Solving Application Oriented Graph Theoretical Problems with DNA Computing

Important features of networks, e.g., length of shortest paths, centrality, are defined in a graph theoretical way. Bipartite graphs are frequently applied, not only in computer science (e.g. PetriNets), but in other sciences as well. They are used to represent various problems, for example, in medicine or in economy. The relations between customers and products can be represented by bipartite graphs. Genes and various diseases can also form a bipartite graph. Here we present a DNA computing approach for solving the mentioned graph theoretical problems. As a counterpart of [1], we present an algorithm that computes all shortest paths between all pairs of nodes in a graph (that may represent a social network, or some other network). From the results of the algorithm we can compute the centrality and eccentricity of a vertex, and also the centrality of an edge. In medical sciences bipartite graphs are used to denote the connection between diseases and causes, or genes and characteristics, etc. [2]. Thus scientists are interested in the direct paths from vi to vj , or all direct paths from vi to all other vertices, or all direct paths between all vertices. We show a graph transformation algorithm: Starting from a bipartite graph we obtain its projection [3] by DNA computation, i.e., a graph with labeled edges having only vertices from one of the sets of the bipartition. Applications related to economy, e.g., marketing are also presented. We present another algorithm for projection that produces only paths of the desired length (and so it is faster and more efficient).