In this paper we study unavoidable sets of types of 3-paths for families of planar graphs with minimum degree at least 2 and a given girth X15003702&_mathId=si28.gif&_user=111111111&_pii=S0012365X15003702&_rdoc=1&_issn=0012365X&md5=df893f77acc66b46eab1c8d90c0f9e66" title="Click to view the MathML source">g. A 3-path of type (i,j,k) is a path uvw on three vertices u, v, and w such that the degree of u (resp. v, resp. w) is at most i (resp. j, resp. k). The elements i,j,k are called parameters of the type. The set S of types of paths is unavoidable for a family 15e1100ca0c2" title="Click to view the MathML source">F of graphs if each graph G from 15e1100ca0c2" title="Click to view the MathML source">F contains a path of the type from S. An unavoidable set S of types of paths is optimal for the family 15e1100ca0c2" title="Click to view the MathML source">F if neither any type can be omitted from S, nor any parameter of any type from S can be decreased.
We prove that the set Sg (resp. S′g) is an optimal set of types of 3-paths for the family of plane graphs having δ(G)≥2 and girth g(G)≥g where