2-connected outerplanar graphs have a unique minimal cycle basis with length 2jEj \Gamma jV j. They are the only Hamiltonian graphs with a cycle basis of this length. Keywords: Minimal Cycle Basis, Outerplanar Graphs AMS Subject Classification: Primary 05C38. Secondary 92D20. The electronic journal of combinatorics 5 (1998), #R16 2 1. Introduction The description of cyclic structures is an important problem in graph theory (see e.g. [16]). Cycle bases of graphs have a variety of applications in science and engineering, among them in structural analysis [11] and in chemical structure storage and retrieval systems [7]. Naturally, minimal cycles bases are of particular practical interest. In this contribution we prove that outerplanar graphs have a unique minimal cycle basis. This result was motivated by the analysis of the structures of biopolymers. In addition we derive upper and lower bounds on the length of minimal cycle basis in 2-connected graphs. Biopolymers, such as RNA, DNA,...

Let K be a subgraph of G. It is shown that if G is 3–connected modulo K then it is possible to replace branches of K by other branches joining same pairs of main vertices of K such that G has no local bridges with respect to the new subgraph K. A linear time algorithm is presented that either performs such a task, or finds a Kuratowski subgraph K5 or K3,3 in a subgraph of G formed by a branch e and local bridges on e. This result is needed in linear time algorithms for embedding graphs in surfaces.

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. Key words: Minimal Cycle Basis, Relevant Cycles, Essential Cycles, Biopolymer Graphs. Subject Classification: 05C38, 05C85. 1 Introduction Organic carbon compounds, such as the example shown in Figure 1, may exhibit elaborate polycyclic structures. Biopolymers, such as RNA, DNA, or proteins form well-defined three dimensional structures which are of utmost importance for their biological function. The most sali...

New base exchange properties of binary and graphic matroids are derived. The graphic matroids within the class of 4-connected binary matroids are characterized by base exchange properties. Some progress with the characterization of arbitrary graphic matroids is made. Characterizing various types of matroids by base exchange properties is e.g. important in invariant theory.