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Operator Decomposition of Graphs 152 The International Arab Journal of Information Technology, Vol. 3, No. 2, April 2006 Operator Decomposition of Graphs Ruzayn Quaddoura Computer Science Department, Zarqa Private University, Jordan Abstract: In this paper we introduce a new form of decomposition of graphs, the (P, Q) -decomposition. We first give an optimal algorithm for finding the 1 -decomposition of a graph which is a special case of the (P, Q) -decomposition which was first introduced in [ 21 ]. We then examine the connections between the 1 -decomposition and well known forms of decomposition of graphs, namely, modular and homogeneous decomposition. The characterization of graphs totally decomposable by 1 -decomposition is also given. The last part of our paper is devoted to a generalization of the 1 - decomposition. We first show that some basic properties of modular decomposi tion can be extended in a new form of decomposition of graphs that we called operator decomposition. We introduce the notion of a (P, Q) -module, where P and Q are hereditary graph -theoretic properties, the notion of a (P, Q) -split graph and the closed here ditary class (P, Q) of graphs (P and Q are closed under the operations of join of graphs and disjoint union of graphs, respectively). On this base , we construct a special case of the operator decomposition that is called (P, Q) -decomposition. Such decompos ition is uniquely determined by an arbitrary minimal nontrivial (P, Q) -module in G. In particular, if G ( P, Q) , then G has the unique canonical (P, Q) -decomposition. Keywords: Graph decomposition, hereditary class, split graph . Received October 28 , 2004; accepted August 8, 2005 1. Introduction An isomorphism of triads is defined as an isomorphism of 2 -colored gr aphs. Denote by Tr the set All graphs considered are finite, undirected, without of triads distinguished up to isomorphism of triads. The loops and multiple edges. For all notions not defined action (1) induces the associative binary operation on here the reader is refer ed to [3 ]. The vertex and the Tr . So the set Tr becomes a semigroup .of operators with edge sets of a graph G are denoted by V (G) and E the exact action on the set of graphs. The semigroup Tr (G), respectively, while n denotes the cardinality of V was i ntroduced in [ 21 ]. The following structure theorem (G) and m the cardinality of E (G). We write of the decomposition was presented in the same paper. u v(u ~~ / v) if vertices u and v are adjacent (non - A graph F is called decomposable if there exist a adjacent). For the subsets U, W ⊆ V (G) the notation triad T and a graph H such that F = TH , otherwise F is U ∼W means that u∼ w for all vertices u∈U and w ∈ indecomposable. The decomposition theorem asse rts that every decomposable graph F can be uniquely W, U ~/ W means that there are no adjacent vertices u∈ represented in the form : U and w ∈ W. To shorten notation, we write u ∼ W F = T 1T2…T kF0 (u ~/W) instead of { u}∼ W ({u} ~/W). The subgraph of G Where Ti is indecomposable element of the semigroup induce d by a set A ⊆ V (G) is denoted by G [A]. We Tr and F0 is indecomposable graph. This theorem write G for the complement graph of G. The occurs to be useful instrument for the characterization neighborhood of a vertex v in the graph G is denoted by and enu meration of several graph classes [ 19, 22 ]. On (v) NG (or N (v)), N G (v) = V (G) \ v \ G (vN ) . the base of the theorem , the Kelly -Ulam reconstruction One type of graph decomposition based on the we ll - conjecture was proved for the class of decomposable known notion of split graphs is investigated. A triad T = graphs. A criterium of decomposability of graphs was (G, A, B), where G is a graph and ( A, B) is an ordered presented in [ 23 ]. In the same paper on the base of the bipartition of V (G) into a clique A and a stable set B, is decomposition theorem an exhaustive description of considered as an operator acting on the set of graphs. An unigraphs was obtained. (A graph is called a unigraph operator T acts on a g raph H by formula : if it is determined uniquely up to isomorphism by its degree sequence) . Namely, it was proved that a graph is TH = G ∪ H ∪{ax / a A, x ∈∈ (HV )} (1) a unigraph if and only if all its indecomposable components are unigraphs, and the catalogue of (all edges of the complete bipartite graph with the parts indecomposable unigraphs was given. A and V (H) are added to the disjoint union G H). In this paper , a decomposition theory is developed. U In section 2 , we present the 1 -decomposition which 0perator Decomposition of Graphs 153 was first introduced in [21]. An o (n) algorit hm of bipartition (2), we shall call the triad ( G, A, B) a con structing 1 -decomposition starting from the degree splitting of G or a splitted graph. sequence of the vertices of a graph is presented in Theorem 1 implies that a proper split sequence d A section 3. In section 4 , several examples of the can be divided into two parts d and dB which are the applications of 1 -decomposition are discussed lists of vertex degrees for the upper ( A) and the lower concerning the structure and recognition of some classes (B) parts of its realizations, respectively (one of the of graphs. We study also a connection between the 1 - parts can be empty). The sequense d written in the decomposition and two well known forms of form : decomposition of graphs namely modular and A B d = (d ; d ) homogeneous decomposition. Finally , we characterize by forbidden subgraphs the family of graphs which are is called a splitting of d or a splitted sequence . A A totally decomp osable by the 1 -decomposition. A far - splitted g raph having d and dB as the lists of vertex reaching notion of a more general decomposition that degrees for its upper and lower parts, respectively, is we called ( P, Q)-decomposition is introduced in section called a realization of the splitted sequence d. 5. We conclude this section by giving an example of the The following assertion is obvious. application of ( P, Q)-decomposition. Corollary 1: A proper split n -sequence d with d 〉 m(d ) −1 has the unique splitting 2. Basic Structure Theorem of 1- m(d ) Decomposition (d 1,…, d m (d) + 1,…d n) (4) If , the sequence d has exactly two This decomposition is based on the well -known dm(d ) = m(d ) −1 notion of split graphs. A graph G is called split [ 6] if splittings, , namely,(4) and there is a partition of its vertex set into a clique A and an independent set B. We call this partition a (d1,...,dm(d )−1;dm( ) ,...,dnd ). bipartition. One of the parts can be empty, but not both. The concept of isomorphism of splitted g raphs appears naturally. Let ( G, A, B) and ( H, C, D) be two V G = A B (2) ( ) U splitted graphs , and f :G → H be a graph isomorphism. In what follows , a sequence of length n is called If f preserves the parts, i. e. , f (A) = C and f (B) = D, an n-sequence . The ith member of a sequence d is then f is called an isomorphism of splitted graphs ( G, A, denoted by di. An n-sequence d is graphical if a graph B) and ( H, C, D). In this case we exist s (a rea lization of the sequence d) such that d is write f :(G, A,B) → (H C,, D) and ( G, A, B) ≅ (H, C, its degree sequence (the list of its vertex degrees). A D). It may happen that (G, A, B) ≅/ (H , , DC ), graphical sequence is called split if it has a split although G ≅ H (for example, two splitted graphs realization. A splitness criterium for a graph is formulated in terms of vertex degrees. Therefore , all resulting from K4 - e). realizations of a split sequence are split. An n - In what follows , graph s are considered up to sequence d is called proper if isomorphism, but splitted ones are considered up to isomorphism of splitted graphs. Denote by Σ and Γ the n −1≥ d1 ≥ d2 ≥ ...≥ dn ≥ 0 sets of splitted graphs and of simple graphs, respectively. Define the composition o : Σ × Γ → Γ Obviously , a graphical sequence can be assumed to be composition as follows: proper. If σ ∈ Σ,σ = (G, A B),, H ∈ Γ , then Theorem 1: For a proper graphical n -sequence d put σ o H = (G U H )+ {aν : a ∈ A,ν ∈V (H )} (5) m = m (d) = max {i : d i ≥ i - 1} . The sequence d is split if and only if the following equality holds The edge set of the complete bipartite graph with the parts A and V (H) is added to the disjoint union m n G H If, in addition, H is a split graph with a ∑∑di = m(m −1) + di (3) U i=+=11i m bipartitionV(H) =CUD, then the compositionσ ° H = F For m = 1 and m = n equality (3) has the form : is split as well with the bipartition n n V ( f ) = (A C ) (B D).In this case we suppose ∑∑di = 0or di = n(n −1), U UU i==11i (G, A, B) ° (H, C, D) = (F, AU C, B U D) (6) respectively [12 ].
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