J. Graph Theory
Connectivity, graph minors, and subgraph multiplicity.
D. Eppstein.
Tech. Rep. 92-06, ICS, UCI, 1992.
J. Graph Th. 17: 409–416, 1993.It was known that planar graphs have \(O(n)\) subgraphs isomorphic to \(K_3\) or \(K_4\). That is, \(K_3\) and \(K_4\) have linear subgraph multiplicity. This paper shows that the graphs with linear subgraph multiplicity in the planar graphs are exactly the 3-connected planar graphs. Also, the graphs with linear subgraph multiplicity in the outerplanar graphs are exactly the 2-connected outerplanar graphs.
More generally, let \(\mathcal{F}\) be a minor-closed family, and let \(x\) be the smallest number such that some complete bipartite graph \(K_{x,y}\) is a forbidden minor for \(\mathcal{F}\). Then the \(x\)-connected graphs have linear subgraph multiplicity for \(\mathcal{F}\), and there exists an \((x-1)\)-connected graph (namely \(K_{x-1,x-1}\) that does not have linear subgraph multiplicity. When \(x\le 3\) or when \(x=4\) and the minimal forbidden minors for \(\mathcal{F}\) are triangle-free, then the graphs with linear subgraph multiplicity for \(\mathcal{F}\) are exactly the \(x\)-connected graphs.
Please refer only to the journal version, and not the earlier technical report: the technical report had a bug that was repaired in the journal version.