Nonseparating K4‐subdivisions in graphs of minimum degree at least 4

We first prove that for every vertex x of a 4‐connected graph G, there exists a subgraph H in G isomorphic to a subdivision of the complete graph K₄ on four vertices such that is connected and contains x. This implies an affirmative answer to a question of Kühnel whether every 4‐connected graph G contains a subdivision H of K₄ as a subgraph such that is connected. The motor for our induction is a result of Fontet and Martinov stating that every 4‐connected graph can be reduced to a smaller one by contracting a single edge, unless the graph is the square of a cycle or the line graph of a cubic graph. It turns out that this is the only ingredient of the proof where 4‐connectedness is used. We then generalize our result to connected graphs of minimum degree at least 4 by developing the respective motor, a structure theorem for the class of simple connected graphs of minimum degree at least 4. A simple connected graph G of minimum degree 4 cannot be reduced to a smaller such graph by deleting a single edge or contracting a single edge and simplifying if and only if it is the square of a cycle or the edge disjoint union of copies of certain bricks as follows: Each brick is isomorphic to K₃, K₅, K_{2, 2, 2}, , , or one the four graphs , , , obtained from K₅ and K_{2, 2, 2} by deleting the edges of a triangle, or replacing a vertex x by two new vertices and adding four edges to the endpoints of two disjoint edges of its former neighborhood, respectively. Bricks isomorphic to K₅ or K_{2, 2, 2} share exactly one vertex with the other bricks of the decomposition, vertices of degree 4 in any other brick are not contained in any further brick of the decomposition, and the vertices of a brick isomorphic to K₃ must have degree 4 in G and have pairwise no common neighbors outside that brick.