IQ 1993 ACM 00fW5411/93/ll(I(J-1134. $01.50 .... list of rewrite rules. ...... Fellows and. Langston. [22] show how to find the minimal forbidden minors from an.
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