GraphCliques
 SI: GraphCliques
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On cliquecolouring of graphs with few P_{4}’s
Journal of the Brazilian Computer Society volume 18, pages 113–119 (2012)
Abstract
Let G=(V,E) be a graph with n vertices. A cliquecolouring of a graph is a colouring of its vertices such that no maximal clique of size at least two is monocoloured. A kcliquecolouring is a cliquecolouring that uses k colours. The cliquechromatic number of a graph G is the minimum k such that G has a kcliquecolouring.
In this paper we will use the primeval decomposition technique to find the cliquechromatic number and the cliquecolouring of well known classes of graphs that in some local sense contain few P_{4}’s. In particular we shall consider the classes of extended P_{4}laden graphs, ptrees (graphs which contain exactly n−3 P_{4}’s) and (q,q−3)graphs, q≥7, such that no set of at most q vertices induces more that q−3 distincts P_{4}’s. As corollary we shall derive the cliquechromatic number and the cliquecolouring of the classes of cographs, P_{4}reducible graphs, P_{4}sparse graphs, extended P_{4}reducible graphs, extended P_{4}sparse graphs, P_{4}extendible graphs, P_{4}lite graphs, P_{4}tidy graphs and P_{4}laden graphs that are included in the class of extended P_{4}laden graphs.
Introduction
In this paper we are concerned with the so called cliquecolouring of a graph. To introduce this concept we need the following definitions. A subset K of G is a clique if every pair of distinct vertices of K are adjacent in G. A clique is maximal if it is not properly contained in any other clique of G. A hypergraph is a pair \(\mathcal{H}\)\(=(V, \mathcal{E})\), where V is the set of vertices of \(\mathcal{H}\) and \(\mathcal{E}\) is a family of nonempty subsets of V called edges. A kcolouring of \(\mathcal{H}\) is a mapping c:V→{1,2,…,k} such that for all \(e \in\mathcal {E}\) with e≥2 there exist u,v∈e with c(u)≠c(v), that is, there is no monocoloured edge of size at least two. The chromatic number\(\chi( \mathcal{H})\) of \(\mathcal{H}\) is the smallest k such that \(\mathcal{H}\) has a kcolouring.
We will consider special hypergraphs: hypergraphs arising from graphs. Given a graph G=(V,E), the cliquehypergraph of G is the hypergraph \(\mathcal{H} (G)\)\(= (V, \mathcal{E})\), whose vertices are the vertices of G and whose edges are the maximal cliques of G. A kcolouring of \(\mathcal{H} (G)\) will also be called a kcliquecolouring of G and the chromatic number\(\chi( \mathcal{H})\) of \(\mathcal{H}\) the cliquechromatic number of G. In other words: a cliquecolouring of a graph is a colouring of its vertices such that no maximal clique of size at least two is monocoloured.
Cliquecolouring is harder than ordinary vertex colouring: it is coNPcomplete even to check whether a 2cliquecolouring is valid [7]. The complexity of 2cliquecolourability is investigated in [26] where they show it is NPhard to decide whether a perfect graph is 2cliquecolourable even for those with cliquenumber 3. A valid 2cliquecolouring is not a good certificate, since we cannot verify it in polynomial time. In [27] it is proved that it is \(\varSigma_{2}^{p}\)complete to check whether a graph is 2cliquecolourable, even for oddholefree graphs [15]. However quite general classes of graphs have been proved to be 2clique colourable or 3clique colourable. In [7] it was proved that K_{1,3}free graphs are 2clique colourable and that almost all perfect graphs are 3clique colourable. In [29] it was shown that every planar graph is 3clique colourable and in [26] it was proposed a polynomial algorithm to decide if a planar graph is 2clique colourable. The cliquecolourability of several other classes of graphs has been investigated in [1, 9, 10, 14, 19].
In this paper we consider some classes of graphs that have been characterized in terms of special properties of the unique primeval decomposition tree associated to each graph of the class. The primeval decomposition tree of any graph can be computed in linear time [8] and therefore it is the natural framework for finding polynomial time algorithms of many problems.
Using the primeval decomposition technique, we will determine the cliquechromatic number of graphs with few P_{4}’s. In particular we shall consider the classes of extended P_{4}laden [16], ptrees [3] (graphs which contain exactly n−3 P_{4}’s) and (q,q−3)graphs, q≥7, [2] such that no set of at most q vertices induces more that q−3 distincts P_{4}’s. As corollary we shall derive the cliquechromatic number and the cliquecolouring of the classes of cograph [11, 12], P_{4}reducible graphs [22], P_{4}sparse graphs [20, 24], extended P_{4}reducible graphs [17], extended P_{4}sparse graphs [17], P_{4}extendible graphs [23], P_{4}lite graphs [21], P_{4}tidy graphs [18] and P_{4}laden graphs [16] that are included in the class of extended P_{4}laden graphs. Furthermore we will extend these results to more general classes obtained by substituting vertices of the above classes by homogeneous sets.
In Sect. 2 we give some definitions and preliminary results. In Sect. 3 we show that any graph that it is not a nonseparable pconnected graph is 2cliquecolourable. In Sect. 4 we show that the class of ptrees is 2cliquecolourable. In Sect. 5 we find the cliquechromatic number and the cliquecolouring of the remaining classes of graphs mentioned above. These results lead to polynomial time algorithms for finding the cliquecolouring and the cliquechromatic number of the above classes.
Preliminaries
Basic notions
Throughout this paper let G=(V,E) be a finite simple undirected graph and let V=n and E=m. The complement graph of G=(V,E) is the graph \(\overline{G} = (V, \overline{E})\), where \(uv \in\overline{E}\) if and only if \(uv\not\in E\).
For a vertex v∈V the neighbourhood of v in G is N(v)={uuv∈E} and for U⊂V, N(U) is the set of vertices in V−U that are adjacent to at least one vertex of U. A clique of G is a set of pairwise adjacent vertices of G and a stable set is a set of pairwise nonadjacent vertices of G. Given a subset U of V, let G[U] stand for the subgraph of G induced by U. Let P_{ n } denote the chordless path on n vertices and n−1 edges. Let C_{ n } denote the chordless cycle with n vertices. If n≥4 then C_{ n } is called a hole and its complement an antihole. A graph is called a complete graph if every pair of distinct vertices is connected by an edge. A graph is called split graph if its vertex set can be partitioned in a clique K and a stable set S. A split graph is a spider if and only if K=S≥2 and there exists a bijection f between S and K such that for each v∈S, either N(v)={f(v)} (thin legs) or N(v)=K−{f(v)} (thick legs). The simplest spider is a P_{4}. In a P_{4} with vertices u,v,w,x and edges uv,vw,wx, the vertices v and w are called midpoints whereas the vertices u and x are called endpoints.
A module of G is a set of vertices M of V such that each vertex in V−M is either adjacent to all vertices of M, or to none. The whole V and every singleton vertex are trivial modules. Whenever G has only trivial modules it is called a prime graph. A nontrivial module is also called an homogeneous set. We say that M is a strong module if for any other module A the intersection M∩A is empty or equals either M or A. For a nontrivial graph G, the family {M_{1},M_{2},…,M_{ p }} of all maximal (proper) strong modules is a partition of V(G). This partition is the modular decomposition ofG. We will often identify the modules M_{ i } with the induced subgraphs G_{ i }=G[M_{ i }].
Whenever a graph G has a nontrivial maximal module M, in order to get some of its structural properties, it is useful to contract the module M to one representative vertex m obtaining a new graph H where V(H)=V(G)−M∪{m} and E(H)=E(G/M)∪{ymy∈N(M)}. The graph G′ obtained from G by shrinking every maximal nontrivial module to a single vertex is called the characteristic graph of G. If G is a prime graph the G′=G.
If G is a prime graph we consider the substitution operation which consists of substituting any vertex x of G by any graph H in such a way that all the vertices of H have the same adjacencies of x in G. Therefore in the new graph G′ obtained with this substitution operation V(H) is a homogeneous set.
Let G and G′ be two vertex disjoint graphs. We can define the parallel composition of G and G′ as the graph G∪G′ so that V(G∪G′)=V(G)∪V(G′) and E(G∪G′)=E(G)∪E(G′). The serial composition of G and G′ is the graph G+G′ defined by V(G+G′)=V(G)∪V(G′) and E(G+G′)=E(G)∪E(G′)∪{vv′ for each v∈V(G) and v′∈V(G′)}.
pConnectness and primeval decomposition
Following the terminology of Jamison and Olariu [25], a graph G is pconnected (or, more extensively, P_{4}connected) if, for each partition V_{1},V_{2} of V into two nonempty sets, there exists a chordless path of four vertices P_{4} which contains vertices from V_{1} and V_{2}. Such P_{4} is a crossing between V_{1} and V_{2}. An equivalent characterization of pconnected graphs can be given in terms of pchains, a natural analogue of paths in the context of usual connectedness of graphs. A pchain connecting vertices u and v is a sequence of pairwise different vertices (v_{1},v_{2},…,v_{ k }) such that

1.
u=v_{1}, v=v_{ k }, and

2.
X_{ i }={v_{ i },v_{i+1},v_{i+2},v_{i+3}} induces a P_{4}, for i=1,2,…,k−3.
By Babel and Olariu [5, 6] a graph is pconnected if and only if for every pair of vertices in the graph there exists a pchain connecting them.
The pconnected components of a graph G are the maximal induced pconnected subgraphs. Vertices of G that do not belong to any pconnected component of G are termed weak vertices. A pconnected graph is called separable if its vertex set can be partitioned into two nonempty sets V_{1} and V_{2} in such a way that each crossing P_{4} has its midpoints in V_{1} and its endpoints in V_{2}. For separable pconnected graph the following theorem holds.
Theorem 1
[25] Apconnected graph is separable if and only if its characteristic graph is a split graph.
Separable pconnected components play a crucial role in the theory of pconnectedness and their introduction is justified by the following general theorem.
Theorem 2
[25] For an arbitrary graphGexactly one of the following conditions is satisfied:

1.
Gis disconnected;

2.
\(\overline{G}\)is disconnected;

3.
There is a unique proper separablepconnected componentHofGwith vertex partition (V_{1},V_{2}) such that every vertex outsideHis adjacent to all vertices inV_{1}and to no vertex inV_{2};

4.
Gispconnected.
This theorem implies a decomposition scheme for arbitrary graphs called primeval decomposition.
For disconnected G, the maximal strong modules are the connected components. In this case G=G_{1}∪G_{2}∪⋯∪G_{ p } is called parallel.
If \(\overline{G}\) is disconnected, the maximal strong modules of G are the connected components of \(\overline{G}\). In this case G=G_{1}+G_{2}+⋯+G_{ p } is called serial.
If both G and \(\overline{G}\) are connected, then either G can be decomposed according to condition 3 of Theorem 2 and in this case G is called a decomposable neighbourhood graph or G is a pconnected graph that may be either separable or nonseparable.
By repeating the process we can associate to any nonempty graph G its unique primeval decomposition treeT(G). The root of T(G) is G, the leaves are the pconnected components and the weak vertices of G and the internal nodes of T(G) are labeled with P, S or N (for parallel, serial, or decomposable neighbourhood graph, respectively).
The cliquecolouring of graphs which are not nonseparable pconnected graphs
In this section we will show that the cliquechromatic number of any graph which is not a nonseparable pconnected graph is equal to 2. In the next sections we shall consider classes of graphs that contain nonseparable pconnected graphs of special type.
From now on we will assume that the graphs we are considering are connected, otherwise we could always consider separately each connected component.
First we show that in order to find an optimal cliquecolouration of a graph, it is enough to have an optimal cliquecolouration of its characteristic graph.
Theorem 3
Every graphGhas the same cliquechromatic number of its characteristic graphG′. Given an optimal cliquecolouration ofG′, an optimal cliquecolouration ofGis obtained assigning to every vertex belonging to a maximal strong module ofGthe same colour of its representative vertex inG′.
Proof
Let G be a graph and let G′ be its characteristic graph. Let {M_{1},M_{2},…,M_{ p }} be the family of all maximal (proper) strong modules of G and let {v_{1},v_{2},…,v_{ p }} be the set of the corresponding characteristic vertices in G′. Let us assume that an optimal cliquecolouration of G′ is known. Let \(K' = \{v_{i_{1}},v_{i_{2}},\ldots, v_{i_{s}}\}\) be any maximal clique of G′. By replacing each vertex \({v_{i_{j}}}\) of every K′ with any maximal clique of the corresponding module \({M_{i_{j}}}\) we obtain all the maximal cliques of G. In fact each set of vertices generated is a clique since each pair of vertices are adjacent either by construction or by definition of module. The maximality follows by the maximality of each clique K′ of G′ and the maximality of the cliques replacing each vertex of K′. Now, we can extend the clique colouration of G′ to G by assigning the colour of each representative vertex of G′ to every vertex of the corresponding homogeneous set in G. No maximal clique of size at least 2 is monocoloured since every maximal clique of G contains an induced subgraph isomorphic to a maximal clique of G′ that is not monocoloured by hypothesis. The optimality of the colouration of G follows from the optimality of the colouration of G′. □
Theorem 4
Every serial graph is 2clique colourable.
Proof
Let G=G_{1}+G_{2}+⋯+G_{ p }, p≥2, be a serial graph. The characteristic graph of G is a complete graph that is 2clique colourable. Therefore G is also 2cliquecolourable by Theorem 3. In fact, it is sufficient to colour the vertices of G_{1} with colour 1 and the remaining vertices with colour 2. □
Lemma 1
Every split graph is 2clique colourable.
Proof
Let G be a split graph. Then its vertex set can be partitioned into a maximal clique K and an independent set S. Every maximal clique of G is either K or a subset of K with at most one vertex of S. Choose any vertex of K, say x, and colour it with colour 1. Colour the remaining vertices of K with colour 2. Colour the vertices of N_{ S }(x) with colour 2 and the remaining vertices of S with colour 1. We claim that this colouring is a 2cliquecolouring of G. In fact, if K is a maximal clique of G it has two different colours. Any other maximal clique of G contains vertices of K and only one vertex of S. If it contains x then it is 2cliquecolourable since all its adjacent vertices have colour 2. Else, if it contains only vertices of K coloured with colour 2, then the vertex of S by construction has colour 1. So it is 2clique colourable. □
Theorem 5
Every separablepconnected graph is 2cliquecolourable.
Proof
The characteristic graph of every separable pconnected graph G is a split graph by Theorem 1. Any split graph is 2cliquecolourable by Lemma 1. Therefore G is also 2cliquecolourable by Theorem 3. □
Theorem 6
Every decomposable neighbourhood graph is 2clique colourable.
Proof
Let G=(V,E) be a decomposable neighbourhood graph. Then by condition 3 of Theorem 2 there is a unique proper separable pconnected component H of G with vertex partition (V_{1},V_{2}) such that every vertex in V_{3}=V∖(V_{1}∪V_{2}) is adjacent to all vertices in V_{1} and to no vertex in V_{2}. Then V_{3} is a module in G. The characteristic graph of G is obtained by shrinking V_{3} to a single vertex, and by substituting H with its characteristic graph, which is a split graph by Theorem 1. Then the characteristic graph of G is also a split graph. Therefore G is 2clique colourable by Lemma 1 and Theorem 3. □
The cliquecolouring of ptrees and pforests
The purpose of this section is to show that a special subclass of the P_{4}connected graphs, called ptrees is 2cliquecolourable. This class, introduced by Babel in [3], is provided with structural properties that can be expressed in a quite analogous way to the characterization of ordinary trees. A vertex is called a pendvertex if it belongs to exactly one P_{4}. A pcycle is a pconnected graph with no pendvertices, and which is minimal with this property. The class of ptrees is the class of pconnected graphs without induced pcycles and containing exactly n−3 P_{4}’s. A pforest is a graph whose pconnected components are ptrees [3].
In the following we shall use the characterization of ptrees given in [4], based on the structure of the pchains in a ptree, which we recall for reader’s convenience.
A pchain X=(v_{1},v_{2},…,v_{ k }) is simple if the only P_{4}’s contained in G[{v_{1},v_{2},…,v_{ k }}] are induced by the set of vertices {v_{ i },v_{i+1},v_{i+2},v_{i+3}} for i=1,2,…,k−3. In other words a pchain X is simple if and only if the vertices of the pchain induce precisely k−3 P_{4}’s.
Let P_{ k } be a path with vertex set V={v_{1},v_{2},…,v_{ k }} and edge set E={v_{1}v_{2},v_{2}v_{3},…,v_{k−1}v_{ k }} and let Q_{ k } be a split graph with vertex set V={v_{1},v_{2},…,v_{ k }}, where A={v_{2i−1}∈V1≤2i−1≤k} is a stable set, B={v_{2i}∈V2≤2i≤k} is a clique and the edges connecting each vertex of B to the vertices of A are {v_{2i}v_{2i−1} and v_{2i}v_{2j+1},j>i} (see Fig. 1). The ordered sequence (v_{1},v_{2},…,v_{ k }) of the vertices of P_{ k } (k≥4), Q_{ k } (k≥5) and their complements are simple pchains. It has been proved that graphs whose vertices can be ordered in a simple pchains are ptrees and it turns out that every ptree can be generated starting from a simple pchain extended by a number of pendvertices which can eventually be replaced by cographs [4].
A spikedpchainP_{ k } is a path P_{ k }, k≥5, eventually extended, by introducing two additional vertices x and y such that x is adjacent to v_{2} and v_{3} and y is adjacent to v_{k−1} and v_{k−2}; moreover we request that x and y do not belong to a common P_{4} (see Fig. 2).
A spikedpchainQ_{ k } is a split graph Q_{ k }, k≥6, with eventually, additional vertices z_{2},z_{3},…,z_{k−5} such that
A spiked pchain is shown in Fig. 3. A spiked pchain \(\overline{P}_{k}\) (or \(\overline{Q}_{k}\)) is the complement of a spiked pchain P_{ k } (or Q_{ k }).
It is easy to verify that v_{1},x,y,v_{ k } and v_{1},z_{2},z_{3},…,z_{k−5},v_{ k } are the only pendvertices of spiked P_{ k } (\(\overline{P}_{k}\)) and Q_{ k } (\(\overline{Q}_{k}\)) respectively.
Finally we have the following characterization of ptrees given by Babel.
Theorem 7
[4] A graph is aptree if and only if it is either aP_{4}with one vertex replaced by a cograph or a spikedpchain with thependvertices replaced by cographs.
Theorem 8
Everyptree is 2cliquecolourable.
Proof
From Theorem 7, it follows that the only homogeneous sets of a ptree are the cographs eventually replacing the pendvertices of G. By Theorem 3 graph G has the same cliquechromatic number of its characteristic graph. Therefore it is enough to consider the cliquecolouration of the spiked pchains P_{ k }, Q_{ k } and their complements.
Notice that a prime ptree is separable if and only if it is either a spiked pchain Q_{ k } or \(\overline{Q}_{k}\), (k≥6) or a P_{4}. If G is a separable ptree then it is 2cliquecolourable by Theorem 5. If G is not separable then G is a spiked pchains P_{ k } or \(\overline{P}_{k}\), (k≥5). Let now consider the case when G is a spiked pchains P_{ k }. If the pendvertices x and y are not present then G is a path P_{ k } that can be 2cliquecoloured by assigning to the vertices of the path alternatively colours 1 and 2. If the pendvertices x or y are present they belong to a maximal clique of G that already contains a 2coloured edge so they can be coloured with colour 1 or 2. Let G be a spiked pchain \(\overline{P}_{k}\). If the pendvertices x and y are not present then G is a \(\overline{P}_{k}\). Let v_{1},v_{2},…,v_{ k } be the sequence of the vertices of P_{ k }. The vertex set of a maximal clique of \(\overline{P}_{k}\) is any maximal ordered sequence of vertices \(S= v_{i_{1}},v_{i_{2}},\ldots, v_{i_{s}}\) such that for each two consecutive vertices \(v_{i_{p}}\) and \(v_{i_{q}}\), i_{ q }−i_{ p } is either 2 or 3. Since S is a maximal sequence then \(v_{i_{1}}\) must belong to the set {v_{1},v_{2}} and \(v_{i_{s}}\) to the set {v_{k−1},v_{ k }}. Then every maximal clique of \(\overline{P}_{k}\) contains one of the edges v_{1}v_{ k } or v_{1}v_{k−1} or v_{2}v_{ k } or v_{2}v_{k−1}. Therefore, in order to obtain a 2clique colouration of \(\overline{P}_{k}\) it is enough to colour the vertices v_{1} and v_{2} with colour 1, the vertices v_{ k } and v_{k−1} with colour 2 and any other vertex with colour 1 or 2. If the pendvertices x or y are present they belong to a maximal clique of G that already contains a 2coloured edge so they can be coloured with colour 1 or 2. □
Theorem 9
Everypforest is 2cliquecolourable.
Proof
A pforest G is either a serial graph or a decomposable neighbourhood graph or a ptree. Then G is 2cliquecolourable by Theorems 4, 6 and 8 respectively. □
The cliquecolouring of graphs with few P_{4}’s
In this section we shall consider graphs which, in a local sense, contain only a restricted number of P_{4}’s. It all started with the class of cographs, which is a class of graphs where no P_{4} is allowed to exist. These graphs have been investigated in [11, 12] and many nice structural results are known. In particular it has been shown in [11] that every connected cograph is a serial graph and therefore it is 2cliquecolourable by Theorem 4.
The study of cographs has been extended by many authors. Hoáng [20] introduced the class of P_{4}sparse graphs, which is the class such that no set of five vertices induces more than one P_{4}. The nontrivial leaves of its associated primeval tree are spiders [24]. Jamison and Olariu [21–23] introduced the class of P_{4}reducible graphs, P_{4}extendible and P_{4}lite. The P_{4}reducible graphs are the graphs such that no vertex belongs to more than one P_{4}, and its pconnected components are P_{4}’s. The P_{4}extendible are graphs where each pconnected component consists of at most five vertices. Each pconnected component is either P_{5} or \(\overline{P_{5}}\) or C_{5}, or P_{4} with one vertex eventually substituted by a homogeneous set with cardinality two. The P_{4}lite are graphs such that every induced subgraph with at most six vertices either contains at most two P_{4}’s or is isomorphic to a spider. The pconnected components of a P_{4}lite graph are either a spider (possibly with one vertex replaced by a homogeneous set of cardinality 2) or one of the graphs P_{5}, \(\overline{P_{5}}\). The P_{4}laden [16] are graphs such that every induced subgraph with at most six vertices either contains at most two P_{4}’s or it is isomorphic to a split graph. The pconnected components of a P_{4}laden graph are spiders (possibly with one vertex replaced by a homogeneous set of cardinality 2), split graphs, P_{5}’s or \(\overline{P_{5}}\)’s [16]. The above classes are ordered as follows:
Another generalization of the above classes are the extended P_{4}reducible [17], extended P_{4}sparse graphs [17], P_{4}tidy graph [18] and the extended P_{4}laden graphs [16]. These classes are obtained from P_{4}reducible and P_{4}sparse graphs, P_{4}lite and P_{4}laden respectively, by also allowing C_{5}’s as pconnected components. All the previously mentioned classes are included in the class of extended P_{4}laden graphs.
Using the primeval decomposition we have the following.
Theorem 10
Every extendedP_{4}laden graphGis 2cliquecolourable with the exception ofC_{5}which is 3cliquecolourable.
Proof
If G is a serial graph or a decomposable neighbourhood graph, it is 2cliquecolourable by Theorem 4 and Theorem 6 respectively. If G is a pconnected graph different from C_{5}, then either it is isomorphic to a P_{5} or \(\overline{P_{5}}\) that are trivially 2cliquecolourable or it is a separable pconnected graph isomorphic either to a spider graph (possibly with one vertex replaced by a homogeneous set of cardinality 2), or to a split graph. The characteristic graph in both cases is a split graph and therefore it is 2cliquecolourable by Lemma 1. If G is a C_{5} then G is 3cliquecolourable. □
Corollary 1
Every cograph, P_{4}reducible, P_{4}sparse andP_{4}lite graphs are 2cliquecolourable.
Corollary 2
EveryP_{4}extendible, extendedP_{4}reducible and extendedP_{4}sparse graphs are 2cliquecolourable with the exception ofC_{5}which is 3cliquecolourable.
A generalization of some of the above classes was given by Babel and Olariu. They proposed to call a graph a (q,t)graph if no set of at most q vertices induces more than t distinct P_{4}’s. In this terminology, the cographs are precisely the (4,0)graphs, the P_{4}sparse graphs coincide with the (5,1)graphs. In particular the (7,4)graphs properly contain all cographs, P_{4}reducible, P_{4}sparse, ptrees and pforests. The (9,6)graphs additionally contain all P_{4}extendible, extended P_{4}reducible and extended P_{4}sparse graphs.
The pconnected (q,q−3)graph has been characterized as follows.
Theorem 11
[4] LetGbe apconnected graph withnvertices.

(i)
Ifn≥7, thenGis a (n,n−3) graph if and only ifGis aptree.

(ii)
Ifn>q, q∈{7,9}, thenGis a (q,q−3)graph if and only if precisely one of the following conditions holds

(a)
Gis aptree;

(b)
Gis a hole or antihole;

(c)
Gis a spider.

(a)

(iii)
Ifn>q, q=8, orq≥10, thenGis a (q,q−3)graph if and only if precisely one of the following conditions holds

(a)
Gis aptree;

(b)
Gis a hole or antihole.

(a)
In order to find the cliquecolouration of (q,q−3)graphs we will need the following result.
Theorem 12
Every antihole\(\overline{C}_{k}\), withk>5 is 2cliquecolourable.
Proof
Let v_{1},v_{2},…,v_{ k }, k≥6, be the sequence of the vertices of C_{ k }. Any 2colouring where v_{1},v_{2},v_{3} have colour 1 and v_{4},v_{5},v_{6} have colour 2 is a 2cliquecolouring. In fact a clique that misses 3 consecutive vertices v_{ i },v_{i+1},v_{i+2} is not maximal because we could include v_{i+1} obtaining a bigger clique. Hence, any clique has (at least) one vertex in {v_{1},v_{2},v_{3}} and (at least) one vertex in {v_{4},v_{5},v_{6}} and it is 2cliquecoloured. □
Note that Theorem 12 can be obtained also as a particular case of results contained in [1, 7].
Now we are ready to prove the following theorem.
Theorem 13
Letq≥7 be a fixed integer andGa (q,q−3)graph withnvertices, n≥q. Gis 2cliquecolourable, unless it is aC_{ n }withnodd (which is 3cliquecolourable).
Proof
If n=q then G is a ptree, which is 2cliquecolourable by Theorem 8. If n>q then G is either a serial graph or a decomposable neighbourhood graph or a pconnected graph. In the first two cases it is 2cliquecolourable by Theorems 4 and 6 respectively. In the last case, G has more than 7 vertices and it is either a ptree, or a hole, or an antihole, or a spider by Theorem 11. Every ptree is 2cliquecolourable by Theorem 8. Every spider is a split graph and therefore it is 2cliquecolourable by Lemma 1. Every antihole is 2cliquecolourable by Theorem 12. Finally every hole is 2cliquecolourable if it is even, otherwise is 3cliquecolourable. □
Final remarks
We would like to point out that, in order to study the cliquecolouration of a graph, it is enough to consider the cliquecolouration of its characteristic graph that can be obtained in linear time through the modular decomposition of that graph [13, 28]. We have showed that every serial graph, every separable pconnected graph and every neighbourhood decomposable graph is 2cliquecolourable. Furthermore we have solved the cliquecolouring problem for classes of graphs that contain very special nonseparable pconnected graphs. Moreover, if the cliquecolouration of a prime graph G is known, then any graph obtained by substituting any vertex of G with any graph H has the same cliquecolouration of G. The problem remains open for general nonseparable pconnected graphs.
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Acknowledgements
The authors thank Sylvain Gravier for having introduced the clique colouration subject to the authors and Célia Picinin de Mello for valuable conversations concerning this problem, which helped to improve the presentation of the ideas contained in this work.
This research was partially supported by CNPq and FAPERJ.
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Klein, S., Morgana, A. On cliquecolouring of graphs with few P_{4}’s. J Braz Comput Soc 18, 113–119 (2012). https://doi.org/10.1007/s1317301100533
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Keywords
 Graphs
 Cliquecolouring
 Primeval decomposition
 P_{4}structure