

=~’[r[+l(H)l, =~’[lj(H)],

G =f[H]

= f’[P,(H,

G =~[Hl

=f’[S,(

In each case above, the inductive

K)], . . ..H.

the depth

hypothesis

. .)1. of ~’ is less than

that

there

is a ~’

that of ~. In case (i), we have by

such that

G = Y,”( f’l(rl+

l(H),

K’)).

But this can also be written r,: [(1’, + ,(H, 1:+ 1(K’))) where H occurs as required. Likewise, we have for case (ii) that G = r~~ ,( P,+ ,(1~+ ,(H ), K’)) = rl*( F’,( H, K“ )), where K“ is obtained from K‘ by removal of the jth source (it follows from Corollary 4.9 that any source of a graph can be removed). For case (iii), we have G = rl*(F’l(P,( H, K), K’)) = r,*(P, ( H, K“ )), where K“ = P,( K, K ‘). Finally,

We can take Now

the

rewriting 3—we

the total of

a graph

system

analogously

size than

the

characterization

to

THEOREM

4.7.

function. reduction Remark.

It

operators

proves

the

from

that

So it follows

from

sets of graphs (HR)

graph

following

is our

at most k.

L

is generated

MA. This

This

algebra

L = L( R, K)

by Fk.

We

has a size

for

some finite

4.7 are definable

by hyper-

❑

K of L.

L as in Theorem grammars.

The

set K c L.

3.3 that

subset

Section

side of every rule

of treewidth

4.1 that

Proposition

a graph

in

system R and some finite

3 for the Fk-algebra

R and some finite

edge replacement

side.

of

systems.

set of graphs

reduction

that

systems

the left-hand

reduction

Proposition

in Section

from

rewriting

right-hand

for graph

for some graph

follows

The

and require

as the size of a graph.

follows

algebraic

Let L be a recognizable

the results system

the

system

corresponding

result

Then L = L( R, K] PROOF.

of other

of edges and vertices

reduction

only add a size function

consider

of S, in terms

number

notion

has larger main

the expression

❑

last case (iv).

follows

from

the closure

prop-

erty of HR sets of graphs with respect to intersection with recognizable sets and from the fact that the set of graphs with tree-width at most k is HR. Some HR sets of graphs are definable by reduction without being recognizable. An example can be constructed from the nonrecognizable context-free language by the reduction system {a’b: ~ ab} with the {a’’b”In > 1},which is defined accepting

word

ab.

PROPOSITION 4.8. G = f ‘[H] PROOF.

Let

G = f [ H ] where

for a context f‘[ By the proof

] generated

of Proposition

G and H are gelzerated

by Fk

and k’

by F~. Then

< 2k.

4.6 and Corollary

4.4, such a graph

G

can be written as r,*(P, (H, K)) for some i not greater than k. Consider a tree-decomposition of width at most k of G, (T, {X,,},,. ~ ). Consider an X,l that contains one source s of H (and thus also of K). Add every source of K vertices of H from n‘ ● N, and remove all non-source except s to every K., every X~. This results

in a tree-decomposition

of width

at most

k + i – 1
4. R and

minors)

but the corresponding

So L is MS-definable

K of Theorem

one could effectively not enough to know

(with

4.7. It appears

MS-formula,

the formula difficult

@~, is not known

p A @L) but we cannot

to find

@L—once

find

it is available

find the minimal forbidden minors for partial k-trees. It is an algorithm deciding the membership in L to be able to

construct

WI . One must also know at least an upper bound on the number of equivalence classes of -~ , see also Lengauer and Wanke [27]. Fellows and Langston [22] show how to find the minimal forbidden minors from an algorithm deciding -~ or one of its finite refinements, for a minor-closed family L and when a tree-width bound is known for these minors. In

the

Instead,

examples we start

combinations

follow, the

of old values

time a new value to one previously to conclude

that with

that

has been obtained, two graphs

we

nullary

not

derive

operators

do

and

as arguments

the

classes

generate

to the operators

new

automatically. values

using

of the algebra.

Each

obtained, we must decide if it is congruent by -~ and if so generate a reduction rule. Here, it is fatal are congruent

when

they are not, but the opposite

mistake only results in more classes than strictly necessary. The congruences must ultimately be proved, but often it turns out that there is a small set of congruent pairs that generates all congruences. Only if one infinitely often fails to identify two congruent values as such does this procedure fail to terminate. 4.4 AN EXAMPLE. We illustrate the theory with the class Lz of partial 2-trees and the algebra generates

exactly

the class Lz by Proposition

a simple example. Consider Mz as defined above. Fz

4.1. But

since we allow

contexts

S. ARNBORG ET AL.

1146 not generated trivial

by Fz to distinguish

one with

consider

values,

-~,

is a finer

congruence

only one class of each sort. By-Proposition

than

the

4.8 it is sufficient

to

the

class Lz in the algebra 114~. We find the following The class of graphs classes of -L, , given by their representatives. in L~ for any-context is omitted.

equivalence that are not

o

g,):

g,:

1 g?: 2,ez, Here,

Kj

K~

sources

(KJ

minus

and deleting

an edge) is obtained

the edge between

sort

g~ that we do not list since we will

not

necessary

generates

to consider

the whole

L generated

congruence F‘

There

not need classes

class Lz. The reason

by a subsignature

by regarding

them.

two vertices

are quite

them.

not

classes of

We also note

generated

for this is that

of F, then

of KJ as

many

if a graph

it is always

that

it is

by Fz, since

F,

is in a class

possible

to parse

it

with respect to F‘ and to reduce it using only rewrite rules whose both members are generated by F’. In our example, the last class of sort gz cannot be generated

by Fz and need

not be further

considered.

This

class, however,

constitutes the context that under parallel composition, distinguishes the first two classes. The procedure given in the proof of Proposition 3.1 leads to the familiar (Pz(el,

series-parallel ec ), ez ), (Sl(l),

reduction 1), ( Kl, 0)), with

system, four

tion, parallel edge elimination, pendant elimination, respectively. The construction also

a number

above,

like

of

rewrite

(Sz(l~(l),

rules

that

1~(1)), i;(l))

are

(removal

rules

R for

=

(( SJ(ez,

degree

2 vertex

e2),

ez),

elimina-

edge removal and isolated vertex process of Proposition 3.1 generates immediate

consequences

of an isolated

vertex

of

those

on condition

that there are two more vertices in the graph), and the same rule can be generated several times. It will often be convenient to consider a graph algebra generating only i-sourced graphs, for some i, and consider such a graph “equivalent” to the corresponding representatives

graph

with

sources

(or all minimum

gO if we have them for sort g,: representative (or every minimum then the members)

removed

size members)

by the

operator

of all congruence

rl*. We

can find

classes of sort

If S, is a set containing a minimum size size member) from each class of sort g,,

set S contains a minimum size representative (all minimum size from each class of gO, where S is the union over i of the set r,*(Sl )

of graphs from S1 with sources removed, and the set of graphs with fewer than i vertices. The latter set is not finite, since there is no limit on the multiplicity of edges. But in every finite congruence, a minimum size member of a class has a finite bound (usually, 1 or 2) on the multiplicity of an edge. 4.5 A LINEAR-TIME deciding membership

DECISION METHOD. We now describe an algorithm for in a recognizable set L of graphs with treewidth bounded

by some known number k. The algorithm is based on a graph reduction system that is used to successively update a data structure initially representing a given graph. The results of applications of reduction rules to identified subgraphs isomorphic to left-hand side of rewriting rules are recorded in the data structure. (Each such instance is called a redex and the resulting graph a reduct.) When no redex can be found, the membership of the irreducible reduct graph in the finite set of accepting graphs determines the membership of the original graph in L.

An Algebraic The

Theov

of Graph

distinguishing

property

linear-time

algorithm

composition

of internally

an i-sourced

graph

its source

is

of a special

that

each

connected

is internally

vertices

do not

1147

Reduction graph

reduction

left-hand graphs

connected

constitute

side

with

of

system

identical

is

sources.

if the underlying

a separating

admitting

a rule

a

a parallel

We say that

graph

is connected,

set, and there

are no edges

between sources except when the graph itself is an edge between two sources. The internally connected parts of an i-sourced graph are obtained from the components

into

the original

source

which

the graph

vertices

is split

as sources.

by the source

Moreover,

vertices,

in a special

together graph

with

reduction

system not too many of its left-hand-side components with the same source sequence can be composed in parallel without constituting a redex. Finally, let us call a reduction system auto-reduced if none of its left-hand sides properly contains

a redex.

summarize, (i)

Every There

(iii)

reduction

is an upper parts

irreducible,

and

graphs,

bound from

system

R is special

side is a parallel

i-sourced

connected

composition

for some

also

auto-reduced.

To

of a number

of internally

i, O < i s k,

on the number

left-hand

is

if

of parallelly

sides of R whose

composed parallel

internally

composition

is

R is auto-reduced.

~EIVIMA 4.9. defined

ELWV recognizable

by a special

PROOF. a given

reduction

all

construct

set of graphs

smallest

construct

reduction

class L. For this purpose,

D using

elements

of

the method

D‘

and

L of bounded

treewidth

can be

system.

We show how such a special

recognizable

contains shall

special system

left-hand

connected (ii)

The

a reduction

every

congruence

described

R, by adding

system can be obtained

assume

class

in the proof

elements

starting

from

that the set D originally of

W~ . (We

of Proposition from

can

3.3.) We

the empty

set. Let

c be the size of a largest element in D. Repeat the following process until it results in no more changes of R and D‘: Construct all sort-compatible from F~. from D U D‘ and operators expressions f(dl, ..., dp ) with operands Consider

the

connected

value

parts.

d of such

For

an index

an expression i denoting

and

split

a subset

{di,, Ii < ~ s J,} be the set of all internally connected same sources. The parallel composition d, of these some d, in D. If every

rule

(e, f)

each

such

rule

Id, I = I ~1, then from

internally-connected (di, ~)

R and parts,

to R and apply

add ~to D’. This procedure

process

in R such that

will

if

then

Ie‘ I > If I and

i or d. Otherwise,

by {(d,, ~)}

to some

e‘ is a parallel

in R to obtain

since no element

its internally

sources

of

d, let

parts of d having graphs is congruent

add ( e‘, f ) to R, else add

all rewrites terminate,

the next

e is reduced

d into

of the

a normal

first

find

e‘. Remove

composition

e‘ to D:. form

the to

Now

of add

~ of d, and

of D‘ will be larger

than

c2k

(each of the 2~ source subsets can receive a reduct of size at most c), a rule once removed from R will not be added again, and left-hand sides of rules are obtained from expressions of the form f(d,, ..., dP ), with the operands from D u D’. The class L is defined by L = L(R, (D U D’) n L), and the reduction system is of the required type. Moreover, R can be augmented so that no left-hand side of R consists of more than c internally connected parts and so that

every

set of c internally

has a subset

among

connected

the left-hand

components

sides of R.

❑

from

lefthand

sides of R

S. ARNBORG ET AL.

1148 By

a pmf

internally

in

a

connected

graph

reduction

part

of a left-hand

system

R,

we

subsequently

side of R. By a partiul

mean

match

in

an

G, we

mean a part and its isomorphic subgraph of G together with the isomorphism. The following lemma shows that we do not have to find and explicitly remove those partial match indicators that have become invalidated (i.e., are no longer applicable after a reduction step), because a vertex matching one of its sources has disappeared and we never find the same separator in the reduced graph. LEMMA a graph

4.10. G.

invalidates appear

Let S be a set of partial

An

application

of a red~lction

exact~ those partial

in the reduct.

SLlbSL?qLle?W

redex

PROOF.

matches

Thlis,

matches the imalid

rule

of a special

LlpOtZ discoLe~

in S that hale partial

reduction

system in

of a redex in S

a source lerte.x that does not

matches

will

not be refen-cd

to in

searches.

Restating

the lemma,

let us assume

that

a vertex

L’ of G matches

an internal vertex of a part RI of a redex completed by some partial matches from S. Let us assume further that L matches also a vertex of a part R2, not a part of the redex, in another partial match ot’ S. We have to show that there is a vertex }V in G that matches an internal vertex of RI and a source of RJ. In the subgraph that

match

of G isomorphic sources

of R ~, then

to R,, consider

of R,. If none

the vertices

of them

of the redex

all paths

contains

are internal

from

a vertex

L to the vertices matching

for RQ, which

a source

contradicts

the

auto-reduced property of the special reduction system. Since the parts R, and Rz are internally connected, if the vertices matching sources are identical, so are the matching Algorithm Input:

4.1:

Outputi Data Structures:

parts.

❑

Membership Decision. Graph G, given by its adjacency list, Special reduction system R, given by the list of rules and the set of accepting irreducible graphs. YES if the irreducible reduct is an accepting graph, NO otherwise. Access structures A,, 1 < ~ < k, indexed by t-tuples identifying vertex sequences matched to sources in partial matches, Current reduct graph, List of low degree vertices, Array of current vertex degrees.

Method: 1. Initialize data structure: G becomes current reduct graph and all vertices of low degree are put on the List. z. while non-empty List do Let L’ be a vertex on the List. 2.1 while partial match with [ not found do Attempt to match 1’ against an internal vertex not already matched of every internally connected component of a left-hand side of every rule in R. Z.Z if a partial match is found,

against

then record it in the appropriate A, with the appropriate source index. if the partial match completes a redcx recognition, perform the reduction Z.Z. 1 changing the structure of the reduct graph. 2.z.z initializing the new vertex adjaccncics and degrees, 2.2.3 updating the source degrees. and 2.2.4 inserting new vertices of low degree in the List. 2.3 if no match is found, remove L’ from the List. 3. Match the irreducible rcduct graph against the accepting graphs. if there is a match, output YES, otherwise NO.

L’

by

An Algebraic LEMMA

Tkeory

We

result

Reduction

The Algorithm

4.11.

PROOF. desired

of Graph

show upon

The invariant

that

4.1 is correct.

the

algorithm

the exit from

of the inner

1149

maintains

the outer

loop

loop

an invariant

implying

the

(2).

(2.1) states that

only the vertices

on the List,

together with the partial matches recorded in the A,s, can match a left-hand side of a reduction rule. This follows easily from the finiteness of the reduction system (and thus bounded degree of the internal vertices of the left-hand sides of its rules). The invariant of the outer loop states that the current reduct congruent with the input graph and that the data structures are ❑ maintained. This follows from Lemma 4.10. For

our

vertices of

decision

algorithm

of the input

vertex

indices

operations

are

to work

graph,

we shall

is used Store(Z

as key

ualue),

in time

need to

to the number

an access structure

store

Read(tl,

proportional the

and

necessary

Remolje.

graph is correctly

where

information.

The

size

of

The

the

stored is bounded by a constant that depends only on the set L description by a reduction system. First, we investigate the performance able. The method for the readers LEMMA mole

is known

4.12.

et al. [1, exercise

2.12]. We recall

it here

A data

structure

with

access operations

so that each operation

Store,

takes 0(1)

Read,

and Re-

time on a RAM

with

cost measure.

PROOF.

The data structure

each ranging ing $dices

Aho

item

and its achiev-

convenience.

can be implemented

the uniform

from

of

a p-tuple

over

O(n)

of initialized

consists

values,

of an O(n’)

and an O(n)

elements

array

indexed

size “verification

and stored

values.

An array

by p indices,

table”

contain-

element

indexed

iP ) is retrieved by accessing the fih entry of the array. An by [= (ii,..., initialized entry of the array contains the offset in the verification table where the index of the entry and the stored range or the pointed index does not then

this entry

an element

consists

consists

in the verification LEMMA

linear

PROOF. vertex

first

is recorded. If this offset is out of with the index of the array entry,

noise (nothing then

either

of the data structures

step (1) of the

and adjacency

the inlariants

or adding

decision

used in Algotithm of Lemma

algorithm

lists of the instance

Storing

an element

The

invariants

of the outer

and the inner

The

total

of this phase

is O(n).

❑

loops

4.1 takes a

4.11.

is to store

graph

graph. During this phase it is not necessary to explicitly, since there are numbers a and b, O s a < upper bound on the size of a minimum member of m-allel that b parallel edges are congruent to ~ parallel edges need ever be stored in the ( ~u ! table. are initialized in constant time (cf. Lemma 4.12). The placed in the List, which becomes non-empty unless time

has been stored).

changing

❑

of time and establishes

The

degrees

table.

reading,

Initialization

4.13.

amount

of uninitialized

of first

value agree

the

as the current

edges, reduct

store all multiple edges b s c (recall that c is an a congruence class), such edges, that is, at most b The access structures A, vertices of low degree are the graph is irreducible.

are thus

trivially

established.

S. ARNBORG ET AL.

1150 LEMMA

4.14.

Deciding

whether

a Lertex 1 of a gil)etl

internal LIerte.r w of a part takes only constant consideredconstant and G Laries). The

PROOF.

part

matching

to vertices

vertices

matching

adjacency

the

but only

constantly

constantly

number

only on the vertices of

rewriting the part

LEMMA

4.15.

matching

with

Deciding

whether

reduction

bounded

sources

a match

system

the

be matched

isomorphisms

by a constant

that

in of a

depends

graph that match internal than some constant and of the

of a part

(as stated

the

to search

can also only be matched

One has to explore

match

is

of the

identifying

part will

an

the part

vertices

then

be necessary

system. All vertices of the must have degrees less that

and

in G. A symmetrical

degrees

G matches

(when

the internal

never

ways. A vertex

graph

of time

degrees,

it will

vertex

of vertices

rede.x of a special

pattern

need

completes

in Lemma

not

be

a search for

4. 14) takes

a

constant

of time.

PROOF.

The

that

concerns

access

is, a given

each match defined by counts

Thus,

many,

of vertices

other adjacencies ❑ explored.

entry,

degree

by first

appropriate

many ways in all (to all parts).

constant

amount

the

sources.

list of a high

in many,

is done

in G with

amount

structures

sequence

~,

should

of sources,

type with the same sequence left-hand sides of the special

graph

vertices

are checked

corresponding

against

the

maintain,

an array

each match

of sources. The match reduction system and

by a match

required

for

of partial

counts

type of the

to part

initialized counts

for

types are the count

vertices.

The

sides

every

left-hand

time a new partial match is found. By the property (ii) of the special reduction system, the number of these partial matches is bounded by a constant adding only

a constant

amount

when rediscovering partial matches).

4.16.

to state

Let

when

requires

our main

Membership

graphs of bounded treewidth on a RAM with the uniform PROOF.

needed

(which

recording

updating

a partial

of the counts

match,

and

and lists of

❑

We are now ready THEOREM

of time

a redex

algorithmic

result.

monadic

second

in eleiy

can be decided cost nleasure.

w be the treewidth

in time linear

of the graph

order

definable

in the six

set of

of the graph

class and let n be the number

of vertices of the graph. Clearly the size of the graph is 0(n) since it has fewer than wn edges. We know that every MS-definable set of graphs is recognizable (Theorem 4.5). Hence, if it is also of bounded treewidth, it is recognized by a graph reduction system (Theorem 4.7), and moreover by a special one (Lemma 4.9). By Lemma 4.11, Algorithm 4.1 based on a special graph reduction system R is correct. It remains to show that the algorithm decides class membership of a graph in linear time. By Lemma 4.13, the initialization the number the graph, it required for By finding

step can be performed

in linear

time.

Since

of reductions made by a reduction system is linear in the size of now suffices to show that there is a constant bound on the time each reduction step. the maximum number of nonsource vertices c1 in a right-hand

side of R, we find vertices used during

a bound n(l + c1 ) on the number the reduction process. We introduce

of original a numbering

and new (internal

Atl Algebraic naming)

Theory of Graph

for

the parts

Reduction

of left-hand

1151

sicles of reduction

rules,

and introduce

an

array A,, indexed by p-tuples for every sort g,, of such a part, where we store the number of partial matches and the indices of the matches to its nonsource vertices.

One

current

reduct

such

table

has 2 indices

Now we use the arrays in the graph

and

is used

to store

the

edges

of the

connected

parts

graph. in linear

to the degree

to store

time.

partial

matches

We examine

of some internal

vertex

those

to internally

vertices

of a part.

with

By Lemmas

a degree

identical

4.14 and 4.15, this

takes a linear amount of time. As soon as the multi-set of matches with a given set of source vertices contains a left-hand side of a reduction rule, the reduction can be performed regardless of possibly overlapping matches (cf. Lemma 4. 15). The reduction is done tion

as follows: rule’s

The

left-hand

are removed

vertices

corresponding

side are removed

and the adjacency

is of bounded

degree

the neighbors’

to internal

vertices

the graph:

Edges

lists are updated.

so its adjacency

adjacency

from

Note

that

list is of constant

lists, we know

which

element

of the reducinvolving

them

the removed

vertex

length.

When

to remove,

is involved. Next, the right part of the rule is introduced graph by allocation of new vertex indices for its internal

updating

so no search

in the current reduct vertices and its edges

are introduced in the edge array. The vertices matched into the sources must also be considered as having their neighborhood altered and thus must be regarded as new. Since suffices reach

the initialization time is O(n) for graph size to reduce the graph size by a constant amount,

a normal

form

reject

it (as

graphs

( D U D’)

there.

not

with

respect

belonging

to

to R is O(}z ). If this graph

L);

n L in constant

n and constant work the time required to

otherwise,

time

it

is compared

and is accepted

is too large, to

the

list

we of

if and only if it is found

❑

For a given k, the class of partial k-trees is recognizable. This follows, for example, from the fact that it is minor-closed and thus it has a finite set of minimal forbidden minors (by [29]), and thus it is MS-definable and, by Theorem

4.5, recognizable.

CO ROI.I.ARY

decidable Remark.

4.17.

in linear The

Hence.

the following

For et ‘cty k, m(~nlbemllip

corollary: in the class of partial

k-trees

is

time. corresponding

algorithms

are known

only

for

k