Relevant Cycles in Biopolymers and Random Graphs

Short cycles are an important characteristic of molecular graphs in organic chemistry as well as in structural biology. Minimum cycle bases are of particular interest, despite the fact that they are usually not unique. Hence, one sometimes resorts to the set relevant cycles, defined as the union of all minimum cycles bases. Here we introduce the set of essential cycles as the intersection of a graph's minimum cycle bases and provide an algorithm for their computation. Furthermore, we extend previous bounds on the length of minimal cycles bases to certain book-embeddable graphs.