Stud.Cercet.Ştiint., Ser.Mat., 17 (2007), Supplement Proceedings of CNMI 2007, Bacău, Nov. 2007, pp. 255-260 A RECOGNITION ALGORITHM FOR TRIVIALLY PERFECT GRAPHS MIHAI TALMACIU, ELENA NECHITA Abstract: A graph G is trivially perfect if for each induced subgraph H of G, the number of maximal cliques of H is equal to the maximum size of an independent set of H. In this paper we give an algorithm for trivially perfect graphs. Keywords and phrases:Trivially perfect graphs, weakly decomposition,recognition algorithm. (2000) Mathematics Subject Classification: 54 1 Introduction Let G=(V,E) be a graph with n vertices and m edges. Our algorithm rely on a new characterization of trivially perfect graphs. The paper is organized as follows. In Section 2 we give notations and definitions. For the unity of the paper, in Section 3 we shortly remind the weakly decomposition [5]. In Section 4 we present the necessary and sufficient conditions for a graph to be an trivially perfect graphs, in order to determine the number clique and the stability number. 2 Notations and definitions Throughout this paper [1] G=(V,E) is a simple (i.e. finite, undirected, without loops and multiple edges) graph. Let co-GG= denote the complement graph of G. For U ⊆ V let [U] denote the subgraph of G induced by U. By G-X we mean the graph [V-X], whenever X⊆ V, but we shall often denote it simply G- v (∀ v∈V) when there is no ambiguity. If v∈V is a vertex in G, the neighborhood NvG ( ) denotes the vertices of G-v that are adjacent to v. We shall write N(v) when the graph G appears clearly from the context. The neighborhood of the vertex v in the complement of the graph G will be 255 denoted by Nv(). For any subset S of vertices in the graph G the neighborhood of S is NS( )= ∪−vS∈ Nv ( ) S and N[S]=S ∪ N(S). A clique is a subset of V with the property that all the vertices are pairwise adjacent. The clique number of G, denoted by ω (G) is the size of the maximum clique. By Pn, Cn, Kn we mean a chordless path on n ≥ 3 vertices, the chordless cycle on n ≥ 3 vertices, and the complete graph on n ≥ 1 vertices. If e=xy∈E, we shall also write x ∼ y, and x ≠ y whenever x, y are not adjacent in G. A set A is totally adjacent (non adjacent) with a set B of vertices (A ∩ B=φ ) if ab is (is not) edge, for any a vertex in A and any b vertex in B; we note with A ∼ B (A ≠ B). A graph G is F-free if none of its induced subgraphs is in F. 3 The weakly decomposition Here we recall the notions and the results (see [3], also [5]) that are necessary in the next section. For this we define the notion of weakly component and give a characterization for the weakly decomposition of a graph. Definition 1. [3], also [5]) Let G=(V,E) be a graph. A set of vertices, A, is called weakly set if NG(A)=V-A and the induced subgraph by A is connected. If A is a weakly set, maximal with respect to the inclusion, the subgraph induced by A is called weakly component. For simplification, the weakly component G(A) will be denoted with A. The name of "weakly component" is justified by the next result. Theorem 1. ([3], also [5]) Any connected and incomplete graph G=(V,E) admits a weakly component A such that GV(−= A ) G (())(()) N A + G N A . Theorem 2. ([3], also [5]) Let G=(V,E) be a connected and incomplete graph and A ⊂ V. Then A is a weakly component of G if and only if G(A) is connected and N(A)~N(A) . Definition 2. ([3], also [5]) A partition (A,N(A),V-A ∪ N(A)), where A is a weakly set, is called weakly decomposition of graph G in relation to A. We call: A the weakly component, N(A) the minimal cutset, and V-N(A) the remote set. The next result insures the existence of a weakly decomposition in a connected and incomplete graph. Theorem 3. ([3], also [5]) If G=(V,E) is a connected and incomplete graph then the set of vertices V admits a weakly decomposition (A,B,C) such that G(A) is a weakly component and G(V-A)=G(B)+G(C). 256 Theorem 2 provides an O(n+m) algorithm for building a weakly decomposition for an incomplete and connected graph. Algorithm for the weakly decomposition of a graph Input: A connected graph with at least two nonadjacent vertices, G=(V,E). Output: A partition V=(A,N,R) such that G(A) is connected, N=N(A), A~R=N(A). begin A := any set of vertices such that A ∪ N(A) ⊂ V N:=N(A) R:=V-A∪ N(A) while ( ∃ n∈N, ∃ r∈R such that nr∉E ) do A:=A∪ n N:=(N-{n}) ∪ (N(n) ∩ R) R:=R-(N(n) ∩ R) end One can observe that [A ]G is connected, NNA= G ( ) , R not is φ is an invariant of the algorithm. Corollary 1. If G is a connected graph and (A,N,R) a weakly decomposition with A weakly component then the following holds: ω (G)=max{ω ([N])+ω ([R]),ω ([A ∪ N])}. 4 Trivially perfect graphs 4.1 Basic properties In this subsection we establish the necessary and sufficient conditions for a graph to be an trivially perfect graph. Definition 3. (4) A graph G is trivially perfect if for each induced subgraph H of G, the number of maximal cliques of H is equal to the maximum size an independent set of H. Theorem 4. (4) A graph is trivially perfect if and only if it contains no vertex subset that induces P4 or C4. In [2, Brandstadt et al. establish: Trivially pergect graphs are exactly the chordal cographs. 257 A vertex ordering α = (v1 ,v2 ,…,vn ) of a graph G is a universal-in-a- component ordering (uco) if for 1 ≤ i ≤ n , the vertex vi is universal in the connected component of G({vi ,vi+1 ,…,vn }) that vi belongs to. A graph is trivially perfect if and only if it has a uco. A cograph G is a trivially perfect if and only if, in the cotree T of G, every 1-node has at most one child that is a 0-node. Theorem 5. Let G=(V,E) be connected with at least two nonadjacent vertices and (A,N,R) a weakly decomposition with A weakly component. G is {P4 ,C4 }-free graph if and only if: i) A∼N∼R ii) N is clique iii) G(A), G(R) are {P4 ,C4 }-free graphs. Proof. If G is {P4 ,C4 }-free graph then G(A), G(N), G(R) are {P4 ,C4 }-free graphs and A∼N∼R Because G is C4 -free results that G is triangulated, so N is clique. We suppose that i), ii) and iii) holds. Since A∼N∼R, N is clique and G(A), G(R) are {P4 ,C4 }-free graphs it follows that G is P4 -free. Since G(R) is {P4 ,C4 }-free it follows that G(R) is triangulated. If G-R is not triangulated graph then there exists Ck ⊆ G − R (k ≥ 4) . If k ≥ 5 then G-R it contains P4 . So C4 ⊆ G − R . Because N is clique and G(A) is C4 -free it follows that C4 ∩ A ≠ φ , C4 ∩ N ≠ φ . Because A∼N it follows that ⎟C4 ∩ A ⎟=⎟C4 ∩ N ⎟=1. 4.2 The recognition algorithm. Theorem 5 leads to the following recognition algorithm. Input: A connected graph with at least two nonadjacent vertices, G=(V,E). Output: An answer to the question: is G a trivially perfect graph begin L:={G}; {L a list of graphs While ( L ≠ φ ) Extrac an element H from L; Find a weakly decomposition (A,N,R) for H; 258 If ( A ≠ N or N ≠ R) then Return: G is not trivially perfect Else if N is not clique then Return: G is not trivially perfect Else introduce in L the connected components of G(A), G(R) incomplete Return: G is trivially perfect end From Theorem 5 results: Corollary 2. If G is a connected graph and (A,N,R) a weakly decomposition with A weakly component then G is trivially perfect if and only if A∼N∼R N is clique G(A), G(R) are {P4 ,C4 }-free graphs. Corollary 3. If G is a connected graph, trivially perfect and (A,N,R) a weakly decomposition with A weakly component then the following holds: α(G) =max{α(G(A)),α(G(GR))} ω(G) =⎟N⎟+max{ω(G(A)),ω(G(R))}. References [1] Berge, C., " Graphs", Nort-Holland, Amsterdam (1985) [2] A.Brandstadt, V.B.Le, and J.P.Spinrad, Graph classes: A survey, Philadelphia, SIAM, 1999. [3] C. Croitoru, E. Olaru, M. Talmaciu, "Confidentially connected graphs", The annals of the University "Dunarea de Jos" of Galati, Suppliment to Tome XVIII (XXII) 2000, Proceedings of the international conference "The risk in contemporany economy. [4] M.C.Golumbic, Trivially perfect graph, Discrete Math 24: 105-107, 1978. 259 [5] Talmaciu, M., " Decomposition Problems in the Graph Theory with Applications in Combinatorial Optimization", Ph.D.Thesis, University "Al.