Alternating cycles in soliton graphs MIKLÓS KRÉSZ Department of Computer Science University of Szeged, Teacher Training College 6725, Szeged, Boldogasszony sgt. 6. HUNGARY Abstract: Soliton graphs having an alternating cycle are characterized with the help of a shrinking procedure. This characterization leads to a method testing the existence of an alternating cycle in a soliton graph. The suggested algorithm runs in O(n 3 ) time, where n is the number of vertices in the graph. Key-Words: Combinatorial Problems, Graph Matchings, Alternating cycles, Soliton graphs, 1-extendable graphs 1 Introduction One of the most ambitious goals of research in mod- ern bioelectronics is to develop a molecular computer. Inspired by this research soliton automata were intro- duced in [4] to serve as a mathematical model for cer- tain molecular switching devices. Many interesting special cases of soliton automata have been described (see e.g [5]), but it was not until [1] that matching theory was recognized as the fundamental theoretical background for the study of this model. The underlying object of a soliton automaton is a so called soliton graph. Such a graph is the topologi- cal model of a hydrocarbon molecule chain. In order for the graph to act as an automaton we need to de£ne its states. To reach this goal we use certain matchings, called perfect internal matchings, of the graph, where by a matching of graph G we mean a set of edges, with- out two incident ones to the same vertex. A soliton graph must have a perfect internal matching, which is a matching covering all vertices with degree at least 2. These vertices – called internal – model carbon atoms, whereas vertices with degree one – called external – represent a suitable chemical interface with the out- side world. Because of the chemical background, the name state is also used as a synonym for perfect inter- nal matching. In addition to possessing a state, a soli- ton graph is also expected to have an external vertex. The edges of a soliton graph G are also distinguished between, such as allowed (contained in some state of G) or forbidden (not contained in any state of G). The analysis of soliton automata is a complex task, and the general case is still open. Therefore it is a cen- tral problem to describe the structure of soliton graphs with respect to their states. In [2] a decomposition of soliton graphs into elementary components – maximal connected subgraphs spanned by allowed edges only – was worked out, and these components were grouped into pairwise disjoint families based on how they can be reached by alternating paths starting from external vertices. From a practical point of view the most im- portant special case is the class of deterministic soli- ton automata, and consequently the graphs associated with them, called deterministic soliton graphs. The graph-theoretic characterization of deterministic soli- ton graphs is given in [3], where it was proved that a soliton graph G is deterministic iff it does not con- tain an alternating cycle with respect to any state of G. However, this characterization does not provide a di- rect method to solve the important practical problem of checking the determinism of a soliton graph ef£ciently, as a graph might have an exponential number of states. In this paper we show that testing the existence of an alternating cycle, and thus testing the determinism, can be solved in O(n 3 ) time, where n denotes the num- ber of vertices. To reach this goal £rst we show in Sec- tion 3.1 that the general problem can be reduced to 1- extendable graphs, which are connected graphs with- out forbidden edges. Then in Section 3.2 an ear de- composition of soliton graphs is worked out to serve as a technique for their structural description. The main result is based on the shrinking operation presented in Section 3.3, by which certain internal vertices with de- gree 2 are eliminated. With the help of the shrink- ing operation we obtain a characterization of soliton graphs with alternating cycles, which directly leads to an ef£cient algorithm. 1