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We define a new framework for rewriting graphs, called a formal graph system (FGS), which is a logic program having hypergraphs instead of terms in first-order logic. We first prove that a class of gr...aphs is generated by a hyperedge replacement grammar if and only if it is defined by an FGS of a special form called a regular FGS. In the same way as logic programs, we can define a refutation tree for an FGS. The classes of TTSP graphs and outerplanar graphs are definable by regular FGSs. Then, we consider the problem of constructing a refutation tree of a graph for these FGSs. For the FGS defining TTSP graphs, we present a refutation tree algorithm of $O left(log^2 n+log m \right)$ time with $O left(n+m \right)$ processors on an EREW PRAM. For the FGS defining outerplanar graphs, we show that the refutation tree problem can be solved in $O left(log^2 n \right)$ time with $Oleft(n+rn)$ processors on an EREW PRAM. Here, n and m are the numbers of vertices and edges of an input graph, respectively.show more
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