A sufficient condition for cactus-expandability of 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 non-cactus graph can be equivalently transformed into a cactus graph by using a conventional technique for simplifying block diagrams; 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. As a solution, we present a sufficient condition based on two graph characteristics, called the nucleus graphs and doubly bidirectionally-connected pairs.