Planar graphs without 4-, 7-, 9-cycles and 5-cycles normally adjacent to 3-cycles

A graph is (I,F)-partitionableif its vertex set can be partitioned into two parts suchthat one partIis an independent set, and the otherFinduces a forest. A graphisk-degenerateif every subgraphHcontains a vertex of degree at mostkinH.Bernshteyn and Lee defined a generalization ofk-degenerate graphs, which is calledweakly k-degenerate. In this paper, we show that planar graphs without 4-, 7-, 9-cycles,and 5-cycles normally adjacent to 3-cycles are both (I,F)-partitionable and weakly2-degenerate. (c) 2024ElsevierB.V.Allrightsarereserved,includingthosefortextanddatamining,AItraining,andsimilartechnologies.