The notion of a recognizable set of finite graphs is introduced. Every set of finite graphs, that is definable in monadic second-order logic is recognizable, but not vice versa. The monadic second-order theory of a context-free set of graphs is decidable.

The monadic second-order logic of graphs. I. recognizable sets of finite graphs

A partial k -arboretum of graphs with bounded treewidth

Publisher Summary This chapter presents three mathematical tools that can be used to describe graph-grammars and the sets generated by them. The chapter describes graph properties by logical formulas and presents a comparison of the powers of several logical languages. The chapter discusses first-order logic, second-order logic, and monadic second-order logic, together with a few variants and restrictions of these three languages. It presents algebraic techniques that are useful for defining and studying graph rewriting rules and context-free graph-grammars. The chapter also presents the use of category theory for specifying graph rewriting rules in a precise and concise way, and for properly defining the least or rather the initial solution of a system of graph equations. It presents some links between the context-free graph-grammars and monadic second-order logic. Some applications to the definition of sets of graphs by forbidden configurations and to the theory of NP-completeness are also presented in the chapter.

Graph Rewriting: An Algebraic and Logic Approach.

Handle-rewriting hypergraph grammars

Graph rewriting: an algebraic and logic approach

A data structure, decomposition trees, is introduced, which enables graphs to be represented in a certain structured way, and which leads to simple, recursive algorithms for many difficult graph problems.

Decomposition Trees: Structured Graph Representation and Efficient Algorithms

A number of different graph grammar types have been called ”context-free” in the literature We consider two recent such formalisms, boundary node-label controlled (BNLC) and hyperedge replacement (HR) grammars, from a complexity-theoretical point of view It is shown that all HR languages, the members of which satisfy a certain separability restriction, are contained in LOGCFL, the class of sets which are log-space reducible to context-free (string) languages In particular, this implies the existence of efficient sequential as well as parallel recognition algorithms for these languages Since HR grammars can simulate a large class of BNLC grammars, the same holds for an according class of BNLC languages Thus, in a sense, a large class of BNLC and HR languages are ”close” to context-free string languages

Efficient algorithms on context-free graph languages

A number of different graph grammar types have been called ”context-free” in the literature. We consider two recent such formalisms, boundary node-label controlled (BNLC) and hyperedge replacement (HR) grammars, from a complexity-theoretical point of view. It is shown that all HR languages, the members of which satisfy a certain separability restriction, are contained in LOGCFL, the class of sets which are log-space reducible to context-free (string) languages. In particular, this implies the existence of efficient sequential as well as parallel recognition algorithms for these languages. Since HR grammars can simulate a large class of BNLC grammars, the same holds for an according class of BNLC languages. Thus, in a sense, a large class of BNLC and HR languages are ”close” to context-free string languages.

Clemens Lautemann

Papers

Decomposition Trees: Structured Graph Representation and Efficient Algorithms

Efficient algorithms on context-free graph languages

Efficient Algorithms on Context-Free Graph Grammars