Path parity and perfection

DISCRETE MATHEMATICS ELSEYIER

Discrete Mathematics

1651166 (1997)

233-252

Path parity and perfection Hazel Everett a, Celina M.H. de Figueiredob,*, Clkdia Linhares-Sales ‘J, FrkdQic Maffray d, Oscar Port0 e, Bruce A. Reedf a UniversitP du QuPbec ri MontrPal, Departement d’informutique, Montreal, QuP., Canada H3C‘ SPK ’ Uni~ersidade Federal do Rio de Janeiro, Instituto de Matemdtica, Caixa Postal 68530, 21944 Rio de Janeiro, RJ, Brazil. ’LSD2-IMAG, Grenoble, France d LSM-IMAG, BP 53, 38041 Grenoble Cedex 9, France ’PUC-Rio, Departamento de Engenharia El&rira, Rua Marquis de SZo Vicentr 225. Predio Cardeal Leme, 22453, Rio de Janeiro, RJ, Brazil ’C.N.R.S., Universit& Pierre et Marie Curie, Paris, France

Abstract Two nonadjacent vertices x and y in a graph G form an even pair if every induced path between them has an even number of edges. For a given pair {x,y} in a graph G, we denote by GAYthe graph obtained from G by contracting x and y. In 1982, Fonlupt and Uhry proved that if G is perfect then so is G,,. In 1987, Meyniel used this fact to prove that no minimal imperfect graph contains an even pair. In the last eight years, even pairs have become an important tool

for proving that certain classes of graphs are perfect and for designing optimization algorithms on special classes of perfect graphs. This paper surveys results of these types. It also discusses numerous related concepts including odd pairs.

1. Contracting

even pairs

Two nonadjacent

vertices x and y in a graph G form an even pair if every induced

path between them has an even number of edges. For a given pair {x, y} in a graph G, we denote by GXYthe graph obtained by deleting x and y and adding a new vertex xy adjacent to precisely those vertices of G --x - y which were adjacent to at least one of x or y in G. We say that GXYis obtained by contracting on {x, y}. The following two easy facts motivate our interest in this contraction operation.

* Corresponding author. E-mail [email protected]; ’ On leave from UFF, Niteroi, Brazil

partially

0012-365X/97/$17.00 Copyright PII SOOl2-365X(96)00174-4

Science B.V. All rights reserved

@ 1997 Elsevier

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Fact 1. Zf {x, y} is an even pair in G then the largest clique in GxY has the same size as the largest clique in G. Proof. No clique of G contains

both x and y. If C is a clique in G containing

neither

x nor y then C is also a clique in GXY. If C is a clique of G containing precisely one of x and y then C - x - y + xy is a clique in GXu of the same size as C. Thus the largest clique in G is no bigger than the largest clique in GXY. It remains to show the converse. Let C be a clique in GXY. If xy is not in C then C is also a clique in G. If xy is in C then every vertex in C - xy is adjacent to at least one of x or y in G. Recall that {x, y} is an even pair. Thus there cannot be two vertices a and b in C - xy such that a is adjacent to x and not to y but b is adjacent to y and not to x. It follows that one of x or y is adjacent to all of C - xy. So, there is a clique of G with the same size as C. Thus the largest clique in G is no smaller clique in GXY. 0

than the largest

Fact 2. Zf {x, y} is an even pair in G then the chromatic number of G is equal to

the chromatic number of G,,. Proof. To each colouring of GXY with k colours, there corresponds a colouring of G with k colours in which x and y receive the same colour (this is the colour which was given to xy, all other colours remain unchanged). Conversely, to any colouring of G in which x and y receive the same colour there corresponds a colouring of G,.. with the same number of colours. Now, consider a colouring of G in which x and y receive different colours. We can assume that x is assigned colour 1 and y is assigned colour 2. Let H be the bipartite graph induced by the vertices of colours 1 and 2. Since {x, y} is an even pair, x and y are in different components of H. We can obtain a new colouring by swapping colours 1 and 2 in the component of H containing x. In this new colouring, x and y have the same colour. It follows that there is a corresponding colouring of G,, with the same number of colours. The above remarks imply that G and GXYhave the same chromatic number. 17 We note that the proof of Fact 1 yields a simple procedure which given a largest clique in Gnr finds a largest clique in G. Similarly the proof of Fact 2 yields a simple procedure which given a k-colouring of GXY yields a k-colouring of G. As we shall see, these procedures can be used to develop fast algorithms for finding a largest clique and an optimal colouring in certain kinds of graphs. To illustrate the technique, we consider the sequence of graphs GO,. . . , Gj depicted in Fig. 1. For i < j - 1, Gi+i is obtained from Gi by contracting the even pair {Xi, yi}. As Gj is a clique, it is trivial to obtain an optimal colouring and find a largest clique within it. We can then work backwards to find a largest clique and an optimal colouring of GO. We note that the size of the largest clique in GO is equal to its chromatic number. We will often iteratively contract even pairs to obtain optimal colourings which use the same number of colours as there are vertices in a largest clique. This is one of the links between even pairs and

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235

fQyo- xlfyp_ xIyly2 Fig. 1. Sequence

perfect

graphs

(defined

below).

of even pair contractions.

Before discussing

this relationship

further,

we recall

some salient facts about perfect graphs. We use x(H) to denote the chromatic number of H and co(H) to denote the size of the largest clique in H. We use cc(H) to denote the size of the largest stable set in H and B(H) to denote the minimum number of cliques needed to cover H. Note that ‘x(H) = o(k) and 6(H) = x(H). In 1960, Berge introduced the notion of a perfect graph: a graph G is perfect if for each induced subgraph H of G we have x(H) = o(H). A graph is called minimal imperfect if it is not perfect but all its proper induced subgraphs are. Denote by C,, an induced cycle on n vertices. It is easy to see that for k at least 2, C’zk+t and Clk+t are minimal imperfect graphs; hence no perfect graph can contain them as induced subgraphs. No other minimal imperfect graphs have, to date, been found. When he first introduced two conjectures (see [3]):

perfect graphs, Berge made the following

Strong Perfect Graph Conjecture (SPGC). no C,,,t or C&f,, for k>,2.

A graph is perfect if and only if it contains

Weak Perfect Graph Conjecture (WPGC). plement is perfect.

A graph is perfect if and only if its com-

A graph is called Berge if it contains no &k+t or C2k+t, for k>2. Berge graphs are perfect is equivalent to proving the SPGC. In 1972, Lo&z proved the following theorem [35]: Lovisz’s

Theorem.

A graph G is perfi?ct if and only if o(H)a(H)

Proving

that all

3 1V(H)I, for u/l

induced subgruphs H of G. It follows

immediately

Perfect Graph Theorem.

that the WPGC

is true; so we have:

A graph is perfect if and only if its complement is perfect.

Thus, if G is perfect then for all induced subgraphs H of G we have cc(H) = B(H). The problems of computing a(G), o(G), x(G) and B(G) are in general NP-complete [ 171. Grbtschel et al. have shown that all these problems can be solved in polynomial time for perfect graphs [20]. Their algorithm is based on the ellipsoid method and is quite complicated. Research in perfect graph theory has centered around the SPGC,

H. Everett et al. IDiscrete Mathematics 1651166 (1997) 233-252

236

finding

fast combinatorial

algorithms

for optimizing

on perfect graphs, and developing

a polynomial-time recognition algorithm for the class of perfect graphs. The link between even pairs and perfect graphs is made clearer by the following two facts. Fact 3 (Fonlupt

and Uhry [16]). rf {x, y} IS an even pair in a perfect

graph G then

Gxr is perfect. Proof. Let {x, y} be an even pair in a perfect graph G. Consider

any induced subgraph

H of G,,. If xy is not a vertex of H then H is also an induced subgraph of G and hence X(H) = w(H). If xy is a vertex of H then let H’ be the induced subgraph of G with vertex set V(H) - xy +x + y. Since G is perfect, x(H’) = o(H’). So, by Facts 1 and 2, x(H) = w(H). Thus, GXY is perfect as required. 0 Fact 4 (Meyniel [38] and Bertschi tains an even pair.

and Reed [6]). No minimal

imperfect

graph con-

Proof. Assume the contrary and let x and y form an even pair in a minimal imperfect graph G. Consider a proper induced subgraph H of GXY. If H is also a subgraph of G then H is perfect. Otherwise there is a proper induced subgraph F of G such that H = FxY. We know that F is perfect thus, by Fact 3, so is H. So, every proper induced subgraph of GXYis perfect. Now, x( GXY)= x(G) > o(G) = o( G,,) and so GXY is minimal imperfect. By Lovisz’s Theorem we know that 1V(G)1 - 1 = a(G)o(G) and /VG,,)( a(G,,)).

- 1 =a(G,,)w(G,,). Thus 1 =(IV(G)lW(lWx,)I This implies that w(G) = 1, a contradiction. 0

- l>=dG)(4G)-

Note that the converse of Fact 3 is not true, as the first even pair contraction in Fig. 1 shows. Following Bertschi [5], we call a graph G even contractile if there is a sequence Ga = G, Gi, . . , Gj such that Gj is a clique, and, for i < j - 1, Gi+i is obtained from Gi via the contraction of an even pair. Again following Bertschi, we call a graph perfectly contractile if each of its induced subgraphs is even contractile. Facts 1 and 2 imply that if H is even contractile then x(H) = w(H). Thus, if G is perfectly contractile then G is perfect. It turns out that many classical classes of perfect graphs are perfectly contractile. In the next section, we discuss the proofs that various classes of graphs are perfectly contractile. The classes discussed include weakly triangulated graphs, Meyniel graphs and perfectly orderable graphs as well as subclasses of these classes such as triangulated graphs, comparability graphs, parity graphs and clique separable graphs. We also enumerate what we believe is a list of all the minimal nonperfectly contractile graphs. Proving that a class of graphs contains only perfectly contractile graphs not only proves that every graph in the class is perfect, it also suggests a natural combinatorial algorithm which will find optimal colourings and largest cliques for graphs in the

H. Everett et al. I Discrete Mathematics 1651166 (1997) 233-252

237

Fig. 2. Berge graph with no even pair

class. We simply need an efficient procedure which, given a graph G in the class, finds a sequence of even pair contractions which transforms G into a clique. In the next section, we will describe the classes we discuss.

fast algorithms

which find such a sequence

for some of

A naive innocent might hope to prove the SPGC by proving that every Berge graph is perfectly contractile. The Berge graph of Fig. 2 shows this approach is doomed to failure as it is not a clique and yet contains no even pair. In Section 4, we discuss various conjectures of a similar flavour, some of which are still open. In fact, it is the authors’ hope that the SPGC can be proved along these lines (although Rusu’s recent results, see [46,47] indicate that such a proof will not be straightforward). In the remainder of this section we give several definitions and prove a few useful facts about even pair contractions. The length of a path is the number of edges it contains. A path is odd if its length is odd and even otherwise. We use Pk to denote an induced path on k vertices and Ck to denote the induced cycle on k vertices. A triangle is a cycle of length three. We often confound a set of vertices with the induced subgraph on that set. For k at least 5, we denote by Dk a graph isomorphic to a cycle of length k with exactly one chord such that this chord forms a triangle with two edges of the cycle. A hole is an induced cycle of length a least five. An antihole is an induced subgraph isomorphic to the complement of a hole. We say x is a neighhour of y, or n sees y if xy is an edge of G. The neighbourhood of x, denoted by N(x) is the set of neighbours of x. If xy is not an edge of G then we say x misses y. We often use n to represent the number of vertices in a graph and m to represent the number of edges. An arc is an edge with an order on its endpoints. A directed graph consists of a set of vertices and a set of arcs. An orientation of a graph G is a directed graph obtained by choosing for each edge e of G one of the two possible arcs corresponding to e. Fact 5. IJ’{x, y} zs an even pair in u graph G which contains contains no odd holes.

no odd holes then G,,.

Proof. Consider an even pair {x, y} in a graph G. If GXY contains an odd hole H which does not contain xy then H is also an odd hole in G. If GXY contains an odd hole H with xy in H then one of H - xy +x or H - xy + Y is an odd hole in G as otherwise, H corresponds to an induced odd path between x and y in G. c1

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Fact 6. If {x, y} 1s an even pair in a graph G with no antiholes no antiholes Proof.

of length difSerent from

then GXY contains

six.

Let x and y form an even pair in a graph G which contains

no antiholes.

Note

that C, is isomorphic to C,. So, by Fact 5 to prove this fact we need only show that GXy contains no antihole of length seven or greater. Assume the contrary and let A be an antihole

of length seven or greater in GXy. Obviously, xy is a vertex of A. We enumerate the vertices of A as a0 = xy, al,. . . , ak in such a way that ai ‘misses aj if and only if i - j is congruent to +l or to - 1 modulo k + 1. Now obviously, in G, both x and y miss both al and ak, Furthermore, no vertex of A - a0 - al - ak can miss both x and y in G. So, we can split A - a0 -al - ak up into three sets X, Y and 2 such that every vertex in X sees x but not y, every vertex in Y sees y but not x, and every vertex in 2 sees both x and y. There is no edge from a vertex in X to a vertex in Y as otherwise there would be an induced path of length 3 from x to y in G. But now, it is easy to verify that there exists some i and j with 1
Similar to Fact 6, the details are left to the reader.

Fact 8. If x and y form Proof. Follows

2. Perfectly

0

an even pair in a Berge graph then GXY is a Berge graph.

from Facts 5 and 7.

contractile

then Gxr contains

??

graphs

We begin this section with proofs that various classes of graphs are perfectly contractile. Most of the classes in which we are interested are hereditary, i.e., every induced subgraph of a graph in the class is also in the class. Thus, we can apply the following simple lemma. Lemma 1. If A is a hereditary class of graphs and every graph in A is either a clique or contains an even pair whose contraction yields a graph in A then every graph in A is perfectly contractile. Proof. Since A is hereditary, we need only show that every graph in A is even contractile. Assume the contrary and let G be a smallest graph in A which is not even contractile. Clearly, G is not a clique so by assumption there is an even pair {x, y} in G such that GXy is in A. Now, by the minimal&y of G, there exists a sequence of graphs HO = GXY,HI,. . ,Hk such that Hk is a clique and for i between

H. Ewrrtt

1 and k, H; is obtained GO=G,G,=Ho,GZ=H

,,...,

et al. IDiscrete

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from Hi-1 by contracting an even pair. Now, the sequence Gk+,- -H k shows that G is even contractile, a contradiction.

In what follows we say a graph is nontrivial if it is not a clique. Note that in order to prove that a hereditary class of graphs contains

only perfectly

contractile graphs, it is not sufficient to prove that every nontrivial graph in the class has an even pair. We must show that each nontrivial graph in the class contains an even pair whose contraction

yields

a smaller

graph in the class. To drive this point

home, we consider the class of Meyniel graphs. A graph is Meyniel if every odd cycle of length at least five has two or more chords. Obviously, a graph is Meyniel if and only if it contains no Clk+l (k32) and no &k&i (k 32). Meyniel [37] proved that these graphs are perfect. Later [38], he showed that every such graph is either a clique or contains an even pair. However, there are nontrivial Meyniel graphs which contain no even pair whose contraction yields a Meyniel graph. One such graph can be obtained by substituting every vertex in a Cg by a pair of adjacent vertices. (This example was found by Sarkossian, and independently by Hougardy. A larger example was found by Bertshi and appears in his thesis.) Thus, we cannot hope to apply Lemma 1 directly to prove that Meyniel graphs are perfectly contractile. Hertz [23], sidestepped this problem by defining a slightly larger class of graphs to which he could apply the lemma. Specifically, a graph G is quasi-Meyniel if (i) it contains no &+I, and (ii) for some x of G, the chord of every &+I in G have x as an endpoint. Given a quasi-Meyniel graph G we call a vertex x, which is endpoint of every chord of a &+I, a tip of G. We shall say that every vertex of a Meyniel graph G is a tip of G. Note that every Meyniel graph is quasi-Meyniel. In addition, if G is quasi-Meyniel but not Meyniel, then G has at most two tips and if x is a tip of G, then G - x is Meyniel. Hertz proved that every nontrivial quasi-Meyniel graph contains an even pair whose contraction yields a quasi-Meyniel graph. Thus, by Lemma 1, quasi-Meyniel graphs are perfectly contractile. Actually,

Hertz proved the following

lemma.

Lemma 2. Let x he a tip of a quasi-Meyniel of G - x, and let y he a nonneighbour

graph

G which is not adjacent

of x maximizing

y _f?wm an even pair in G, G,,. is quasi-Meyniel,

IN(x)

n N(y)l.

to air’

Then x und

and xy is a tip CI~G,?..

Proof. Let G, x and y be as in the statement of the lemma. We show first that x and y form an even pair in G. Otherwise, there is an odd induced path P from x to y in G. We enumerate the vertices along P as po =x, ~1,. . , p2k+l = y. Note that pl sees p2 but not y. Thus by our choice of y there is a vertex z which sees x and y but not ~2. Now, z sees both endpoints of P but not all of it. Since z + P is not bipartite, it contains an induced odd cycle which must be a triangle since G is quasi-Meyniel. In

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fact, we can think of P as being the concatenation of a set of subpaths each of which forms an induced cycle with z. Each of these paths is either even or an edge. Since at least one but not all of these subpaths are edges, there is an edge subpath and a nonedge

subpath

with z induce

which have a common

a Dzk+l. Furthermore,

endpoint.

But then these two paths together

the chord of this DZk+l has as endpoints

an interior vertex of P. This contradicts

z and

our choice of x. Thus, x and y are indeed an

even pair. We now show that GXY is quasi-Meyniel

with tip xy. By Fact 5, GXY contains

odd hole so we need only show that it contains

no

no &+i

which does not have xy as an endpoint for its chord. Assume GXY does have such a subgraph D. Let d be the unique vertex of D adjacent to both endpoints of the chord. If xy = d then either one of D -xy+ y or D -xy fx is an odd cycle with one chord in G contradicting the fact that x is a tip, or there is a path of length three from x to y in G through the chord of D contradicting the fact that x and y form an even pair. So, we can assume that xy is not d. Now, D-d is an induced cycle C of GXY.If neither D-xy+y nor D-xyfx is an odd cycle in G contradicting the fact that x is a tip then C corresponds to a path P from x to y in G. Let w be the vertex on this path which has a common neighbour with x on but not w. P + d + z with tip x.

P. Now, by our choice of y, there is a vertex z which sees both x and y Using arguments similar to those in the above paragraph we can show that contains an odd cycle which contradicts the fact that G is quasi-Meyniel We omit the details. ??

Corollary 1. Every nontrivial quasi-Meyniel traction yields a new quasi-Meyniel graph.

graph contains an even pair whose con-

Proof. If G is Meyniel and not a clique then we can apply Lemma 2 to any vertex x of G with fewer than n - 1 neighbours and hence find the desired even pair. If G is quasi-Meyniel but not Meyniel then any tip must have fewer than n - 1 neighbours so again we can apply Lemma 2. 0 We turn now to another class of graphs, the weakly triangulated graphs. A graph is weakly triangulated if it contains no holes and no antiholes. Hayward [21] proved that every weakly triangulated graph is perfect. The key lemma in his proof was: Hayward’s Lemma. Zf C is a minimal cutset in a weakly triangulated graph and c is connected then every component of G - C contains a vertex adjacent to all of C. With the help of this lemma, Hoang and Maffray [28] proved that every weakly triangulated graph contains an even pair. Now, it is not true that contracting an even pair in a weakly triangulated graph always yields a new weakly triangulated graph. For instance a C2k is obtained by contracting the endpoints of a Pzk+t (ka3). In order to apply Lemma 1 to weakly triangulated graphs we first define a special kind of even pair whose contraction cannot create holes or antiholes. To wit, two vertices x and y

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H. Everett et al. I Discrete Mathematics 1651166 (1997) 233-252

in a graph G form a two pair if every induced path between them has two edges. It is not hard to prove, particularly given Facts 5 and 6, the following fact. Fact 9. Contracting triangulated Hayward

a two pair in a weakly

triangulated

graph yields a new Iraeakly

graph. et al. [22] proved:

Lemma 3. Every pair.

weakly

triangulated

graph

Combining Lemma 3, Fact 9 and Lemma graph is perfectly contractile.

which is not a clique contains

1 we see that every weakly

a taco

triangulated

We turn now to the class of perfectly orderable graphs. A perfect order on the vertices of a graph G is an order < such that there is no set of four vertices {w,x, y,z} with {WX,XY,_~z>C E(G), { wz,zx, wy} C E(G) and w < x,z < y. A graph is perf&ctl}, orderable if its vertex set permits a perfect order. This class of graphs was introduced by Chvital [9] who proved that all such graphs are perfect. Meyniel [38] later proved that every non-trivial perfectly orderable graph has an even pair (see also [24]). It is easy to extend Meyniel’s result to prove: Lemma 4. Every nontrivial perfectly orderable graph has an even pair whose con-. traction yields a new perfectly orderable graph. Proof. Let G be a perfectly orderable graph. Let < be a perfect order on the vertices of G. Let x be the first vertex in this ordering with less than n - 1 neighbours. Let y be the first vertex in the ordering not adjacent to x. Then {x, y} is called a minimal pair with respect to <. It is easy to see that every perfectly orderable graph which is not a clique contains a minimal pair. We shall show that any such pair is an even pair whose contraction yields a new perfectly orderable graph. So let {x, y} be the minimal pair with respect to <. We show first that {x, y} is an even pair. Suppose, by way of contradiction, that there is an odd induced path from pz,,,, pz,,,+l = y. Since p1 misses y, it does not have n - 1 x to Y, say PO =x,PI,..., neighbours and hence x < ~1. Since < is a perfect order, it follows that p2 < p3 and more generally p2j < pzj+i, for j between 1 and m. But since y was the first verte.x under the order missed by x, it follows that x must see pz,,,, a contradiction. Now, we claim that G,, is perfectly orderable. To see this we consider the order <’ defined as equalto < onG-x-yandsuchthatforzinG-x-y,wehavez<’xyifand only if z < x. The routine verification of the fact that <’ is a perfect order is left to the interested reader. The following three facts are all that are needed. (i) < is a perfect order, (ii) every vertex which comes before xy under < ’ has n - 2 neighbours in G,,., and 0 (iii) every vertex of G which misses x comes after y under <.

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graphs are perfectly

contractile.

Other classical classes of perfect graphs which have been proven to be perfectly contractile include the triangulated graphs [19] (which are contained in all three of the classes mentioned above), the comparability graphs [19] (which are perfectly orderable),

the parity and i-triangulated

separable

graphs [ 181. For an alternative

graphs [8] (which are Meyniel),

and the clique-

proof that these last three classes are perfectly

contractile see [5]. Having proved that all these graphs are perfectly contractile, we would like to design fast and simple optimization algorithms which take advantage of the contraction sequences to optimally colour and find maximum cliques within them. We first consider weakly triangulated graphs. We note that two nonadjacent vertices in a graph G form a two pair if and only if they are in different components of the graph obtained from G by deleting

the intersection

of their neighbourhoods.

We can test this condition

for

a particular pair of vertices in O(m + n) time. Thus, we can find a two pair in a graph which contains one in 0(n2m) time. It follows that for any weakly triangulated graph G we can find, in 0(n3m) time, a sequence of two pair contractions which reduce G to a clique. As described in the introduction, we can use such a sequence to find an optimal colouring and largest clique in G. In fact, given the sequence of contractions, this can be done in O(nm) time. In [22], Hayward et al. formally describe O(n3m) algorithms which solve the maximum clique and minimum colouring problems on weakly triangulated graphs in this manner. As the complement of a is weakly triangulated their algorithms can also be used to cover and maximum stable set problems on this class. They similar flavour to solve the weighted versions of these four

weakly triangulated graph solve the minimum clique developed algorithms of a problems in 0(n4m) time.

Arikati and Pandu Rangan [2] developed an O(nm) algorithm to find a two pair in a graph which has one. Using their algorithm yields a corresponding speedup in the optimization algorithms. Even pair contractions

can also be used as an optimization

tool in Meyniel

and

quasi-Meyniel graphs. Recall that Lemma 2 describes the even pair to be contracted in that case. Thus given a quasi-Meyniel graph G and a tip x of G which is not adjacent to all of G - x we can find, in O(m) time, a y such that G,, is a quasiMeyniel graph and xy is one of its tips (we simply chose a nonneighbour y of x maximizing IN(x) n N( y)l ). Actually, if x is adjacent to all of GXYthen G obviously must be Meyniel, so every vertex of G is a tip and if G is not a clique we can choose some tip z of G which is not adjacent to all of G - z. It follows that given a quasiMeyniel graph G which is not a clique and a tip x of G we can find, in O(m) time, an even pair {z, y} whose contraction yields a quasi-Meyniel graph GXY with tip zy. Recursively applying this procedure yields an O(nm) algorithm which given a quasiMeyniel graph G and a tip ,\: of G provides a sequence of even contractions reducing G to a clique. Since we can quickly find an optimum colouring and maximum clique of G by working through this sequence backwards, this yields an O(nm) algorithm for these two optimization problems given a quasi-Meyniel graph with a specified tip.

H. Everett

et al. I Discrete

Since every vertex of a Meyniel two optimization This algorithm would

Mathematics

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graph is a tip, this yields an O(nm)

algorithm

problems on Meyniel graphs. for optimizing on Meyniel graphs was first developed

like to informally

describe

how to extend

Hertz’s

algorithm

for the

by Hertz. We

so that it works

for all quasi-Meyniel graphs. Given a quasi-Meyniel graph, we just pick an arbitrary vertex and assume it is a tip. We then apply the algorithm which yields a sequence of nonadjacent vertex pair contractions which reduce G to a clique. Note that these contractions

may not be even pair contractions

because x may not have been a tip for

G. However, we can still attempt to work backwards through the sequence to find a clique C of G and a colouring of G with ]C] colours. If our backtracking procedure fails at some step, we actually find an odd path between two of the vertices we contract. Hertz’s

proof

suggests

an O(nm)

algorithm

to find an odd cycle with at most one

chord in the graph in which the contraction is being carried out. Actually, repeated applications of Hertz’s technique yield a cycle with at most one chord in G. Since G is quasi-Meyniel, this cycle has exactly one chord. Now, one of the two endpoints of this chord must be a tip of G so two further applications of the algorithm will yield an optimal colouring and maximum clique. No one, as of yet, has been able to develop similar optimization algorithms class of perfectly orderable graphs. The difficulty does not lie in determining pairs of vertices

in a perfectly

orderable

for the which

graph form even pairs, this can be done in

polynomial time. The problem is that it is NP-complete to determine if a graph is perfectly orderable [40]. Thus, once we have contracted on an even pair in a perfectly orderable graph we cannot check directly if the resultant graph is perfectly orderable. There may be a sophisticated way of finding quickly an even pair in a perfectly orderable graph whose contraction yields another perfectly orderable graph, but this problem seems hard. The algorithm for finding even pairs in perfectly orderable graphs works on a much larger class of graphs. A pe[f:fect orientation of a graph G is a choice of direction for each edge under which there are no four vertices {w,x, y,z} such that {~vY~ ,vz> C E(G), { wz,zx, wy} C E(c) and wx is directed towards x while ,VZ is directed towards y. A graph is perfectly orientable if it has a perfect orientation. We note that every perfect order corresponds to a perfect orientation, we simply direct each edge so that if xy is directed towards x then y
H. Eoeretr et al. IDiscrete Mathematics 1651166 (1997) 233-252

244

orientable graph G then under any orientation we can find vertices a and b of P such that the path P can be broken into a directed a to x path, a directed b to y path and a path from a to b in which every vertex is a source or a sink. This fact, which follows immediately from the definition, allows them to develop a dynamic programming method to check for the existence of odd and even induced paths between all pairs of vertices given a perfect orientation. We note that perfectly orientable graphs are not perfectly contractile because odd holes are perfectly orientable. Thus we cannot hope to apply Lemma 1 to perfectly orientable perfectly

graphs. There is however

a class of graphs between

perfectly

orientable

and

orderable

which seems like a natural candidate for our optimization technique. If {WXJY, YZ) C E(G), and { wz,zx, wy} C E(G) then we say the arc wx directly forces the arc yz. We say that wx forces yz if there is a sequence of arcs al =wx, a2,. . , ak=yz such that for i between 1 and k - 1, ai directly forces ai+,. A directed cycle in an orientation is forced if there is some arc xy of the orientation which forces every arc of the cycle. A strong perfect orientation is a perfect orientation which contains no forced cycle. A graph is strongly perfectly orientable if it permits a strong perfect orientation. We can easily check in polynomial time if a graph permits a strong perfect orientation. Clearly, every perfectly orderable graph is strongly perfectly orientable. It is also trivial to show that every strongly perfectly orientable graph is Berge. Reed conjectured that every strongly perfectly orientable graph is perfect and, more strongly, perfectly contractile. In fact, he conjectured that in every nontrivial strongly perfectly orientable graph there is an even pair whose contraction yields another strongly perfectly orientable

graph. If this is true, then the algorithm

for determining

if two vertices

form

an even pair in a perfectly orientable graph can be combined with the recognition algorithm for strongly perfectly orientable graphs to obtain a polynomial-time algorithm for optimally colouring and finding a largest clique in strongly perfectly orientable graphs. We hope that the reader’s

interest

in perfectly

contractile

graphs has been piqued

by the results discussed above. We close this section with some suggestions for future research directions. To begin we mention a few classical classes of perfect graphs which may be perfectly contractile. A graph is strongly perfect [4] if each of its induced subgraphs contains a stable set meeting all maximal cliques. It is easy to see that a graph is perfect if and only if each of its induced subgraphs contains a stable set meeting all maximum cliques. Thus, strongly perfect graphs are perfect. A graph in alternately orientable if it permits an orientation in which no induced cycle contains a directed path with two edges, i.e., each cycle alternates. Alternately orientable graphs were shown to be perfect by Hoang [25]. A star cutset is a cutset C containing a vertex x adjacent to all of C - x. A graph is in Bip* [lo] if each of its induced subgraphs either is Bipartite or contains a star cutset. Chvkal [lo] proved that no minimal imperfect graph contains a star cutset from which it follows that the graphs in Bip* are perfect. The following three questions are all open: ?? Are strongly perfect graphs perfectly contractile? ?? Are alternately orientable graphs perfectly contractile? ?? Are the graphs in Bip* perfectly contractile?

245

H. Everett et al. IDiscrete Mathematics 165/166 (1997) 233 -252

Fig. 3. Two odd stretchers.

Of course antiholes and graph, we mean an induced triangles and three vertex stretcher is odd if all these

odd holes are not perfectly contractile. By a stretcher in a subgraph whose edge set can be partitioned into two disjoint disjoint paths, each with an endpoint in both triangles. A

three paths are odd and ecen if all three paths are even. Two examples of odd stretchers are depicted in Fig. 3. Notice that a stretcher containing no odd hole is either odd or even. It is not hard to prove that odd stretchers are minimal not-perfectly contractile. A notable fact is: Any possible sequence of even contractions starting jiom an odd stretcher leads to c, itself (this fact needs some checking; see [33] for a formal proof). In 1993, Everett and Reed made the following Conjecture 1. A graph is perfectly contractile no odd holes and no odd stretchers.

eight conjectures

[43]:

if and only if it contains

no antiholes,

We say that G is a Grenoble graph if it contains no antiholes, no odd holes and no odd stretchers. We say that G is an Artemis graph if it contains no antiholes, no odd holes and no stretchers. We note that to prove Conjecture 1 we must show that all Grenoble graphs are perfectly contractile. Now, stretchers and antiholes are neither alternately orientable nor strongly perfectly orientable. Furthermore, stretchers are not in Bip* and odd stretchers are not strongly perfect. Thus, Conjecture 1 implies: Conjecture 2. If G is alternately orientable, strongly entable, or in Bip* then G is perfectly contractile.

perfect,

strongly

perfectly

ori-

The following weakening of Conjecture 1 would still imply that alternately orientable graphs, strongly perfect graphs, strongly perfectly orientable graphs, and the graphs in Bip* are perfectly contractile. Conjecture 3. Artemis We also believe:

graphs are perfectly

contractile.

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H. Everett

et al. IDiscrete

Mathematics

1651166 (1997) 233-252

Conjecture 4. Every nontrivial perfectly contractile contraction leaves the graph perfectly contractile. Conjecture 5. There is a polynomial-time contractile

algorithm

graph contains

an even pair whose

which given a nontrivial

graph G finds an even pair {x, y} of G such that GXYis perfectly

Since we believe Conjecture

1 we can make these last two conjectures

We say that an even pair {x, y} in a graph G is a strong

even pair

perfectly contractile.

more precise. if there is no

induced subgraph of G whose edge set can be partitioned into two vertex disjoint triangles and four vertex disjoint paths such that: two paths have an endpoint in each triangle, one has s as endpoint and an endpoint in a triangle, one has t as an endpoint and an endpoint in a triangle and one of s or t is not contained in the union of the two triangles (see Fig. 4 for two examples of the forbidden configuration). It is not hard to see that if {x, y} is an even pair in a Grenoble graph then GXY is a Grenoble graph if and only if {x, y} is a strong even pair. Thus the following conjecture implies both Conjectures 1 and 4. Conjecture 6. Every nontrivial Similarly,

the following

Grenoble

conjecture

We provide support for these conjectures conjecture

Conjecture 8. Every bull-free fectly contractile.

a strong even pair.

implies both Conjectures

Conjecture 7. There is a polynomial-time graph G returns a strong even pair.

a proof of the following

graph contains

algorithm

1 and 5.

which given a nontrivial

in the next section. In particular

Grenoble

we discuss

made by Everett and Reed in 1993 [43]:

graph containing

no antiholes

and no odd holes is per-

3. Recent progress In this section we give a survey of recent attempts to characterize perfectly contractile graphs. It is not known whether the characterization by forbidden induced subgraphs proposed in Conjecture 1 leads to a polynomial-time recognition algorithm for this class of graphs. Bienstock [7] proved that it is NP-complete to decide whether a graph admits an odd hole containing a specified vertex; he also established the co-NP-completeness of deciding whether a given graph admits an even pair. The SPGC has been established for many subclasses of graphs, including planar graphs [48], claw-free graphs [42], and bull-free graphs [13]. For each of these three subclasses of Berge graphs the corresponding recognition problem was solved, respectively, in [31, 14, 451.

t 0 cl__

H. Everett et al. I Discrete Mathematics 1651166 11997,i 233 -252

t

s

s

Fig. 4. Two forbidden

247

configurations.

Recent progress established Conjecture 1 for planar graphs [33], claw-free graphs [32] and bull-free graphs [ 151. In each case, the proof consists of a sequence of even pair contractions that turns an input graph with none of the forbidden induced subgraphs into a clique. In addition, in each case, a decomposition theorem leads to a polynomial-time algorithm that recognizes the subclass of perfectly contractile graphs. A little result turned out to be quite useful. When a graph G contains a clique-cutset C, and the components of G - C are denoted Br, . . . ,Bk (k 22), it is well-known that G is perfect if and only if all the induced subgraphs G[B, U C] are perfect 1191, The same holds for perfectly contractile graphs.

Indeed, each G[Bi U C] can be even-contacted into a clique as any even pair of G[B; UC] is an even pair of G. We then get a graph G’ in which any vertex of f sees all vertices of G - C and each component of G - C is a clique. Now G’ is triangulated and so perfectly contractile. 3.1. planar graphs We note that any antihole graphs Conjecture

Theorem 1 (Linhares-Sales

stretcher is per-e&

with at least eight vertices

1 reduces to the following

is not planar. Thus for planar

statement:

et al. [33]). A plannr graph with no odd hole and no odd

contractile.

The proof consists of a polynomial algorithm which finds an O(G)-coloring for G through a sequence of even pair contractions. We assume that the graph is drawn in the plane. Since G can be assumed to have no clique cutset by Lemma 5, every face of G is an even hole or a triangle. Then, every nontriangular face (if any) can be shown to contain an even pair at distance two along the face. The contraction of such a pait obviously keeps the graph planar; it also preserves the absence of odd holes and of odd

248

stretchers.

H. Everett et al. IDiscrete Mathematics 1651166 (1997) 233-252

Finally,

when every face of G is a triangle,

an argument

similar to that used

in [31] implies that G is a comparability graph and so is perfectly contractile [24]. The corresponding recognition problem - does a planar graph contain an odd hole or an odd stretcher?

-

can be solved by a revised

tree for planar perfect graph recognition 3.2. Claw-free

version

of Hsu’s decomposition

[31].

graphs

The claw is the graph K~,J and a claw-free graph is a graph having no induced claw. Chvatal and Sbihi [14] presented a polynomial-time algorithm for claw-free Berge graph recognition. They showed that a claw-free graph is perfect if and only if either it has a clique-cutset, or it is ‘elementary’, or it is ‘peculiar’. A graph is elementary if it can be edge-coloured with two colours in such a way that every chordless path on three vertices is bi-coloured. A graph G is peculiar if it can be constructed as follows: take a complete graph K whose set of vertices is split into six pairwise disjoint nonempty sets Al, B1, Al, Bz, A3, B3; for each i = 1,2,3 remove at least one edge between A, and B l+lmod3; add pairwise disjoint nonempty cliques K,,Kz, K3 and, for each i = 1,2,3, make each vertex in K, adjacent to all vertices in K - (A, U Bi). Now our problem of characterizing claw-free perfectly contractile graphs reduces by Lemma 5 to elementary graphs and to peculiar graphs. The latter case, namely, that every peculiar graph with no antihole is perfectly contractile, is a consequence of a stronger result of Chvatal [12] which states that all peculiar graphs with no antiholes are perfectly orderable. We note that Chvatal’s proof is such that it is possible to test in polynomial-time if a given peculiar graph contains an antihole and, if it does not, to build a perfect ordering. By the results in Section 2 we then find a sequence of even pair contractions

that turns G into a clique.

Finally, we are left with the case of elementary graphs. They can be dealt with using a decomposition theorem given in [36]. We note first that a result of Chvatal [ 121 states that all co-bipartite graphs with no antihole are perfectly orderable. Furthermore, Chvatal’s proof allows us to construct in polynomial time a perfect ordering given a perfectly orderable co-bipartite graph. Line-graphs of bipartite graphs are also easy to deal with. Any line-graph of bipartite graph with no odd stretcher contains a nice vertex, that is, a vertex whose neighbourhood is a stable set of size two. In a graph with no odd hole the two neighbours of a nice vertex are necessarily an even pair. Moreover, in a line-graph of bipartite graph with no odd stretcher the contraction of such a pair and the elimination of the nice vertex (now pendant) gives again a linegraph of bipartite graph with no odd stretcher. The decomposition theorem of [36] states that any elementary graph G is obtained from a line-graph of bipartite graph HG by replacing each vertex by a clique and each edge by a co-bipartite graph. Most of these co-bipartite graphs are complete but some independent set M of edges (i.e., A4 forms a matching) may be replaced by arbitrary co-bipartite graphs. This result allows us to find in G a set of even-pair contractions ‘near’ a nice vertex of HG which reduce G to another elementary graph, yielding:

H. Everett et al. I Discrete Mathematics 165/166 (1997)

24’)

233-252

Theorem 2 (Linhares-Sales and Maffray [32]). Euery claw-free graph with no odd hole, no antihole, and no odd stretcher is perfectly contractile. Cl In addition, the decomposition given in [36] is used in [32] to exhibit an odd stretcher in case the elementary graph is not perfectly contractile and its co-bipartite augments contain

no antihole.

3.3. Bull-free graphs A bull is a graph with five vertices a, b, c, d, e and five edges ab, bc, cd, be, ce, and a bull-free

graph is a graph having

no induced

bull. Bull-free

graphs are interesting

be-

cause they generalize Pd-free graphs and bipartite graphs. It was proved by Chvatal and Sbihi that all bull-free Berge graphs are perfect [13]. We note that any odd stretcher not isomorphic to c, contains a bull. Thus for bull-free graphs, Conjecture 1 reduces to the statement that every bull-free Berge graph with no antihole is perfectly contractile [ 151. This statement is a consequence of the following structural result. A homogeneous set in a graph G is a subset S of vertices such that every vertex in G - S sees either all or none of the vertices of S. Theorem 3 (de Figueiredo et al. [15]). Euery bull-free Berge graph with no antihole either is weakly triangulated, or contains a homogeneous set which is not a clique, or is perfectly orderable.

0

It is easy to see that if S is a homogeneous set then an even pair of the induced subgraph G[S] is an even pair of G, and that its contraction gives a Berge (resp. antihole-free) graph whenever G was. As a preliminary step, we eliminate all incomplete homogeneous sets from the graph, as by induction the algorithm obtains a sequence of even pair contractions that turns each homogeneous set H into a clique of size w(H). Moreover, the final graph where all homogeneous sets are reduced to a clique is also bull-free. Hence Theorem 3, and the fact that both weakly triangulated and perfectly orderable are subclasses of perfectly contractile graphs, yields the desired result. Moreover, we obtain again a polynomial-time algorithm which finds an (C)(G)colouring for G through a sequence of even pair contractions. Note that the validity of Conjecture I for bull-free graphs together with the existence of a polynomial-time algorithm for bull-free Berge graph recognition [45] gives a polynomial-time algorithm for bull-free perfectly contractile graphs.

recognition

4. Recent conjectures The study of Conjecture 1 has generated many results on perfect graph theory. According to Meyniel [38] a graph such that every nontrivial induced subgraph admits an even pair is called strict quasi-parity. A graph such that for each of its induced sub-

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H. Everett

et al. IDiscrete

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graphs H, either H or Z? is strict quasi-parity

1651166 (1997) 233-252

is called quasi-parity. In the case of bull-

free graphs, a new proof of the validity of the SPGC was obtained by proving that every bull-free Berge graph is quasi-parity [ 151. In the case of claw-free graphs, the structural description of elementary graphs also provided a new proof of the validity of the SPGC for all claw-free graphs [36], now a mere corollary of the I&rig-Hall theorem. Note that both for planar graphs and for claw-free

graphs, the polynomial-time

recog-

nition algorithm polynomial-time

for the corresponding subclass of Berge graphs actually provides a algorithm for testing for the existence of an odd hole. The same is

true for bull-free

graphs. Sbihi and Reed’s algorithm

bull-free

graphs in fact solves this problem

although

We conclude by mentioning some open problems the study of even pairs. A question, which appeared

[45] for testing the perfection

of

this is not stated in the paper. related to Conjecture 1 and to in the study of bull-free perfect

graphs, is: Conjecture 9. If every proper induced subgraph of G is perfectly has a homogeneous set, then G is perfectly contractile. We note that Conjecture 9 is implied by Conjecture 1. In the study of bull-free perfect graphs we also established conjecture

a particular

and G

case of a

of Chvatal:

Conjecture 10 (Chvatal fectly orderable. Hougardy

contractile,

proposed

[ll]).

Every

an analogue

bull-free

of Conjecture

Berge graph with no antihole

is per-

1 for the class of strict quasi-parity

graphs. Conjecture 11 (Hougardy [29]). Every minimal non-strict quasi-parity an odd hole, an antihole, or the line-graph of some bipartite graph.

graph is either

This conjecture has also been confirmed for the classes of planar graphs [34], clawfree graphs [32] and bull-free graphs. In particular, for bull-free graphs, the classes of perfectly contractile and strict quasi-parity coincide which confirms Conjecture 11 for bull-free graphs [ 151. The results cited give evidence for the following conjecture: Conjecture 12 (Reed [44]). Even pair testing is polynomial of perfect graphs.

when restricted to the class

The validity of the SPGC implies that a minimal imperfect graph does not contain two nonadjacent vertices such that all induced paths between them have the same length parity. Two vertices x, y of a graph G form an odd pair if all induced paths between x and y have an odd number of edges.

H. Everett et al. I Discrete Mathematics 165l166 (1947)

Conjecture 13 (Meyniel contains an odd pair.

and Olariu

251

233-252

[39] and Reed [44]). No minimal

imperfect

graph

Vertices x, y are said to be antitwins if every vertex distinct from x and y is adjacent to precisely no minimal no minimal vertices

one of them. Clearly, antitwins form an odd pair. Olariu [41] proved that imperfect graph contains antitwins. More recently, Hoang [27] proved that imperfect

graph contains

such that all chordless

a three pair, i.e., a pair of two non-adjacent

paths between

them contain

precisely

three edges.

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