Interchangeability of Relevant Cycles in Graphs

Abstract

The set ${\cal R}$ of relevant cycles of a graph $G$ is the union of its minimum cycle bases. We introduce a partition of ${\cal R}$ such that each cycle in a class ${\cal W}$ can be expressed as a sum of other cycles in ${\cal W}$ and shorter cycles. It is shown that each minimum cycle basis contains the same number of representatives of a given class ${\cal W}$. This result is used to derive upper and lower bounds on the number of distinct minimum cycle bases. Finally, we give a polynomial-time algorithm to compute this partition.

Computing cyclic invariants for molecular graphs

“…Moreover, the number of cycles from each equivalence class W κ ⊆ R is the same for every minimum cycle basis B. Hence, we obtain the (slight) generalization of [20,Thm. 11] to the weighted case: Lemma 6.…”

“…of 0 with I = ∅ or ∅ = I ⊆ R κ and every nontrivial subfamily 3 of I ∪ C, C is linearly independent. The following lemma shows that the above definition is equivalent to [20,Def. 6].…”

“…The set of essential cycles is a subset of R. The following characterization generalizes [20,Lemma 5] to the weighted case.…”

Convex cycle bases

“…Cycle bases with special properties have been investigated in much detail in the literature. Examples include minimum cycle bases [2,17,19,29,44], (strictly) fundamental cycle bases [20,32,38], or (quasi) robust cycle bases [26,40]. Here, we consider convex cycle bases.…”

Minimum cycle bases of lexicographic products

“…MCBs play a role e.g. in the analysis of electrical circuits, periodic scheduling problems in traffic planning, graph drawing [18,14], mechanical frame analysis [13], biopolymer structures [17,4,16], and computational chemistry [2,15]. Four standard graph products have received considerable attention [9,8]: the Cartesian, direct, strong and lexicographic product of graphs.…”