Saturation numbers for families of graph subdivisions

For a family of graphs, a graph G is -saturated if G contains no member of as a subgraph, but for any edge in , contains some member of as a subgraph. The minimum number of edges in an -saturated graph of order n is denoted *. A subdivision of a graph H, or an H-subdivision, is a graph G obtained from H by replacing the edges of H with internally disjoint paths of arbitrary length. We let denote the family of H-subdivisions, including H itself. In this paper, we study when H is one of or , obtaining several exact results and bounds. In particular, we determine exactly for and show for n sufficiently large that there exists a constant such that *. For we show that will suffice, and that this can be improved slightly depending on the value of *. We also give an upper bound on for all t and show that *. This provides an interesting contrast to a 1937 result of Wagner (Math Ann, 114 (1937), 570–590), who showed that edge-maximal graphs without a K5-minor have at least edges. © 2012 Wiley Periodicals, Inc. (Research supported in part by UCD GK12 project, NSF award no. 0742434. The author acknowledges support of the National Science Foundation through a fellowship funded by the grant “EMSW21-MCTP: Research Experience for Graduate Students” (NSF DMS 08-38434).)