Cycle Intersection Graphs and Minimum Decycling Sets of Even Graphs

TL;DR

This paper introduces the cycle intersection graph concept for even graphs, providing new bounds on their decycling number and linking cycle decompositions to graph properties, with implications for optimization.

Contribution

It defines the cycle intersection graph, establishes improved upper bounds for the decycling number of even graphs, and explores cycle decomposition optimization.

Findings

New upper bounds for decycling numbers of even graphs

Links between cycle intersection graph structure and decycling number

Optimization framework for cycle decomposition in even graphs

Abstract

We introduce the cycle intersection graph of a graph, an adaptation of the cycle graph of a graph, and use the structure of these graphs to prove an upper bound for the decycling number of all even graphs. This bound is shown to be significantly better when an even graph admits a cycle decomposition in which any two cycles intersect in at most one vertex. Links between the cycle rank of the cycle intersection graph of an even graph and the decycling number of the even graph itself are found. The problem of choosing an ideal cycle decomposition is addressed and is presented as an optimization problem over the space of cycle decompositions of even graphs.