Excluding a large theta graph
A theta graph, denoted θa,b,c, is a graph of order a+b+c−1 consisting of a pair of vertices and three internally-disjoint paths between them of lengths a, b, and c. In this paper we study graphs that do not contain a large θa,b,c minor. More specifically, we describe the structure of θ1,2,t-, θ2,2,t-, θ1,t,t-, θ2,t,t-, and θt,t,t-free graphs where t is large. The main result is a characterization of θt,t,t-free graphs for large t. The 3-connected θt,t,t-free graphs are formed by 3-summing graphs without a long path to certain planar graphs. The 2-connected θt,t,t-free graphs are then built up in a similar fashion by 2- and 3-sums. This result implies a well-known theorem of Robertson and Chakravarti on graphs that do not have a bond containing three specified edges.
Publication Source (Journal or Book title)
Journal of Combinatorial Theory. Series B
Ding, G., & Marshall, E. (2018). Excluding a large theta graph. Journal of Combinatorial Theory. Series B, 132, 47-79. https://doi.org/10.1016/j.jctb.2018.03.002