Let a, b, c, d be four vertices in a graph G. A K4-minor rooted at a, b, c, d consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of G, respectively containing a, b, c, d. We characterise precisely when G contains a K4-minor rooted at a, b, c, d by describing six classes of obstructions, which are the edge-maximal graphs containing no K4-minor rooted at a, b, c, d. The foll...

In 1991, Bollobás and Frieze showed that the threshold for Gn,p to contain a spanning maximal planar subgraph is very close to p = n−1/3. In this paper, we compute similar threshold ranges for Gn,p to contain a maximal bipartite planar subgraph and for Gn,p to contain a maximal planar subgraph of fixed girth g.

The ultimate energy-harvesting efficiency (93%) is made possible by violating Kirchhoff's law of radiation. Using a planar nonreciprocal structure, complete violation can be achieved. In addition, this planer structure does not require any coupling beyond the light cone.