AbstractPerfectly orderable graphs were introduced by Chvátal in 1984. Since then, several classes of perfectly orderable graphs have been identified. In this paper, we establish three new results on perfectly orderable graphs. First, we prove that every graph with Dilworth number at most three has a simplical vertex, in the graph or in its complement. Second, weprovide a characterization of graphs G with this property: each maximal vertex ofG is simplical in the complement of G. Finally, we introduce the notion of a locally perfect order and show that every arborescence-comparability graph admits a locally perfect order

Hoáng, C.T.

Mahadev, N.V.R.

Elsevier - Publisher Connector 

Discrete Mathematics 74 (1989) 77-84 
North-Holland 
77 
A NOTE ON PERFECT ORDERS 
C.T. HOANG 
Department sf Computer Science, Rutgers University, New Brunswick, New Jersey 08903, USA 
N.V.R. MAHADEV 
Department of Mathematics, Northeastern University, Boston, MA 02115, USA 
Perfectly orderable graphs were introduced by Chvfital in 1984. Since then, several classes of 
perfectly orderable graphs have been identified. In this paper, we establish three new results on 
perfectly orderable graphs. First, we prove that every graph with Dilworth number at most 
three has a simplicial vertex, in the graph or in its complement. Second, we provide a 
characterization of graphs G with this property: each maximal vertex of G is simplicial in the 
complement of G. Finally, we introduce the notion of a locally perfect order and show that 
every arborescence-comparability graph admits a locally perfect order. 
1. Introduction 
It is well known that the problem of determining the chromatic number x(G) 
of a graph G is NP-hard. However, the problem is polynomial13 solvable for 
many restricted classes of graphs. One such class of graphs is the class of 
“triangulated” graphs: a graph is triangulated if each of its cycles with more than 
three vertices contains a chord. A theorem of Dirac [6] asserts that every 
triangulated graph contains a simpliciul vertex, i.e. a vertex whose neighbourhood 
forms a clique. It follows from Dirac’s theorem that one can order the vertices of 
a triangulated graph G into a sequence vl, v2, . . . , v, such that each TJ~ is 
simplicial in the subgraph Gi of G induced by { ur, u2, . . . , Vi}; such a sequence is 
called a simplicial order of G. 
Now, an optimal colouring of G can be found by applying the following 
“greedy” algorithm: 
(1) Scan the vertices of G in the order given by the sequence vl, v2, . . . , v, 
(2) Assign to each vi the smallest positive integer assigned to no neighbour Vj 
of vui with j < i. 
To see that the greedy procedure produces an cptimal colouring of the 
triangulated graph G, one only needs observe that if vi is assigned integer k, then 
ui belongs to a clique of size k: vi is adjacent to at least k - 1 vertices in 
1 211, V2,’ l l 9 Vi} with colours 1,2, . . . , k - 1. Since the order is simplicial, it 
follows that l/i and its neighbours in {V 1, v2, . . . , Vi} form a clique of size 
precisely k. 
0012~365X/89/$3.50 @ 1989, Elsevier Science Publishers B.V. (North-Holland) 
78 C. T. H&&g, N. V.R. Mahadev 
The above remark shows that for a triangulated graph G we have x(G) = w(G) 
(the number of vertices in a largest clique of G). Thus, triangulated graphs are 
perfect in the sense detied by Berge [2]: a graph G is perfect if X(H) = w(H) for 
each induced subgraph H of 6. 
We are interested in studying graphs G with the following property: (*) G 
admits an order < such that the greedy algorithm, when applied to any induced 
subgraph H of G, produces an optimal colouring of H. The above property was 
first studied by Chvatal [3], and he proved that (*) holds for an ordered graph 
(6, C) if and only if no chordless path with vertices a, b, c, d and edges ab, bc, 
cd has a <b and d <c (this ordered path is called an obstruction). Chvatal 
proposed to call the order < a perfect order on G, and to call G perfectly 
orderable. Chvatal also proved that every perfectly orderable graph G is strongly 
peqect in the sense of Berge and Duchet [l]: each induced subgraph H of G 
contains a stable set that meets all maximal cliques of %I. In this paper, we shall 
study several interesting properties of some classes of perfectly orderable graphs. 
2. Good orders 
Let us define an order u1 < u2 c l l l <v, on a graph G to be good if for any 
induced subgraph H of G, either the largest vertex of (H, C) is simplicial or the 
smallest vertex of (H, 0 is simplicial in fi. A graph is good if it admits a good 
order. (If x is simplicial in G, then we shall call x a cosimpkial vertex of G.) 
Remark 2.1. Euery good order is a perfect order. 
Proof. By induction on the number of vertices. Let P = vu1 c v2 < v2- l l < v, be a 
good order of a graph G. If 21, is cosimplicial in G then no obstruction in P can 
contain v,; by induction v2 c l l l < v, is perfect, and so P is perfect. Similarly, 
if v, is simplicial in G, then no obstruction of P can contain v,; but by induction 
v,<===QJ,._, is perfect, and so P is perfect. Cl 
Chvatal defined a graph G to be brittle if each induced subgraph H of G 
contains a vertex that is not the endpoint or the midpoint of any Ps (the chordless 
path on four vertices) of Zf. It is easy to see that good graphs are brittle. At 
present no characterization (by minimal forbidden induced subgraphs) of brittle 
graphs (and good graphs) is known. But by Dirac’s theorem, we know that 
triangulated graphs and their complements are good and therefore brittle. The 
purpose of this section is to present a new class of good graphs. 
Let G be a graph and let X, y be two vertices of G. It is said that x dominates y 
if {x} U N(x) 2 N(y)(N(t) stands for the set of neighbours of t); if x dominates y 
or y dominates x then it is said that x and y are compw-able. Now, the Dilworth 
Perfect orders 79 
number of a graph is the largest number of pair-wise incomparable vertices of the 
graph. (Note that the domination relation is a transitive order.) 
Preissmann first proved that graphs with Dilworth number at most three are 
strongly perfect. Later, it is proved by Chvatal, Hoang, lVIahadev and deWerra 
[4] that these graphs are perfectly orderable. Now the following theorem shows 
that graphs with Dilworth number at most three admit good orders. 
Theorena 2.2. If G is a graph with Dilworth ntRmber at most three, then either G 
contains a simplicial vertex or G contains a cosimplicial vertex. 
Proof. Let G be a graph with Dilworth number at most three. Then by 
Dilworth’s theorem [5], the vertex set of G can be partitioned into three sets X0, 
X1, X2 (one or more of the Xi’s may be empty) such that the vertices of each Xi 
can be linearly ordered by <i SO that a Ci b only if a is dominated by b in G. 
For each nonempty Xi, define (with respect to <i) 
(i) ai to be the largest vertex in Xi, 
(ii) ci to be the smallest vertex in Xi, 
(iii) 6, to be the smallest vertex in Xi with the property that all vertices (of Xi) 
larger than bi form a clique. 
Note that within each Xi, no vertex smaller than bi ir adjacent to bi and that the 
vertices smaller than bi form a stable set. Also the vertices ai, bi, ci need not 
be distinct. 
Let {i, i, k} = (0, I, 2). If Ci is not adjacent to any vertex outside Xi then ci is 
simplicial. Thus, we may assume that Ci is adjacent to some vertex in Xi. If either 
Xk is empty, or ck is adjacent to some vertex in Xi, then clearly aj is a cosimplicial 
vertex of G. 
From the above remark, we may assume now that each c, has a neighbour in 
XI+* and no neighbour in XI+2 (obviously the subscripts t are taken modulo 3). 
Now, if some c, is not joined to any vertex smaller than b,, , in Xl+, , then clearly 
c, is a simplicial vertex. Hence we may assume that each c, is adjacent to some 
vertex smaller than b,+l in X,+, . Now, we claim that each a, is a cosimplicial 
vertex of G. To justify our claim, note that c~+~ is adjacent to some vertex in Xt; 
therefore at is adjacent to c,+~ and hence to all vertices of Xt+2. Clearly, each 
vertex nonadjacent to a, is either smaller than b,, or b,+!. Thus a, is cosimplicial 
as claimed. Cl 
Let 6 be a graph with Dilworth number at most three. Let n and m be the 
number of vertices, and the number of edges of G, respectively. Suppose we are 
given a partition of G into sets X0, X1, X2 as described in the proof of Theorem 
2.2. Then we can construct a good order of G in O(n . m) steps as follows. First 
find a vertex x which is either simplicial or cosimplicial, by testing all six vertices 
a- 1) ci (this takes O(m) steps: clearly checking if a vertex is simplicial or 
80 C. T. Ho@, N. V.R. Mahadev 
cosimplicial takes O(m) steps, and we have to check only six vertices of G). Now, 
recursively construct a good order x1 < l l l < x~_~ of G - X. If x is simplicial then 
return the good order nl < l l l <x,,+ <x, otherwise return the good order 
X <X* c l l l C&-p 
Perhaps we should remark that the Dilworth number of a graph can be 
determined in O(n3) steps. Given a graph G, we construct a graph D(G) as 
follows. The vertices of D(G) are the vertices of G and two vertices are adjacent 
in D(G) if and only if one of them dominates the other in G. Clearly, D(G) can 
be constructed in O(n”) steps (testing if a vertex x dominates a vertex y takes 
O(n) steps, and we have to perform O(n”) such tests). Note that D(G) is a 
“comparability” graph (definition will be given later). Also, the Dilworth number 
of G is the “stability” number of D(G) (the stability number of a graph is the 
largest number of pairwise nonadjacent vertices of the graph); and a partition of 
the vertices of D(G) into a smallest number of cliques gives a partition of the 
vertices of G into a smallest number of sets such that the vertices in each set are 
pairwise comparable. Finally, it is well known that the stability number of a 
comparability graph and a partition of its vertices into a smallest number of 
cliques, can be determined in 0(n3) steps, by solving certain network flow 
problem (see [s]). 
3. Maximal vertices 
Define a vertex x of a graph G to be maximal if x is not strictly dominated by 
any vertex of G, i.e. for each vertex y we have either N(x) - N(y) - y # 0 or else 
N(y) - N(x) -x = ia. 
We are interested in the following question: (**) what graphs G have the 
properties that each maximal vertex of G is a cosimplicial vertex? This question 
arises from the proof of Theorem 2.2: one may ask if it is true that for a graph 
with Dilworth number at most three, each maximal vertex is cosimplicial, and 
each minimal vertex (defined in the obvious way) is simplicial. (The answer is 
“no”: consider the graph 2K2, i.e. the union of two disjoint edges.) Fortunately, 
it turns out that there is a simple characterization of the graphs described in (**) 
(in terms of minimal forbidden induced subgraphs). Note that these graphs must 
be complements of triangulated graphs. 
As usual we shall let Pk (respectively C,) denote the chordless path (respec- 
tively cycle) on k vertices. Define the graph 4 (respectively F2) to be the graph 
obtained from a P7 with vertices I.J~, v2, . . . , u7 by adding the edge v1 v3 
(respectively, the edges 24~~ and v7v5). 
Theorem 3.1. A graph G has the property that each maximal vertex is cosimplicial 
if and only if G contains no induced subgraph isomorphic to any of the following 
graphs: 2&, C5, C6, C7, P;, F;, I$. 
Perfect orders 81 
Proof’. The “only if’ part is obvious. To prove the “if” part, suppose that x is a 
maximal vertex of G that is not cosimplicial. This means that there are adjacent 
vertices a, b such that both a and b are not adjacent to X. For simplicity, let us say 
that a vertex r sees (respectively misses) a vertex s if r is adjacent (respectively 
nonadjacent) to s. 
Since x is maximal, there is a vertex a’ that sees_ z _lbut eked 6. I’o avoid 
having a 2K2, a must see a’. Similarly,-- %CI~ is a vertex 6’ that sees X, misses a, . 
and sees b. To avoid having a C5, a’ must see b ‘. Now, since x is maximal, there 
must be a vertex a’ that sees x and misses a’; similarly, there is a vertex b’ that 
sees x and misses b ‘. 
Next, we claim that a’ sees b’; otherwise, we would have a c6 (if a” sees a, b), 
or a C, (if a’ sees a and misses b, or if an misses a and sees b),or a 2K, (if a’ 
misses both a and b). By the same reasoning, we kno-w that a’ sees b’. 
Now a must see a’ or else the set {a’, x, a’, a, b} induces a C, or it contains a 
2K2. Similarly, b must see b’ or else the set {b’, x, b’, b, a} induces a C, or it 
contains a 2&. 
Finally, note that all edges in the subgraph H of G induced by 
{x, a, a’, a’, b, b’, 6”) are determined except for the folio-wing edges ba’, ab”, 
a’b” which may or may not be present in _H. If a’b’ is not an edge, then H 
contains the complement of a chordless cycle with five, or six, or seven vertices. If 
a’b” is an edge, then H contains the complement of a graph isomorphic to a P,, 
or the graph 4 with i = 1 or 2. Cl 
Let C be a colouring of a graph G, and let x be a vertex of G. Define d;(x, C) to 
be the number of colours (of C) appearing in the neighbourhood 1’!(x) of x. 
Preissmann [lo] defined a colouring C of a graph G to be ioealiy perfect if 
F(x, C) = o@/(x)) for each vertex x of 5. A graph is called locally perfect if each 
induced subgraph admits a locally perfect colourkg. Preissman proved that 
locally perfect graphs are perfect and that triangulated graphs are locally perfect. 
Hertz [9] proved that a graph is locally perfect if each of its odd cycles with at 
least five vertices contains two chords. + 
We propose to call an order < on a graph G locally per&t if the greedy 
4. Local perfection 
algorithm, applied to any induced subgraph (H, C) of (G, <), produces a locally 
perfect colouring of H. Qbserve that a locally perfect order can not contain an 
obstruction; but (as we shall see later) a perfect order is not necessarily locally 
perfect. This observation led us to study locally perfect orders that are defined in 
terms of ordered P3’s. 
Let (H, <) be an ordered P3 with vertices a, b, c and edges ab, bc. We shall say 
that(H,<)isoftypelifa<b<c, (H,<)isoftype2ifa<b,c<b, (H,<)is 
of type 3 if b <a, b < c. Any ordered P3 is of one of the above three types. Let S 
82 CT. Hoiing, N. V. R. Mahadev 
be a subset of (1,2,3); define + to be an order such that every ordered P3 is of 
one of the types belonging to S. Call S admissible if and only if any order es (if it 
exists) on any graph G is locally perfect. 
Theorem 4.1. A set S is admissible if and only if S = { 2) or S = {3}. 
Before proving Theorem 4.1, perhaps we should remark that an order is 
usually called trunMve if it contains no P3 of type 1, and a graph is called a 
compurubility graph if it admits a transitive order. In her pioneering paper on 
locally perfect graphs, Preissmann constructed a comparability graph that is not 
locally perfect (see Fig. 3 in [lo]). In our terms, she showed that the set (2,3) is 
not admissible. 
Note that the order < {3) is both transitive and simplicial. Wolk [ll] proved that 
a graph G admits a transitive and simplicial order if and only if G contains no C, 
and no P4. Such graphs are called “arborescence-comparability” graphs [7]. 
Call a graph locally perfectly orderable if it admits a locally perfect order. 
Theorem 4.1 shows that arborescence-comparability graphs are locally perfectly 
orderable. Since these graphs are triangulated graphs, it is natural to ask whether 
all triangulated graphs are locally perfectly orderable. At present we do not know 
the answer to the above question. Perhaps a more interesting problem is to 
characterize locally perfectly ordered graphs by minimal forbidden ordered 
subgraphs. Now, to conclude our paper, we shall prove Theorem 4.1. 
Proof of Theorem 4.1. We remarked previously that the set {2,3} is not 
admissible. Hence, to prove the “only if” part of the theorem, we only need show 
that if 1 E S then S is not admissible. For this purpose, consider a graph with 
seven vertices 1,2, . . . ,7 and with exactly three maximal cliques: { 1,2,3,4}, 
{4,5,6}, and {5,6,7}. Now order the vertices as 1 < 2C - l l < 7. Clearly, every 
P3 is of type 1, but this order is not locally perfect. 
To prove the “if” part, let (G, Cs) be an ordered graph with S = (2) or 
S = (3). For simplicity, we shall let c denote cs. Note that < is a transitive 
order. For each vertex x we shall let A(x) denote the set of neighbours y of x with 
y <x, and let B(x) = N(x) - A(x). 
Lemma 4.2. Zf a vertex x is assigned integer k then A(x) contains a clique C with 
k - 1 vertices uch that the vertices of C are assigned integers 1,2, . . . , k - 1. 
The above lemma is a special case of a theorem of ChvBtal [3]. For the 
sake of completeness, we shall use ChvBtal’s idea to give a short proof of this 
lemma here. Let i be the smallest integer such that there are vertices v(i), 
v(i + l), . . . , v(k - 1) in A(x) satisfying the following properties: (i) v(i) < 
v(i+l)+==iv(k- l), (ii) each v(j) is assigned integer j (i sj s k - l), (iii) 
the vertices v(i), v(i + l), . . . , v(k - 1) form a clique. If i = 1 then we are done. 
Peflect orders 83 
If i > 1 then (by our way or colouring) v(i) has a neighbour v(i - 1) with assigned 
integer i - 1 such that v(i - 1) c v(i). But by transitivity, v(i - 1) is adjacent to x 
and all vertices v(j) (i 6 j s k - 1), contradicting our choice of i. Hence, the 
Lemma holds. (Incidentally, note that Lemma shows that the greedy algorithm 
uses precisely o(G) colours.) 
Now, to prove Theorem 4.1, we shall show that for each X, we have 
F(x, C) = o@!(a)). For this purpose, let k be the integer assigned to X, and let y 
be a neighbour of x with the largest (assigned) integer k’. Note that by our way of 
colouring, we have k - 1 s F(x, C); furthermore if k’ > k then F(x, C) s k’ - 1 
(because no neighbour of x can receive integer k), and if k’ <k then 
F(x, C) = k - 1 (because k’ 2 k - 1). 
If k’ <k then by Lemma 4.2 we have o(N(x)) = k - I= F(x, C). 
Thus we may assume that k’ > k. Now, observe that no vertex t of A(x) can be 
assigned an integer rt > k, otherwise in A(t) there is a vertex z with integer k. But 
then the ordered P,(s, t, x) is of type 1, a contradiction. 
From the above observation, we may assume that x < y. If S = (3) then A(y) is 
a clique (the order is simplicial); thus we have F(x, C) = k’ - 1 = IA(y)1 = 
Now, we may assume that S = (2). By our previous observation, in A(x), only 
integers m <k can appear, and thus by Lemma 4-2 lo(A(x))l = k - 1. Next, 
observe that (i) t sees z whenever t E A(x), z E B(x) and (ii) B(x) is a clique (if (i) 
fails then G has a P3 of type 1 f and if (ii) fails then G has a P3 of type 3). Hence, 
iI== F(x, C). Cl we have o(N(x)) = (k - 1) + IBW 
Acknowledgement 
We would like to thank an anonymous referee for pointing out several errors in 
our original manuscript. 
Refereuces 
PI 
PI 
PI 
PI 
PI 
C. Berge and P. Duchet, Strongly perfect graphs, in Berge, C. and V. Chvdtal, Ed., Topics on 
Perfect Graphs (North-Holland, Amsterdam, 1984) 57-61. 
C. Berge, F&bung von Graphen, deren s8;mtliche bzw. deren ungerade Kreise starr sind 
(Zusammenfassung). Math.-Natur. Reihe 114 (Wiss. Z. Martin-Luther-Univ., Halle- 
Wintenberg, 1961). 
V. Chvatal, Perfectly ordered graphs, in C. Berge and V. Chvatal, Ed., Topics on Perfect 
Graphs (North-Holland, Amsterdam, 1984) 63-65. 
V. Chvatal, C.T. HoBng, N.V.R. Mahadev and D. de Werra, Four classes of perfectly orderable 
graphs, J. Graph Theory 11,4 (1987) 481-495. 
R. P. Dilworth, A decomposition theorem for partially ordered sets, Ann. Math. 51 (1950) 
161-166. 
84 C. T. Ho&g, N. V.R. Mahadev 
[6] 6. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961) 71-76. 
[7] P. Duchet, Classical graphs, in C. Berge and V. ChvBtal, Ed., Topics on Perfect Graphs 
(North-Holland, Amsterdam, 1984), 67-96. 
[8] MC. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 
i980). 
[9] A. Hertz, Slim graphs, Report ORWP 87/l, Swiss Federal Institute of Technology in Laussane 
(1987). 
[lo] M. Preissmann, Locally perfect graphs, manuscript. 
[ll] E. S. Wolk, The comparability graph of a tree, Proc. Amer. Math. Sot. 3 (1962) 789-795. 


A note on perfect orders 

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