Quantitative characteristics of cycles and their relations with stretch and spanning tree congestion

Abstract

The main goal of this article is to introduce new quantitative characteristics of cycles in finite simple connected graphs and to establish relations of these characteristics with the stretch and spanning tree congestion of graphs. The main new parameter is named the support number. We give a polynomial approximation algorithm for the support number with the aid of yet another characteristic we introduce, named the cycle width of the graph.

Geodesic Cycle Length Distributions in Delusional and Other Social Networks

“…To find the longest geodesic cycle in a graph, Algorithm 4.1, "LIC -Longest Isometric Cycle" from Lokshtanov (2009) was implemented in Python in the longestIsometricCycle.py script. It is important to note that, as described in Catrina et al (2021), Lemma 3.6 of Lokshtanov (2009) is erroneous, and that therefore the algorithm is only correct for even-length cycles. For odd-length cycles, the conditions of this Lemma may be met for a particular cycle of length 𝑘 (for an odd value of 𝑘), and yet there is no cycle of length 𝑘 in the graph, and therefore the algorithm will incorrectly find that there is a geodesic cycle of length 𝑘.…”

“…As noted in Catrina et al (2021), there is an error in part of this proof, in the case of odd-length cycles, but the proof can be corrected using the even-length case with an auxiliary graph construction.…”