Counting K_4-Subdivisions
Tillmann Miltzow, Jens M. Schmidt, Mingji Xia

TL;DR
This paper investigates the minimum number of $K_4$-subdivisions in 3-connected graphs, establishing a cubic lower bound, its tightness, and the computational hardness of counting these subdivisions.
Contribution
It proves a tight cubic lower bound on the number of $K_4$-subdivisions in 3-connected graphs and shows counting them is $ ext{ extsterling}P$-hard.
Findings
There are at least $oldsymbol{ ext{ extsterling}n^3}$ $K_4$-subdivisions in every 3-connected graph on $n$ vertices.
The cubic bound is tight for infinitely many graphs.
Counting the exact number of $K_4$-subdivisions is $ ext{ extsterling}P$-hard.
Abstract
A fundamental theorem in graph theory states that any 3-connected graph contains a subdivision of . As a generalization, we ask for the minimum number of -subdivisions that are contained in every -connected graph on vertices. We prove that there are such -subdivisions and show that the order of this bound is tight for infinitely many graphs. We further investigate a better bound in dependence on and prove that the computational complexity of the problem of counting the exact number of -subdivisions is -hard.
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