%
NODE-AND
EDGE-DELETION NP-COMPLETE
PROBLEMS
Mihalis Y a n n a k a k i s
PRINCETON UNIVERSITY
Princeton, N e w Jersey
ABSTRACT
If ~ is a graph property, the general
node(edge) deletion p r o b l e m can be stated
as follows:
Find the m i n i m u m n u m b e r of
nodes(edges), whose d e l e t i o n results in
a subgraph s a t i s f y i n g property ~.
In this
paper we show that if ~ belongs to a rather b r o a d class of p r o p e r t i e s (the class of
p r o p e r t i e s that are h e r e d i t a r y on induced
subgraphs) then the n o d e - d e l e t i o n p r o b l e m
is NP-complete, and the same is true for
several r e s t r i c t i o n s of it.
For the same
class of properties, r e q u i ~ i n g
the rem a i n i n g graph to be c o n n e c t e d does not
change the N P - c o m p l e t e status of the
problem; m o r e o v e r for a certain subclass,
finding any "reasonable" a p p r o x i m a t i o n is
also NP-complete.
E d g e - d e l e t i o n problems
seem to be less amenable to such generalizations.
We show h o w e v e r that for
several common p r o p e r t i e s (e.g. planar,
outer-planar, line-graph, transitive digraph) the e d g e - d e l e t i o n p r o b l e m is NPcomplete.
the m a x clique, the feedback-node set,
the f e e d b a c k - a r c set [K], and the simple
m a x - c u t p r o b l e m [GJS]) can be formulated
in an obvious w a y as node or edge deletion problems, s p e c i f y i n g a p p r o p r i a t e l y
the p r o p e r t y ~.
Furthermore, K r i s h n a m o o rthy and Deo showed in a recent paper
[KD] that the n o d e - d e l e t i o n p r o b l e m for
several o t h e r p r o p e r t i e s is also NPcomplete.
(For an e x p o s i t i o n of NPcompleteness, see [GJ]).
Since Cook's i n t r o d u c t i o n of the concept of N P - c o m p l e t e n e s s , the list of
N P - c o m p l e t e problems has expanded rapidly, with more and more individual problems
from various areas b e i n g added to it [GJ]
Sections 2 and 3 are c o n c e r n e d with nodedeletion problems.
O u r aim is to show,
h o w this set of similar problems (with
p r o p e r t i e s , drawn from a certain large
class of properties) can be treated in a
s y s t e m a t i c fashion in o r d e r to prove the
N P - c o m p l e t e n e s s of all the members of the
set.
(A similar approach was taken also
i n d e p e n d e n t l y by Lewis, Dobkin and Lipton
in [LDL] and [L].
H o w e v e r our results are
s i g n i f i c a n t l y more general than theirs.).
In this paper we consider p r o p e r t i e s that
are h e r e d i t a r y on induced subgraphs, i,e.
if G is a graph (or digraph) with p r o p e r t y
~, then deletion of any node does not
produce a graph v i o l a t i n g ~. We call a
p r o p e r t y n o n t r i v i a l if it is true for a
single node and is not satisfied by all
the graphs in a given input domain.
c l e a r l y the n o d e - d e l e t i o n p r o b l e m makes
sense o n l y for n o n t r i v i a l properties.
W e will require ~ to be easy (i.e. in
p ) at least to recognize (although o u r
results are valid even if , cannot be
r e c o g n i z e d in n o n d e t e r m i n i s t i c
p o l y n o m i a l time.
In this case, the 'NPcomplete' clause should be replaced by
'NP-hard').
KEYWORDS¢
approximation, c o m p u t a t i o n a l
complexity, edge-deletion, graph, graphproperty, hereditary, m a x i m u m subgraph,
node-deletion, N P - c o m p l e t e , p o l y n o m i a l
hierarchy.
i.
INTRODUCTION
The general node(edge) deletion p r o b l e m
can be stated as follows:
Given a graph
or digraph G find a set of nodes(edges)
of m i n i m u m cardinality, whose deletion
results in a subgraph or subdigraph
s a t i s f y i n g the p r o p e r t y ~.
(For the
standard graph theory t e r m i n o l o g y the
reader is referred to [H] or [Bg.
Several of the w e l l - s t u d i e d polynomial g r a p h - p r o b l e m s ~ u c h as the connectivity of a graph [EV, TI], the a r c - d e l e t ion [K], the m a x i m u m m a t c h i n g and the bm a t c h i n g problems led]), as well as NPcomplete problems (such as the node cover,
%work D a r t i a { l y
Mcs-76-15255
N o w suppose that ~ is such a p r o p e r t y
and that there is an upper bound k on the
order of graphs s a t i s f y i n g ~.
Then the
s u p p o r t e d by N S F ' g r a n t
- 253 -
c o r r e s p o n d i n g n o d e - d e l e t i o n p r o b l e m is
p o l y n o m i a l in a t r i v i a l way:
Given a
g r a p h G, e x a m i n e all the (induced) subg r a p h s of G of o r d e r up to k and find the
one w i t h the l a r g e s t o r d e r that s a t i s f i e s
~.
We call a p r o p e r t y i n t e r e s t i n g (on
some input domain) if t h e r e e x i s t s no such
u p p e r b o u n d (for the g r a p h s o f the input
domain), i.e. if t h e r e are a r b i t r a r l y
large g r a p h s s a t i s f y i n g n.
tion (with w o r s t - c a s e ratio 0 ( n l - e ) , for
any e>0, w i t h n the n u m b e r of nodes) of
the c o n n e c t e d n o d e - d e l e t i o n p r o b l e m is
also N P - c o m p l e t e * , and (2) D e t e r m i n i n g
w h e t h e r the l a r g e s t s u b g r a p h s s a t i s f y i n g
are c o n n e c t e d or n o t is in 4 P2 [ N P U co-NP], p r o v i d e d
NP M c o - N P ~ ~
that
The a d d i t i o n a l a s s u m p t i o n , that n is
d e t e r m i n e d by the b l o c k s , is e s s e n t i a l
as e x e m p l i f i e d by the p r o p e r t y n = 'comp l e t e graph' (or ~ = 'star').
This noded e l e t i o n p r o b l e m is e q u i v a l e n t to the
node-cover problem, which has a polynomial 2-approximation.
In S e c t i o n 2 we s h o w that for any n o n trivial
and i n t e r e s t i n g g r a p h - or
d i g r a p h - p r o p e r t y that is h e r e d i t a r y on
i n d u c e d s u b g r a p h s , the n o d e - d e l e t i o n
p r o b l e m is N P - c o m p l e t e .
M o r e o v e r the
r e s t r i c t i o n to p l a n a r g r a p h s (or digraphs)
and to a c y c l i c d i g r a p h s (in case of digraphp r o p e r t i e s ) is also N P - c o m p l e t e .
For the
r e s t r i c t i o n to b i p a r t i t e g r a p h s t h e r e are
few (in a w a y t h a t is d e f i n e d m o r e p r e c i s e ly in S e c t i o n 2) e x c e p t i o n s . F o r e x a m p l e
the node c o v e r p r o b l e m is p o l y n o m f a l (by
K o n i g ' s t h e o r e m and the fact that the
m a x i m u m m a t c h i n g is p o l y n o m i a l [Ed]).
H o w e v e r we s h o w that it is a u n i q u e
e x c e p t i o n a m o n g p r o p e r t i e s t h a t are det e r m i n e d by the c o m p o n e n t s .
(We say that
a p r o p e r t y ~ is d e t e r m i n e d by th___eec o m p o n e n t s - resp. by the b l o c k s - of a graph,
if w h e n e v e r the c o m p o n e n t s - resp. the
b l o c k s - of a g r a p h G s a t i s f y n~ then G
s a t i s f i e s also ,).
In S e c t i o n 4 we t u r n to ~ e d g e - d e l e t i o n
p r o b l e m s ~ w h i c h tend to be e a s i e r to
solve (or h a r d e r to s h o w N P - c o m p l e t e )
than t h e i r n o d e - d e l e t i o n versions.
Note
for e x a m p l e the d i f f e r e n c e for ~ =
'acyclic graph' (tree) or 'degree
constrained'.
W e s h o w the f o l l o w l n g
p r o b l e m s to be N P - c o m p l e t e :
(i) w i t h o u t
c y c l e s of s p e c i f i e d l e n g t h ~ or of any
l e n g t h < ~, (2) d e g r e e - c o n s t r a i n e d w i t h
m a x i m u m d e g r e e r > 2 and c o n n e c t e d ,
(3)
outerplanar,
(4) planar~ (5) l i n e - i n v e r t ible~ (6) c o m p a r a b i l i t y graph, (7)
b i p a r t i t e (max-cut p r o b l e m ) , (8) t r a n s itive digraph.
For p r o b l e m s (5), (6),
(7) we d e t e r m i n e the b e s t p o s s i b l e
b o u n d s on the n o d e - d e g r e e s for w h i c h the
problems remain NP-complete.
In these c o n s t r u c t i o n s the r e m a i n i n g
g r a p h a f t e r the d e l e t i o n of a m i n i m u m
n u m b e r of n o d e s is o f t e n h i g h l y d i s c o n n e c t ed.
O n e m a y w i s h to r e q u i r e the r e m a i n i n g
g r a p h to be c o n n e c t e d and m i g h t e v e n h o p e
that this t a s k c o u l d be easier:
For
e x a m p l e K r i s h n a m o o r t h y and D e o p r o v e d [KD]
that the n o d e - d e l e t i o n p r o b l e m for
= 'nonseparable' is N P - c o m p l e t e .
(If
the r e s u l t i n g g r a p h is d i s c o n n e c t e d t h e y
r e q u i r e that each of its c o m p o n e n t s
s a t i s f i e s ~).
In this case the r e q u i r e m e n t for c o n n e c t i v i t y m a k e s the p r o b l e m
very e a s y (linear): we can find the b l o c k s
of the g r a p h IT2] and then d e t e r m i n e the
b l o c k w i t h the m a x i m u m n u m b e r of nodes.
In S e c t i o n 3 we s t u d y the e f f e c t of the
i n c l u s i o n of a c o n n e c t i v i t y r e q u i r e m e n t .
W e s h o w that for the same class of p r o p e r ties (hereditary~ n o n t r i v i a l and i n t e r e s t ing on c o n n e c t e d graphs) the N P - c o m p l e t e ness of the n o d e - d e l e t i o n p r o b l e m is n o t
affected. (In case of d i g r a p h s
we take
c o n n e c t i v i t y to m e a n w h a t is u s u a l l y c a l l ed w e a k c o n n e c t i v i t y , i.e. c o n n e c t i v i t y
of the u n d e r l y i n g u n d i r e c t e d graph).
Moreo v e r for p r o p e r t i e s that are d e t e r m i n e d by
the b l o c k s (i)
Any nontrivial approxima-
of c o u r s e
A few w o r d s on n o t a t i o n :
we use y (G)
~resp. vC(G)) to d e n o t e the m i n i m u m
n u m b e r o~ n o d e s w h o s e d e l e t i o n r e s u l t s in
a s u b g r a p h (resp. c o n n e c t e d subgraph) of
G that s a t i s f i e s p r o p e r t y ~.
Usually
w h e n no a m b i g u i t y can arise, we d r o p the
s u b s c r i p t ~.
By S0(G) we d e n o t e the
n o d e - c o v e r n u m b e r of G, i.e. S0(G ) =
yw(G), w i t h ~ = ' i n d e p e n d e n t set of
nodes'
2.
THE N O D E - D E L E T I O N P R O B L E M
HEREDITARY PROPERTIES
FOR
T h e o r e m i.
The n o d e - d e l e t i o n p r o b l e m for
nontrivial, interesting graph-properties
that are h e r e d i t a r y on i n d u c e d s u b g r a p h s
is N P - c o m p l e t e .
*
**
- 254 -
For d e f i n i t i o n s c o n c e r n i n g a p p r o x i m a tion a l g o r i t h m s , see [J].
R e g a r d i n g the p o l y n o m i a l - t i m e h i e r a r chy and in p a r t i c u l a r ~ ,
see IS].
Proof:
B H { 8j s a t i s f i e s
For all m , n there is a n u m b e r r(m,n) (the
s o - c a l l e d R a m s e y n u m b e r ) , such that e v e r y
g r a p h w i t h no fewer than r(m,n) n o d e s
contains K
or K . W e c l a i m that e i t h e r
all c l l•q u e sm or a~l i n d e p e n d e n t sets of
n o d e s (or both) s a t i s f y ~.
S u p p o s e , to
the c o n t r a r y / that there are m, n such
that K m and K n do not s a t i s f y W.
Since
is an i n t e r e s t i n g p r o p e r t y , t h e r e is a
g r a p h s a t i s f y i n g ~, w i t h m o r e than r(m,n)
nodes, and since ~ is h e r e d i t a r y on induced subgraphs either K
or K
has to
m
s a t i s f y ~.
Define a complementary property ~ as follows:
a_graph G satisfies
iff its c o m p l e m e n t G s a t i s f i e s ~.
Clearly
s a t i s f i e s also the a s s u m p t i o n s of the
theorem~ (since the c o m p l e m e n t of a subg r a p h is a s u b g r a p h of the c o m p l e m e n t ) ,
and the two n o d e - d e l e t i o n p r o b l e m s are
e q u i v a l e n t (if the input d o m a i n of g r a p h s
is u n r e s t r i c t e d ~ or at least c l o s e d u n d e r
complementation).
c o m p l e t e p - p a r t i t e graph, then J c o n s i s t s
of a s i n g l e edge and k = 2.)
By nont r i v i a l l i t y of ~, there e x i s t s such a
g r a p h J, and f u r t h e r m o r e , since all ind e p e n d e n t sets of n o d e s s a t i s f y ~, J has
at least one c o m p o n e n t of o r d e r no less
than 2.
S u p p o s e from n o w on w i t h o u t loss of
g e n e r a l i t y that all i n d e p e n d e n t sets of
n o d e s s a t i s f y n; o t h e r w i s e c o n s i d e r the
e q u i v a l e n t p r o b l e m for ,.
Let G be a g r a p h w i t h c o n n e c t e d compo n e n t s G I , G 2 ~ . . . ~ G t.
For each G.I take a
c u t p o i n t c. and sort the c o m p o n e n t s of
G. r e l a t i v ~ to c. according to t h e i r
i
.i
orders.
T h i s glves a s e q u e n c e
i]i
(For e x a m p l e
if ~ =
Let Jl,...,J~ be the c o m p o n e n t s of J
t
s o r t e d a c c o r d i n g to t h e i r ~. s, c. the
c u t p o i n t o f Jl that gave =_~
l J^u t~e largest c o m p o n e n t of J. r e l a t i v e to Cl, J. and
,
I
I
J
the g r a p h s o b t a i n e d from Jl and J
r e s p e c t i v e l y b y d e l e t i n g all n o d e s of J0
e x c e p t Cl, and d any• n o d e of J^u o t h e r
than c I.
(There e x l s t s such a node d
since Jl' and c o n s e q u e n t l y J0 too, has
at leas~ 2 nodes).
Now, g i v e n a g r a p h G, input to the
node c o v e r p r o b l e m , let G* c o n s i s t of
n . k i n d e p e n d e n t c o p i e s of G, w h e r e n is
the o r d e r of G.
For e a c h node u of G*
c r e a t e a c o p y of J' and a t t a c h it to u by
i d e n t i f y i n g c I w i t h u.
Replace every
edge (u,v) of G* by a c o p y of Jo' a t t a c h e d
to u and v by its n o d e s c] and d.
(see
Fig. i). (It does not m a t £ e r h o w we
i d e n t i f y the n o d e s Cl,d w i t h the n o d e s u
and v).
s
~.i =A < n i l ' n i 2 ' ' ' ' ' n .
.>, .w i t .h nil
.
~3 i
> n.. ~ and a s s u m e that c. is the c u t -
~.
~
To
N
l
p o i n t of G. that gives the
i c a l l y s m a l l e s t such ~..
lexicograph(If G. is
l
biconnected,
then c.
is any node of
it,
l
and a. = < n. >, w h e r e n. = iGil ) . Sort
l
l
l
.
the s e q u e n c e s of e. 's a c c o r d l n g to the
1
l e x i c o g r a p h i c o r d e r i n g and let
8G = < ~i'
~2'''''~t
~i > ~2 > ~3
E
Fig.
i.
1
>' w h e r e
"'" > st"
E
The s e q u e n c e s 8 G induce a t o t a l ord e r i n g R a m o n g the graphs.
(We m a y
h o w e v e r ~ have~ 8 G = 8_ for two n o n i s o m o r p h i c g r a p h s G and ~).
T a k e J to be
a least g r a p h in this o r d e r i n g that
c a n n o t be r e p e a t e d a r b i t r a r i l y m a n y t i m e s
without v i o l a t i n g ~; i.e. there e x i s t s a
n u m b e r k > i, such that k i n d e p e n d e n t
c o p i e s of J (without any i n t e r c o n n e c t i n g
edges) v i o l a t e ~, k - i i n d e p e n d e n t c o p i e s
of J s a t i s f y ~, and any n u m b e r of ind e p e n d e n t c o p i e s of every H w i t h
Let G' be the r e s u l t i n g graph.
We w i l l show that ~o(G) ~ ~ <==> V (G') ~ nkA.
i).
Let V be a n o d e c o v e r for G, IVl < ~.
D e l e t e V from e a c h copy of G.
Every
c o n n e c t e d c o m p o n e n t of the r e s u l t i n g subg r a p h of G' is e i t h e r (a) a c o m p o n e n t J.
of J o t h e r than J1' or (b) a g r a p h formed
by t a k i n g one cop9 of Ji and s e v e r a l c o p i e s
of J0' d e l e t i n g e i t h e r c I or d from each
copy of J0 and a t t a c h i n g it by the o t h e r
n o d e (d or Cl) to n o d e c I of the copy of
Ji (see Fig. 2 for an e x a m p l e ) , or (c)
Jl - J0 (with c I d e l e t e d ) , or (d)
-{Cl,d) (or
~ em,
in case
the c o n n e c t e d c o m p o n e n t s of
that the c o r r e s p o n d i n g
d e l e t i o n s h a v e d i s c o n n e c t e d them.
However
this does not a f f e c t o u r a r g u m e n t s , since
as it w i l l b e c o m e
o b v i o u s in a minute,
the w o r s t - c a s e is w h e n t h e y are all
- 255 -
connected).
Thus the r e m a i n i n g g r a p h
can be r e g a r d e d as a s u b g r a p h of r e p e t i tions of the f o l l o w i n g g r a p h J*: j* has
t+ s-i c o m p o n e n t s , if s is the n u m b e r of
g r a p h s of the form (b) for the p o s s i b l e
c h o i c e s of the node (c I o r d) d e l e t e d
from each copy of J0: £hese are J 2 , J 3 . . . ,
Jt and the s g r a p h s J[, i = i) ..., s
of the form (b).
For e x a m p l e , if J is
the g r a p h of Fig. 2a) and the m a x i m u m
d e g r e e of G is 3, then J* is as s h o w n
in Fig. 2b.
be a s o l u t i o n to the n o d e - d e l e t i o n problem.
Let m be the n u m b e r of c o p i e s of
G) from w h i c h G ' - V c o n t a i n s J as an
i n d u c e d subgraph.
Since k independent
c o p i e s of J v i o l a t e ~ and since , is
h e r e d i t a r y on i n d u c e d s u b g r a p h s , m < k.
T h a t is, from at least (n-l)k+l c o p i e s
of G) G ' - V does not c o n t a i n J as an
i n d u c e d subgraph.
Let G:i' be such a c o p y
of G and d e f i n e V~ = [ u 6 N i i V c o n t a i n s a
l
n o d e from the copy of J' a t t a c h e d to
v ( p o s s i b l y v itself) or a n o d e from the
copy of J0 that r e p l a c e d and e d g e (v,u)
w i t h v < u (the o r d e r i n g of n o d e s is
arbitrary)].
C l e a r l y IV~I S I V D N i
S u p p o s e that there is an edge (v)u) of
G i such that v) u ~ V~.
T h e n V does not
1
c o n t a i n any n o d e from the c o p i e s of J'
a t t a c h e d to v and u) or from the c o p y
of J0 that r e p l a c e d (v,u) (since o t h e r w i s e the s m a l l e r of v)u w o u l d b e l o n g
to V~) c o n s e q u e n t l y (see Fig. i)
G: - I V N N.] c o n t a i n s J as an i n d u c e d
l
l
s u b g r a p h ( r e g a r d l e s s of h o w the n o d e s
c. and d w e r e i d e n t i f i e d to v and u).
T ~ e r e f o r e V~ is a n o d e c o v e r for G.
Thus
i.
V m u s t c o n t a l n at least Z+I n o d e s from
e a c h of (n-l)k+l c o p i e s of G, i.e.
IVI ~ [(n-l)k+l] (4+1) = n k 6 + Z + 2 + k ( n - l - Z )
~- v(G') > nk~, since n > ~+i.
O
A graph J
Fi@. 2a.
For all i) the c o m p o n e n t s of J~ r e l a t i v e
to c I are (a) t h o s e of J1 e x c e p t jn and
(b) j~ w i t h one of the n o d e s c ,d u
deleted.
S i n c e e a c h c o m p o n e n t l o f the
s e c o n d k i n d has o r d e r less than Ij01 ) the
c u t p o i n t c. g i v e s an ~ - s e q u e n c e for J~
w h i c h is l ~ x i c o g r a p h i c a l l y less than ~ h a t
of Jl) and c o n c e q u e n t l y ~j~ { ~Jl' for
all
C o r o l l a r y i.
The node d e l e t i o n p r o b l e m
r e s t r i c t e d to p l a n a r g r a p h s for g r a p h p r o p e r t i e s t h a t are h e r e d i t a r y on
i n d u c e d s u b g r a p h s ) n o n t r i v i a l and intere s t i n g on p l a n a r g r a p h s is N P - c o m p l e t e .
i.
Proof:
D'j
~3"Z
The c o r r e s p o n d i n g
Fig. 2b.
Therefore
8j, ~ 8j.
For e v e r y n, t h e r e is an r(n) (may take
for e x a m p l e r(n) = 4n)) such that all
p l a n a r g r a p h s w i t h r(n) or m o r e n o d e s
c o n t a i n an i n d e p e n d e n t set of n nodes.
S~ince ~ is i n t e r e s t i n g on p l a n a r graphs,
all i n d e p e n d e n t sets of n o d e s s a t i s f y ~.
T h e node c o v e r p r o b l e m r e s t r i c t e d to
p l a n a r g r a p h s is N P - c o m p l e t e [GJS]. N o w
note, that if the o r i g i n a l g r a p h G and
the g r a p h J d e f i n e d in the p r o o f of
T h e o r e m 1 are p l a n a r , and in a d d i t i o n
the two a t t a c h m e n t p o i n t s c.,d of JAu lie
on a c o m m o n face in an e m b e d d i n g of it
o n the plane, then the r e s u l t i n g g r a p h
G' is also planar.
S i n c e , is n o n t r i v i a l
on p l a n a r graphs, we can c a r r y t h r o u g h
the p r o o f of T h e o r e m 1 and find such a
p l a n a r g r a p h J.
M o r e o v e r we can c h o o s e
n o d e d to lie on a c o m m o n face w i t h c I.
J--3
g r a p h J*
(In o u r e x a m p l e
8j = < < 5 , 3 > , < 4 , 1 > , < 4 > >
and
8 , = < <4,4,4,3>, <4)4)4)3>,
< ~ , 4 , 4 , 3 > , <4,1>, <4>>.)
<4,4,4,3>,
By o u r c h o i c e of J) any n u m b e r of ind e p e n d e n t c o p i e s of J* s a t i s f i e s ~) and
by h e r e d i t a r i n e s s the r e m a i n i n g g r a p h does
so too.
T h e r e f o r e y(G') < n.k.i.
2).
Suppose
that
~o(G)
> I,+i, and
D
T h e o r e m 2.
The n o d e - d e l e t i o n p r o b l e m
r e s t r i c t e d to b i p a r t i t e g r a p h s for g r a p h p r o p e r t i e s that are h e r e d i t a r y on i n d u c e d
let V
- "256 -
consider ~ = 'complete bipartite'
If
G = (N,E) is a bipartite graph with
N = S U T a bipartition of the node set,
let E' = [(u,v) I uES, vET, (u,v) ~E~.
Then it is easy to see that 7,(G) = min
subgraphs, nontrivial on bipartite graphs,
and are satisfied by any independent set
of edges is NP-complete.
Proof:
[~ (G), ~ (G')]. R o w e v e r it can be shown
[y~] that°if property ~E has as its only
forbidden subgraph k + l independent edges,
with k > 2, then the corresponding
node-deletion problem remains NP-complete
even when restricted to bipartite graphs.
There are two cases to be considered.
Case i. All graphs whose components are
stars satisfy ~.
Since ~ is nontrivial
on bipartite graphs we can carry through
the proof of T h e o r e m 1 using as J the Rleast bipartite graph satisfying the
conditions stated there.
Since all graphs
whose components are stars satisfy n, Jl
is not a star.
(Note that the star SZ
has ~-sequence ~S~ = < i,i~...,I>,
and any connected graph H,
with
< ~
is itself a star Sr, with r ~ ~).
Co rpllary 3 [KD].
The node-deletion
p E o b l e m for the following properties
is NP-complete: n = i) planar, 2) outerplanar, 3) line-graph~ 4) chordal, 5) interval, 6) without cycles of specified
length A, 7) without cycles of length
< 4, 8) aegree-contrained with maximum
degree r ~ I, 9) acyclic (forest), I0)
bipartite, ii) comparability graph, 12)
complete bipartite.
Thus there is at least one more node in the
same set with c. in a bipartition of J0"
1
If we choose as d any such node, the graph
G' constructed in the proof of T h e o r e m 1
is bipartite.
Furthermore the restriction to planar
graphs for properties (2)-(12) and to
bipartite graphs for properties (1)-(9)
is also NP-complete.
Case 2. Some graph all of whose components
are stars does not satisfy ~. Then J is
connected~ i.e. has only one component Jl
and this is a star S..
Since all independent sets of edges satisfy ~, i ~ 2. We
distinguish two subcases depending on
whether any number of independent copies
of the graph shown in Fig. 3 satisfies
or not.
For these two subcases we
apply two different reductions from the
SAT-3 problem.
(For a description of the
reductions see [YI]) .
Regarding now digraph-properties note
that the first argument used in the proof
of T h e o r e m 1 does not hold in the case
of digraphs, i.e. it may be the case that
neither ~ n o r ~ is satisfied by an independent set of nodes.
T h e o r e m 3. The node-deletion problem
for nontrivial, interesting digraphproperties that are hereditary on induced
subgraphs is NP-complete.
4-z
V
Proof:
Using Ramsey's theorem we can show that
for all PI~ P2' P3 there is a number
r(Pl' P2' P3 ) such that all digraphs of
Fig.
3.
order at least r(Pl, P2' P3 ) contain as
c o r o l l a r y 2.. with the exception of the node
cover problem, the node-deletion problem
restricted to bipartite graphs for graphproperties that are h e r e d i t a r y on induced
subgraphs, nontrivial on bipartite graphs
and determined by the components is NPcomplete.
If some independent set of edges does
not satisfy ~, there are properties
(besides ~O = 'independent set of nodes')
for which the node-deletion problem becomes polynomial when restricted to bipartite graphs, and properties for which
it remains NP-complete.
For example
- 257 -
an induced subgraph either an independent
set of P1 nodes, or a complete symmetric
(abbreviated c.s.) digraph on P9 nodes,
or a complete antisymmetric transitive
(abbreviated c.a.t.) digraph on P3 nodes.
since w is interesting and hereditary
on induced subgraphs, it is satisfied
either (i) by all independent sets of
nodes, or (ii) by all c.s. digraphs, or
(iii) by all c.a.t, digraphs.
The proof
of T h e o r e m 1 works for cases (i) and
(ii) (in case (ii) the construction is
c a r r i e d out for ~).
It remains therefore
tO show the result for case (iii).
Let s be the largest n u m b e r such that
s i n d e p e n d e n t c.a.t, d i g r a p h s of any o r d e r
s a t i s f y n, i.e. there exist n u m b e r s k.
k2'" " " ' k s + l such that s + l independent'
c.a.t, d i g r a p h s of o r d e r k l , . . . ,k +i
violate ~.
(There exists such a ~u~tber
s if n is not s a t i s f i e d by all i n d e p e n d e n t
sets of nodes, and s >__ i).
Since ~ is
h e r e d i t a r y on i n d u c e d s u b g r a p h s there
exists a n u m b e r k such that s+l
i n d e p e n d e n t c.a.t, d i g r a p h s of o r d e r k
violate ~ (We can take for e x a m p l e k = m a x
v(G)
Given a graph G = (N,E), input to the
node cover problem, w i t h N = [ul,... ,u n]
E = [el,... ,e ], let r = m. (k-l).n.
Form
, m
,
,
a digraph D = (N ,E ) as follows:
N' = {uij I 1 < i < n, 1 < j ! r] U
I 1 <
i _< m,
(If S = 1 there
E' = ((ui~ , Ugh)
and
i < g);
Note
G, with
I J < h, or
and
(j = h a n d
.~E] U
I J < g or
i < f)}.
by s+l i n d e p e n d e n t nodes:
[uig,Ujg] U [efg h I 1 ~ h ~ s-l}
and the
addition
edges.
We c l a i m
< ~.
that
interconnecting
y(D')
< r.~ <
;~ S0(G )
i).
Let V be a n o d e c o v e r for G and V'
the set of the r copies of V.
D'-V'
consists of s i n d e p e n d e n t c o m p l e t e antis y m m e t r i c t r a n s i t i v e digraphs.
The s-i
of them have n o d e set N h = [eij h I
1 < i < m, 1 ~ j < r} and the s-th has
node s~t N = [u.~ I 1 < i < n, 1 < j < r,
.
" -u.
~ Vj.
Ss i n c e V1 3 is a -noae
cover, - - n o -two nodes of N
are coples of a d j a c e n t
nodes of G a n d S t h e r e f o r e <I~ > has the
s
above stated form.
Thus, S0(G ) ~ Z
y(D') < r.6.
Ujg,
since ~ is i n t e r e s t i n g on acyclic
d i g r a p h s we h a v e to c o n s i d e r o n l y cases
(i) and (iii).
For case (iii) the
d i g r a p h D' c o n s t r u c t e d in the p r o o f of
T h e o r e m 3 is c l e a r l y acyclic.
For case
(i), if in the c o n s t r u c t i o n of the p r o o f
of T h e o r e m 1 J is a c y c l i c and in the subs t i t u t i o n of e v e ~
edge (u,v), w i t h
u < v, by J , c I is i d e n t i f i e d w i t h u and
d w i t h v, t~e d i g r a p h G' c o n s t r u c t e d
there is also acyclic.
(Recall that as
we m e n t i o n e d there, the w a y that the
nodes c. and d of J0 are i d e n t i f i e d w i t h
I
the e n d p o i n t s of the edge is irrelevento).
Since , is n o n t r i v i a l on a c y c l i c digraphs,
j can be c h o s e n as the R - l e a s t a c y c l i c
d i g r a p h s a t i s f y i n g the c o n d i t i o n s stated
in the p r o o f of T h e o r e m i.
O
c o r o l l a r y 5.
The n o d e - d e l e t i o n p r o b l e m
for the f o l l o w i n g d i g r a p h - p r o p e r t i e s
is N P - c o m p l e t e : n = i) a c y c l i c (feedbacknode set), 2) t r a n s i t i v e , 3) symmetric,
4) a n t i s y m m e t r i c ,
5) l i n e - d i g r a p h , 6)
w i t h m a x i m u m o u t d e g r e e r, 7) w i t h m a x i m u m
i n d e g r e e r, 8) w i t h o u t cycles of length
~, 9) w i t h o u t cycles of length ~
6.
2).
S u p p o s e that So(G) > 2+1, and let
V be a s o l u t i o n to %he n ~ d e - d e l e t i o n problem.
For an edge ef = (ui,u j) of G,
let Kf = [g I Uig,
n > L+I.
Proof:
(j = g and
that D' is formed by r copies of
every edge ef = (ui,u j) r e p l a c e d
of some
since
C o r o l l a r y 4.
The n o d e - d e l e t i o n p r o b l e m
r e s t r i c t e d to a c y c l i c d i g r a p h s for digraph-properties
that are h e r e d i t a r y on
i n d u c e d subgraphs, n o n t r i v i a l and intere s t i n g on a c y c l i c d i g r a p h s is N P - c o m p l e t e .
1 < h _< s-l]
~re no eij k nodes)
(ui,Ug)
{ (eij h, efg h)
1 _< j _< r.
> r.Z,
The p r o o f of C o r o l l a r y 1 is v a l i d for
digraph-properties
too.
A l s o the result
of T h e o r e m 2 can be e x t e n d e d to d i g r a p h p r o p e r t i e s as well, a l t h o u g h there are
m o r e s u b c a s e s to be c o n s i d e r e d in case 2
a c c o r d i n g to the o r i e n t a t i o n s of the
edges.
{k l,k 2, .... k s + 1 }).
{eij h
h e r e d i t a r y on induced subgraphs, we
m u s t have .IKfl < k-l, if D ' - V is to
s a t i s f y ~.
~ h e ~ e f o r e from at least r-m.
(k-l) = (n-l)m(k-l) copies of G, there
is at least one of the s+l nodes that
r e p l a c e d each edge of G deleted.
Arguing as in the p r o o f of T h e o r e m 1 V m u s t
c o n t a i n at least 6+1 nodes from each of
these copies, i.e. IVl ~ (n-1)m(k-l) .
(~+i)
=
m. (k-l)
[n.t
+n-(~+l)]
- ~.
F u r t h e r m o r e the r e s t r i c t i o n of all
p r o b l e m s to p l a n a r digraphs, and the
r e s t r i c t i o n of p r o b l e m s 2,3,5,6,7
to a c y c l i c d i g r a p h s is also N P - c o m p l e t e .
efgl,...,efg,s_l
~V].
The nodes that r e p l a c e d edge e_ in
the copies of G w i t h index in Kf, fo~m
s+l i n d e p e n d e n t c o m p l e t e a n t i s y m m e t r i c
t r a n s i t i v e d i g r a p h s of o r d e r I K f ! . By
our choice of s and k and since n Is
3.
INCLUSION
MENT.
Theorem
- 258 -
4.
OF A OONNECTIVITY
The
connected
REQUIRE-
node-deletion
p r o b l e m for g r a p h - p r o p e r t i e s that are
h e r e d i t a r y on i n d u c e d (connected) graphs,
n o n t r i v i a l and i n t e r e s t i n g on c o n n e c t e d
g r a p h s is N P - c o m p l e t e .
c o n n e c t e d graphs).
Given a planar graph
G a m a x i m u m (induced) s u b g r a p h of G w i t h
c o n s i s t s of a n o d e v and a m a x i m u m ind e p e n d e n t set of its neighborhood F(v).
S i n c e G is planar, F(v) is o u t e r p l a n a r
for e a c h n o d e v.
S i n c e the m a x i m u m ind e p e n d e n t set of an o u t e r p l a n a r g r a p h
can be found in p o l y m o n i a l time, the same
is true for y~(G).
Proof:
It is e a s y to s h o w that for all Z,n,m,
there is a n u m b e r r(t,n~m) such that all
c o n n e c t e d g r a p h s of o r d e r at least
r(t,n,m) c o n t a i n as an i n d u c e d s u b g r a p h
e i t h e r a star Si or a c l i q u e K n or a p a t h
P of l e n g t h m.
S i n c e ~ is i n t e r e s t i n g
m
o n - c o n n e c t e d g r a p h s and h e r e d i t a r y e i t h e r
all cliques, or all stars or all p a t h s
s a t i s f y ~.
T h e o r e m 4 can be e x t e n d e d to d i g r a p h p r o p e r t i e s as well.
To case 1 (all
c l i q u e s s a t i s f y ~) there c o r r e s p o n d two
cases:(li) all c o m p l e t e s y m m e t r i c dig r a p h s s a t i s f y ~, and (lii) all c o m p l e t e
antisymmetric transitive digraphs satisfy
~.
In case (li) the p r o o f is as in case
1 of T h e o r e m 4.
In case (lii) the p r o o f
is as in case 2.
To case 2 there c o r r e s p o n d t h r e e cases a c c o r d i n g to the o r i e n t a t i o n s Qf the e d g e s o f the stars(Fig. 4
on!ast~
In all 3 cases the p r o o f goes
as in case 2 of T h e o r e m 4.
F i n a l l y if
n o n e of the p r e v i o u s c a s e s is true then
an i n f i n i t e n u m b e r of s e m i p a t h s s a t i s f i e s
~.
(A s e m i ~ t h
is a d i g r a p h w h o s e u n d e r lying g r a p h is a path.)
In this last
case we n e e d to k n o w for e a c h n at least
one g r a p h (for e x a m p l e a semipath) of
o r d e r n s a t i s f y i n g ~ (or at least be able
to g e n e r a t e such a g r a p h in p o l y n o m i a l
time).
U n d e r this last a s s u m p t i o n we
have :
Case I. A l l c l i q u e s s a t i s f y ~.
T h e n the
c o n s t r u c t i o n of G' in T h e o r e m 1 is c a r r i e d
o u t for ~.
If V* is a node c o v e r for G*,
G ' - V * is d i s c o n n e c t e d and c o n s e q u e n t l y
G ' - V * is connected.
C a s e 2.
A l l stars s a t i s f y ~.
Define a
p r o p e r t y ~' as follows: A (not
n e c e s s a r i l y connected) g r a p h G s a t i s f i e s
~' iff the g r a p h G 1 formed by t a k i n g a
n e w node and c o n n e c t i n g it to all n o d e s
of G s a t i s f i e s ~.
C l e a r l y ~' is nontrivial, i n t e r e s t i n g , h e r e d i t a r y on
i n d u c e d s u b g r a p h s and is s a t i s f i e d b y
any i n d e p e n d e n t set of nodes.
A p p l y the
c o n s t r u c t i o n of T h e o r e m 1 for ~'.
From
the r e s u l t i n g g r a p h G' c o n s t r u c t G" by
t a k i n g a n e w n o d e v and c o n n e c t i n g it
w i t h an edge to all n o d e s of G'.
We
c l a i m that yC(G") = ¥_.(G').
Obviously
the g r a p h f o ~ m e d by n~de v and any subg r a p h of G' s a t i s f y i n g ~' is c o n n e c t e d
and s a t i s f i e s ~, thus yC(G") < V~ (G').
T h e o r e m 5.
The connected node-deletion
p r o b l e m for d i g r a p h - p r o p e r t i e s that are
h e r e d i t a r y on i n d u c e d (connected) subgraphs, n o n t r i v i a l and i n t e r e s t i n g on
c o n n e c t e d d i g r a p h s is N P - c o m p l e t e .
C o r o l l a r y 6.
The c o n n e c t e d n o d e - d e l e t i o n
p r o b l e m r e s t r i c t e d to acyclic d i g r a p h s
for d i g r a p h - p r o p e r t i e s that are h e r e d i t a r y
on i n d u c e d (connected) subgraphs, nont r i v i a l and i n t e r e s t i n g on c o n n e c t e d
a c y c l i c d i g r a p h s is N P - c o m p l e t e .
For the o t h e r d i r e c t i o n s u p p o s e that the
o p t i m a l s o l u t i o n V to the (connected)
n o d e - d e l e t i o n ~ p r o b l e m c o n t a i n s node v.
T h e n V m u s t c o n t a i n all the n o d e s from
n k - i c o p i e s of G, if G " - V is to be
connected.
T h u s IVI k (nk-l) n+l.
Since
y , (G') < n k ~ 0(G) < n k ( n - l ) t h i s is
impossible.
T h e r e f o r e V does not c o n t a i n
node v and from the d e f i n i t i o n of ~',
Y~, (G') ~ Y~CG").
F r o m n o w on we w i l l c o n c e n t r a t e on
p r o p e r t i e s that are d e t e r m i n e d by the
b l o c k s of a g r a p h and w i l l not d i s t i n g u i s h
between graph-and digraph-properties.
For
such a p r o p e r t y W that is h e r e d i t a r y on
i n d u c e d s u b g r a p h s , the f o l l o w i n g are
equivalent:
(i)
~ is i n t e r e s t i n g on c o n n e c t e d g r a p h s
(2)
a s i n g l e e d g e s a t i s f i e s ~.
C a s e 3.
S o m e c l i q u e and some star do n o t
s a t i s f y ~.
T h e n all p a t h s h a v e to s a t i s y
~.
In this case we use a r e d u c t i o n from
the S A T - 3 problem. (see [Y2]),
[]
C o r o l l a r y 1 ( r e g a r d i n g the p l a n a r restriction) is no m o r e true if we include a c o n n e c t i v i t y r e q u i r e m e n t .
As an
e x a m p l e c o n s i d e r ~ = 'star' (~ is c l e a r l y
h e r e d i t a r y on c o n n e c t e d i n d u c e d subgraphs~
n o n t r i v i a l and i n t e r e s t i n g on p l a n a r
-
Lemma 1
If ~ is d e t e r m i n e d by the b l o c k s
and is h e r e d i t a r y on i n d u c e d s u b g r a p h s and
there e x i s t s a f o r b i d d e n b i c o n n e c t e d subg r a p h H 1 for ~ w i t h an e d g e e, w h o s e
d e l e t i o n r e s u l t s in a s i n g l y c o n n e c t e d
g r a p h that s a t i s f i e s w~ t h e n (i) The
2 S 9
-
lying truth a s s i g h m e n t for S' satisfies
at m o s t 2 literals in each clause.
F u r t h e r m o r e there is such a truth assignment that satisfies all n o n c o m p l e m e n t e d
variables.
(4) If S is not satisfiable,
then every truth a s s i g n m e n t for S' that
satisfies all n o n c o m p l e m e n t e d variables,
satisfies 3 literals in some clause.
a p p r o x i m a t i o n of the c o n n e c t e d n o d e - d e letion p r o b l e m n w i t h w o r s t - c a s e ratio
0(nl-6), for any ~ > 0, w i t h n the n u m b e r
of nodes is N P - c o m p l e t e , (2)
It is NPh a r d to decide w h e t h e r all largest ind u c e d subgraphs w i t h property ~, of a
given graph are connected.
Proof:
Lemma 3:
If ~ is n o n t r i v i a l on c o n n e c t e d
graphs, d e t e r m i n e d by the blocks and
h e r e d i t a r y on induced subgraphs and there
exists a graph H 2 w i t h at least 5 nodes,
three of w h i c h are d i s t i n g u i s h e d , such
that (i) d e l e t i o n of any d i s t i n g u i s h e d
node results in a g r a p h s a t i s f y i n g ~,
(ii) deletion of at m o s t 2 d i s t i n g u i s h e d
nodes does not d i s c o n n e c t the graph,
(iii) deletion of all 3 d i s t i n g u i s h e d
nodes d i s c o n n e c t s the graph, then the
a p p r o x i m a t i o n of the c o n n e c t e d noded e l e t i o n p r o b l e m ~ w i t h w o r s t - c a s e ratio
0(n l-t) is NP-complete.
(i)
The r e d u c t i o n is from the
SAT-3 ( s a t i s f i a b i l i t y w i t h 3
literals per clause) problem.
For each
clause we form a part of the graph w i t h one
node c o r r e s p o n d i n g to each literal, so
that for each clause at m o s t one such node
can remain if the graph is to s a t i s f y ~.
T h e n we take a cutpoint c of H.-e and
1
connect it to all nodes c o r r e s p o n d i n g to
v a r i a b l e s through copies o f the one
c o m p o n e n t of Hl-e relative to c and to the
nodes c o r r e s p o n d i n g to n e g a t i o n s o f variables through copies of the o t h e r c o m p o n e n t
of Hl-e relative to c. W e add an edge
b e t w e e n any two nodes that c o r r e s p o n d to
c o m p l e m e n t e d literals.
Let G 1 be a graph
with m nodes s a t i s f y i n g p r o p e r t y ~. A t t a c h
a copy of G 1 to all nodes of the p r e v i o u s
c o n s t r u c t i o n that do not c o r r e s p o n d to
literals.
For each clause connect a copy
o f G 1 to the rest of the graph by the 3
nodes c o r r e s p o n d i n g to the literals of
the clause and let G be the r e s u l t i n g
graph.
If the input set of clauses is
s a t i s f i a b l e vC(G) = 2P (with P the n u m b e r
of clauses) w~ereas if it is not satisfiable 7 c > m.
The result follows by
taking m ~ an a p p r o p r i a t e l y high (but
constant for fixed 4) p o w e r of P.
(2)
If the set of clauses is s a t i s f i a b l e
yC(G) = Y (G) = 2P and all s u b g r a p h s w i t h
p ~ o p e r t y ~ of G o b t a i n e d by d e l e t i n g that
few nodes are connected.
If the set is not
s a t i s f i a b l e y~(G) < y~(G).
[]
T h e o r e m 6.
If ~ is n o n t r i v i a l and int e r e s t i n g on c o n n e c t e d graphs, d e t e r m i n e d
by the blocks and h e r e d i t a r y on induced
subgraphs, then the a p p r o x i m a t i o n of the
connected node-deletion problem ~ with
w o r s t - c a s e ratio 0(n I-E) is NP-complete.
Proof:
By showing that for each p r o p e r t y
e i t h e r Lemma 1 or L e m m a 3 can be applied.
O
L e m m a 4:
If , is d e t e r m i n e d by the b l o c k s
of a graph and h e r e d i t a r y on induced subgraphs and there exists a graph H 3 w i t h
at least 4 nodes, three of w h i c h are distinguished, s a t i s f y i n g a s s u m p t i o n s (i)
and (ii) of L e m m a 3 and (iii) H 3 does
not satisfy ~, then it is NP- and co-NPh a r d to d e c i d e w h e t h e r all largest induced s u b g r a p h s w i t h p r o p e r t y ~ are
connected.
If a f o r b i d d e n s u b g r a p h H 1 as above does
not exist, then we w i l l have to connect
c o m p l e m e n t e d literals b y a forbidden subgraph.
But then to get c o n n e c t i v i t y we
will have to k e e p e x a c t l y one of these
two nodes.
proof:
Lemma 2.
G i v e n a set of clauses S =
[C1,..,Cp] w i t h v a r i a b l e s X l , . . . , x n and
e x a c t l y 3 literals per clause, there is
a n o t h e r set of clauses S' = [C~,...,C'
with variables Xl,...,Xn,Xn+l,..
,x r p'
with r and p' linear in p~ and at m o s t
3 literals per clause, such that: (i)
S is s a t i s f i a b l e iff S' is, (2) Each ~
variable, whose c o m p l e m e n t appears in S',
o c c u r s as m a n y times as its complement~
(3)
If S is satisfiable, then e v e r y satis-
W e use r e d u c t i o n s from the SAT-3 problem,
w h e r e we assume that the input set of
clauses consists of a clause c o n t a i n i n g
a single v a r i a b l e x and a set S of clauses
that is satisfiable.
(Note that cook's
o r i g i n a l r e d u c t i o n for the s a t i s f i a b i l i t y
p r o b l e m has e x a c t l y this form [C], if we
take for example as x the variable that
asserts that at the final time the T u r i n g
m a c h i n e is in an a c c e p t i n g state).
We
a p p l y the t r a n s f o r m a t i o n of Lemma 2 to
S and c o n s t r u c t a g r a p h J from the res u l t i n g set S' such that all m a x i m u m subgraphs w i t h ~ ~f J are connected, corresp o n d to a s a t i s f y i n g truth a s s i g h m e n t for
- 260 -
S' and k e e p all nodes
literals.
corresponding
to true
Proposition
Relative
to any set X,
p,X
For the NP-hardness proof we form a
graph as in Fig. 5 where c is a node con
tained in all m a x i m u m subgraphs with
of J and the node b is connected to all
nodes of the 2 copies of J corresponding
to x by copies of H 3.
u
p,X
Thus T h e o r e m 7 gives a class of natural
graph problems that testify to
~ ~ ~ NP U co-NP, in case that NP- ~ co-NP.
A n o t h e r such p r o b l e m is reported in
[Le] .
C o r o l l a r y 7: The conclusions of Theorems
6,7 hold for the following n o d e - d e l e t i o n
problems: planar, outerplanar, bipartite,
chordal, acyclic graph (tree)~ cactus,
acyclic digraph, symmetric, antisymmetric digraph~ without cycles of specified
length £, of gmy length ~ 2..
C
d
Fi~.
5.
For the co-NP-hardness
graph of Fig. 6, where
distinguished nodes of
connected to all nodes
to ~ by copies of H 3.
Fig.
4.
proof we form the
a, and b are
H_, and b is
o~ J corresponding
EDGE-DELETION
PROBLEMS
T h e o r e m 8.
The following edge-deletion
problems are NP-complete.
(i)
"without cycles of specified
length ~", for any fixed £ ~ 3,
(ii)
for even £, the same problem
restricted to bipartite graphs,
(iii)
"without any cycles of length
6" restricted to bipartite
graphs, for fixed £ ~ 4.
T h e o r e m 9.
The e d g e - d e l e t i o n "connected,
with m a x i m u m degreer" p r o b l e m is NPcomplete, for any fixed r ~ 2.
6.
T h e o r e m 7.
If ~ is nontrivial and interesting on connected graphs, determined by
the blocks and hereditary on induced subgraphs then it is NP-and co-NP hard to
decide whether all largest induced subgraphs satisfying property ~ are connected.
Theorem
planar"
i0.
The e d g e - d e l e t i o n "outerp r o b l e m is NP-complete.
Theorem
problem
ii.
The edge-deletion
is NP-complete.
Proof~
The reductions for the two last theorems
are from the H a m i l t o n i a n path problem
(with m a x i m u m degree 3 [GJS] for the
planar case) and are b a s e d on counting
arguments.
We take two copies of the
original graph and two new nodes that
we connect to all the nodes of the
original graphs.
We show that if the
original graph has a Hamiltonian path
then the new graph contains a maximal
o u t e r p l a n a r (resp. maximal planar minus
one edge) subgraph.
Conversely if there
is such a subgraph then e m b e d d i n g it on
the plane andusing properties of maximal
o u t e r p l a n a r (resp. planar) graphs we can
exhibit a H a m i l t o n i a n path of the original graph.
Proofs:
We show that Lemma 4 is applicable, unless
the triangle does not satisfy property ~.
In this case the N P - h a r d part is covered
by Lemma I, and we shall give a separate
proof for the co-NP h a r d part.
[]
The above problem
is easily
"planar"
seen to be
in
~ M (the set of languages recognizable
.
in polynomial time d e t e r m i n i s t i c a l l y by
a query machine with oracle from NP; for
the polynomial-time h i e r a r c h y see [S])
provided that ~ is in NP.
Thus T h e o r e m 7
shows that NP ~ co-NP ==-> ~ ~ NP U co-NP.
H o w e v e r this is not a peculzarity of the
u n r e l a t i v i s e d polynomial hierarchy, i.e.:
-
261-
Theorem ii has been independently
by Geldmacher and Liu.
shown
l
e3
In the next theorems we use a restricted version of the SAT-3 problem.
Lemma 5: The SAT-3 problem is NP-complete
even when each variable appears 3 times.
The requirements of the lemma are in a sense the best possible, since if each variable appears at most twice then theclauses
are trivially satisfiable (assuming that
each clause contains 2 or more literals).
Lemma 5 appears to be useful in proving
the NP-completeness of restricted
problems.
For example from it (rather
from the transformation used) and Karp's
reduction of the SAT-3 problem to the
node cover problem [K], follows a result
of [GJS]: that the node cover p r o b l e m on
graphs with m a x i m u m degree 3 is NP-complete. We use it to determine the best
possible bounds on the node degrees for
the next three theorems.
Theorem 12: The edge-deletion 'line
invertible' graph problem on graphs with
maximum degree 4 is NP-complete.
Fig. 7
Proof:
Remarks:
Given a set of clauses CI,...,C with
variables X I , . . . , X - as in LemmaP5 conS truct the ~ollowlng
"n
graph G = (V,E).
(i)
The restriction of the m a x i m u m
degree to 4 in the T h e o r e m 12 is
best possible, i.e. for graphs with
m a x i m u m degree 3 the p r o b l e m can be
solved in Dolynomial time.
The
algorithm consists in applying
successive transformations to the
input graph (keeping the m a x i m u m
degree 3) until a graph without
triagles results.
Then the p r o b l e m
is reducible to the line-cover
problem.
(2)
If a connectivity requirement is
included,
then the best bound can
be brought down to 3.
V = [ai,bill S i ~ n] U {dijiX i occurs
in Cj} U. [eiji Xi occurs in Cj} U
[d' i, e~i
1 ~ i ~ n] U [Cjll ~ j ~ p] U
{C'jlCj has 2 literals]
E = [(ai,bi)il S i ~ n] U [(ai,dij),
(ai,eik),
(d~, dij),
(e~, eik) i
dij~eik E V] U {(dij , Cj),
(dij'die),
(eij, eie)
(eij,Cj) '
i j ~ e; dij , die ,
eij, eie E V~ U {(Cj, C~)
3
I C[ E V]
3
"
T h e o r e m 13. The e d g e - d e l e t i o n "bipartite"
(simple max-cut) problem on cubic graphs
is NP-complete.
The proof uses Lemmas 2
and 5 (rather the transformations employed there).
The 'NP-completeness of
the simple max-cut problem (without the
restriction on the degrees) was shown in
[GJS], where there was
also m e n t i o n e d
the open status of the p r o b l e m on graphs
with restricted m a x i m u m degree.
The
NP-completeness of the w e i g h t e d version
(i.e. with the edges having weights) was
first shown in Karp's paper [K].
For example
if C 1 = Xl V X 2 V X3' C 2
X2 V X3, C3 = X1 V X 2 V X 3. the graph
G is an in Fig. 7. We show that G has a
line-invertible subgraph obtained by
deleting r edges, where r is the total
number of literal-occurences iff the
clauses are satisfiable.
In our example
the set of edges that are deleted
corresponding to the truth assignment
X 1 = i, X 2 = 0, X 3 = i, is shown with
heavy lines in the figure.
T h e o r e m 14.
The e d g e - d e l e t i o n 'comparability graph' p r o b l e m is NP-complete,
- 262 -
even on cubic graphs.
f e e d b a c k - n o d e set (or any o t h e r p r o b l e m
of the class), w h e r e a s the node cover
p r o b l e m can be e a s i l y a p p r o x i m a t e d within ratio 2, but also b e c a u s e it w o u l d
shed m o r e light into the n a t u r e of
N P - c o m p l e t e p r o b l e m s from the c o m b i n a t o r ial point of view and into their beh a v i o u r w i t h respect to a p p r o x i m a t i o n
algorithms.
T h e o r e m 15.
The e d g e - d e l e t i o n 'transitive-digraph' p r o b l e m is NP-complete.
For the proof of the two last theorems we
first m o d i t y the c o n s t r u c t i o n of T h e o r e m
13 to get a graph w i t h o u t t r i a n g l e s and
then reduce the simple m a x - c u t p r o b l e m to
them.
F i n a l l y we note that for all the above
p r o p e r t i e s the edge d e l e t i o n p r o b l e m rem a i n s N P - c o m p l e t e if we conclude a
c o n n e c t i v i t y requirement.
5.
ACKNOWLEDGEMENT
I w i s h to thank J. D.
U l l m a n for his e n c o u r a n g e m e n t , support,
the p a t i e n t and c a r e f u l reading of
various forms of the m a n u s c r i p t , as well
as m a n y s u g g e s t i o n s that improved the
p r e s e n t a t i o n of the material.
CONCLUSIONS
In S e c t i o n s 2 and 3 we saw h o w a set of
similar p r o b l e m s - the n o d e - d e l e t i o n
p r o b l e m s - can be a t t a c k e d in a s y s t e m a t i c
w a y to prove the N P - c o m p l e t e n e s s of all
the m e m b e r s of the set.
It w o u l d be intere s t i n g to find o t h e r classes of p r o b l e m s
for w h i c h a similar result holds.
In
p a r t i c u l a r it w o u l d be nice if the same
k i n d of t e c h n i q u e s could be a p p l i e d to
the e d g e - d e l e t i o n p r o b l e m s (of course
for an a p p r o p r i a t e l y r e s t r i c t e d class of
properties).
U n f o r t u n a t e l y we suspect
that this is not the case-the r e d u c t i o n s
we found for the p r o p e r t i e s c o n s i d e r e d in
S e c t i o n 4 do not seem to fall into a pattern.
A class of p r o b l e m s w h i c h seems
more likely to be a m e n a b l e to such a
t r e a t m e n t is the class of p o l y n o m i a l and
integer d i v i s i b i l i t y p r o b l e m s [PI,P2],
w h e r e m o s t of the N P - c o m p l e t e n e s s proofs
e m p l o y s i m i l a r reductions.
R e g a r d i n g the class of n o d e - d e l e t i o n
p r o b l e m s two q u e s t i o n s suggest themselves: i) H o w m u c h can we e x p a n d the
class of p r o p e r t i e s for w h i c h the
p r o b l e m remains N P - c o m p l e t e , 2) the
r e d u c t i o n schemes we d e s c r i b e d in
S e c t i o n 2 show that the n o d e - d e l e t i o n
p r o b l e m (without the c o n n e c t i v i t y
requirement) has at least as rich a
s t r u c t u r e (in the c o m b i n a t o r i a l sensesee also [ADP]) as the node cover problem.
It is an immediate c o r o l l a r y of the
proofs that any E - a p p r o x i m a t e a l g o r i t h m
for any of the n o d e - d e l e t i o n p r o b l e m s
could be u s e d to derive an E - a p p r o x i m a t e
a l g o r i t h m for the node cover problem.
W h a t can we say in the o t h e r direction,
and what are the i n t e r r e l a t i o n s h i p s
among the various p r o b l e m s in the class
with respect to their c o m b i n a t o r i a l
structure?
This is very interesting, in
v i e w of the fact that there is for
example no k n o w n a p p r o x i m a t i o n a l g o r i t h m
w i t h b o u n d e d w o r s t - c a s e ratio for the
- 263 -
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