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Theory of 2-structures. II: Representation through labeled tree families. (English) Zbl 0701.05052

This paper continues the investigation of the theory of 2-structures initiated in Part I (see [Zbl 0701.05051]). Hierarchical representations of 2-structures through 2s-labeled tree families are studied.
The main result states that every 2-structure is, in fact, a 2s-labeled family of primitive, complete or linear 2-structures. Hence, each 2- structure can be decomposed into primitive, complete and linear 2-structures.
Given a 2-structure \((D,R)\). Denote by \(2^ D\) the set of all subsets of \(D\). For \(X,Y\in 2^ D\), \(X\), \(Y\) are overlapping if \(X\setminus Y\neq \emptyset\), \(Y\setminus X\neq \emptyset\), \(X\cap Y\neq \emptyset\).
A 2s-labeled tree family is a triple \((D,{\mathcal F},\phi)\), where \(D\) is a finite set, \({\mathcal F}\subseteq 2^ D\) such that \(D\in {\mathcal F}\), \(\emptyset \not\in {\mathcal F}\), \(\{x\}\in {\mathcal F}\) for every \(x\in D\), for every \(X,Y\in {\mathcal F}\), \(X\), \(Y\) are not overlapping and \(\phi (X)\) is a 2-structure for every \(X\in {\mathcal F}\).

MSC:

05C99 Graph theory
05C05 Trees
68R10 Graph theory (including graph drawing) in computer science

Citations:

Zbl 0701.05051
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References:

[1] Berge, C., Graphs and Hypergraphs (1983), North-Holland: North-Holland Amsterdam · Zbl 0523.05040
[2] Ehrenfeucht, A.; Rozenberg, G., Theory of 2-structures, Part I: clans, basic subclasses, and morphisms, Theoret. Comput. Sci., 70, 277-303 (1990), (this issue) · Zbl 0701.05051
[3] Ehrenfeucht, A.; Rozenberg, G., Primitivity is hereditary for 2-structures, Theoret. Comput. Sci., 70, 343-358 (1990), (this issue) · Zbl 0701.05053
[4] (Ehrig, H.; Nagl, M.; Rozenberg, G.; Rosenfeld, A., Graph Grammars and their Application to Computer Science (1987), Springer: Springer Berlin) · Zbl 0636.00013
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