Cactus-Expandable Graphs

Cactus graphs, which are connected graphs with no edge contained in two or more different simple cycles, are known to provide useful properties for network systems. Meanwhile, it has been found that a noncactus graph can be equivalently transformed into a cactus graph by using a conventional technique for simplifying block diagrams. Based on such graph transformation, we can transform a network system on a general graph (where nodes have deterministic dynamics with a multidimensional state variable) into a network system on a cactus graph while preserving the original dynamics, which allows us to apply existing results for network systems on a cactus graph. However, the condition under which such transformation is possible has never been clarified so far. This paper addresses the problem of finding graphs which can be transformed into a cactus graph. We first introduce the notion of cactus expandability, which corresponds to the existence of a cactus graph equivalent to the original graph. A sufficient condition is presented based on two graph characteristics, called the nucleus graphs and doubly bidirectionally connected pairs. Furthermore, we derive a sufficient condition for the interconnection of graphs. Based on these results, we also present a method to obtain a cactus graph equivalent to the original graph and discuss its computational complexity. We finally apply the results to the robustness analysis of biological networks to demonstrate the potential for systems biology.