Convex cycle bases

  title={Convex cycle bases},
  author={Marc Hellmuth and Josef Leydold and Peter F. Stadler},
  journal={Ars Math. Contemp.},
Convex cycles play a role e.g. in the context of product graphs. We introduce convex cycle bases and describe a polynomial-time algorithm that recognizes whether a given graph has a convex cycle basis and provides an explicit construction in the positive case. Relations between convex cycles bases and other types of cycles bases are discussed. In particular we show that if G has a unique minimal cycle bases, this basis is convex. Furthermore, we characterize a class of graphs with convex cycles… 

Figures from this paper

Moore Graphs and Cycles Are Extremal Graphs for Convex Cycles
The equality of convex cycles of a simple graph G is proved and that equality holds if and only if G is an even cycle or a Moore graph thus giving a new characterization of Moore graphs.
Cycle construction and geodesic cycles with application to the hypercube
Construction of cycles in a graph is investigated, where cycles from particular subsets (such as bases) are added together so that each partial sum is also a cycle or each new cycle intersects the
Robust cycle bases do not exist for K n , n if n ≥ 8
A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path
Robust cycle bases do not exist for if
Planar Median Graphs and Cubesquare-Graphs
It is shown that a graph is planar median if and only if it can be obtained from cubes and square-graphs by a sequence of “square-boundary” amalgamations, and considerations lead to an O(n logn)-time recognition algorithm to compute a decomposition of a planar Median graph with n vertices into cubes andsquare- graphs.
Jernej Azarija Some Results From Algebraic Graph Theory
The subject of algebraic graph theory is introduced presenting some general results from this area and how certain algebraic objects such as matrices and polynomials can be used to gain structural information about graphs is shown.
Geodesic Cycle Length Distributions in Delusional and Other Social Networks
This Reply tries to clarify how this situation arose in this specific case, and address some more general issues Martin raises, including the use of nodal covariates, what the authors can learn from ERGMs, and methodological monoculturalism in social network research.
Dynamics of small autocatalytic reaction network IV : Inhomogeneous replicator equations
ter. Statistics of landscapes based on free energies, replication and degradation rate constants of RNA secondary structures.tion of a strange attractor. chaotic dynamics in low dimensional


A note on quasi-robust cycle bases
The concept of quasi-robust bases is introduced as a generalization of the notion of robust bases and it is demonstrated that a certain class of bases of the complete bipartite graphs K m, n with m , n ≥ 5 is quasi-Robust but not robust.
On robust cycle bases
Minimum cycle bases of product graphs
A construction for a minimal cycle basis for the Cartesian and the strong product of two graphs from the minimal length cycle bases of the factors is presented. Furthermore, we derive asymptotic
Interchangeability of Relevant Cycles in Graphs
This work introduces a partition of R such that each cycle in a class W can be expressed as a sum of other cycles in W and shorter cycles and shows that each minimum cycle basis contains the same number of representatives of a given classW.
Union of all the Minimum Cycle Bases of a Graph
A polynomial algorithm is presented that computes a compact representation of the potentially exponential-sized set ${\cal C_R}$ in $O(\nu m^3)$ (where $\nu$ denotes the cyclomatic number).
Is every cycle basis fundamental?
A constructive characterization is given that leads to a algorithm that can be used to determine if a graph has a cycle basis that covers every edge two or more times and an equivalent dual characterization for the cutset space is given.
New Approximation Algorithms for Minimum Cycle Bases of Graphs
This is the first time that any algorithm which computes sparse cycle bases with a guarantee drops below the Θ(mω) bound, and two new algorithms to compute an approximate minimum cycle basis are presented.