Counting K_4-Subdivisions

A fundamental theorem in graph theory states that any 3-connected graph contains a subdivision of K4. As a generalization, we ask for the minimum number of K4-subdivisions that are contained in every 3connected graph on n vertices. We prove that there are Ω(n) such K4subdivisions and show that the order of this bound is tight for infinitely many graphs. We further prove that the computational complexity of the problem of counting the exact number of K4-subdivisions is #P -hard.

[1]  H. Whitney.  Non-Separable and Planar Graphs. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[3]  Wolfgang Mader,et al.  3n − 5 Edges Do Force a Subdivision of , 1998, Comb..

[4]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[5]  D. Barnette.  On Steinitz's theorem concerning convex 3-polytopes and on some properties of planar graphs , 1969 .

[6]  William T. Tutte,et al.  A theory of 3-connected graphs , 1961 .

[7]  Bruce A. Reed,et al.  Recognizing a totally odd K4-subdivision, parity 2-disjoint rooted paths and a parity cycle through specified elements , 2010, SODA '10.

[8]  László Lovász,et al.  Computing ears and branchings in parallel , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).