Simulation of soliton circuits Mikl´osKr´ esz Department of Computer Science, Juh´asz Gyula Teacher Training College University of Szeged, Szeged, Hungary kresz@jgytf.u-szeged.hu Soliton circuits are among the most promising alternatives for molecular electronic devices based on the design of molecular level conventional digital circuits. In order to capture the logical and computational aspects of these circuits, a mathematical model called soliton automaton was introduced by Dassow and J¨ urgensen in 1990. The underlying object of a soliton automaton is a so called soliton graph, which is a finite undirected graph allowed to have loops and multiple edges. In order for the graph to act as an automaton, it must have a perfect internal matching, which is a matching covering all vertices with degree at least 2. Such vertices are called internal , while external vertices are ones with degree 1. Let G be a soliton graph, fixed for our present discussion. The graph G defines an automaton A(G), the states of which are the perfect internal matchings of G. With a slight ambiguity, we shall also say that “M is a state of G”, rather than “M is a state of A(G)”. Inputs to A(G) are pairs of external vertices of G. In state M , a possible transition on input (v 1 ,v 2 ) is carried out by switching along an alternating walk – called soliton walk – connecting v 1 with v 2 . In that case the above transition is expressed by M ′ ∈ δ(M, (v 1 ,v 2 )), where M ′ denotes the induced state and δ denotes the transition function of A(G). From practical point of view it is a fundamental question to develop a simulation method for soliton circuits. Translating the above problem to the language of soliton automata, we consider the following task. Automaton Construction Problem (ACP): Given a soliton graph G. Construct the automaton A(G) associated with G. In order to give a solution for ACP, first we must design a method determining the set S(G) of states of G, then an algorithm for constructing the transition function is needed. The first problem can be solved by adopting an extension of the method suggested by Itai, Rodeh and Tanimoto for bipartite graphs with perfect matchings. It is assumed that a state M of G has been previously found, which can be achieved by a simple mod- ification of any known matching algorithm. Our algorithm uses the straightforward observation that a perfect internal matching is not unique iff it contains a so-called alternating unit by which we mean either an even-length alternating cycle or an alter- nating path connecting two external vertices. The borrowed idea is to define the procedure NEWSTATES (G ′ ,M ′ ,α ′ ,L ′ ) for any nice subgraph G ′ of G – i.e. a subgraph having a perfect internal matching such that every perfect internal matching of G ′ can be extended to a perfect internal matching of G –, perfect internal matching M ′ of G ′ , M ′ -alternating unit α ′ and perfect internal matching L ′ of G \ V (G ′ ). It finds all the additional perfect internal matchings of G ′ . A state of G is obtained by adding the set of edges L to a perfect internal matching of G ′ .