A note on combinatorial type and splitting invariants of plane curves

Splitting invariants are effective for distinguishing the embedded topology of plane curves. In this note, we introduce a generalization of splitting invariants, called the G-combinatorial type, for plane curves by using the modified plumbing graph defined by Hironaka [14]. We prove that the G-combinatorial type is invariant under certain homeomorphisms based on the arguments of Waldhausen [32, 33] and Neumann [22]. Furthermore, we distinguish the embedded topology of quasi-triangular curves by the G-combinatorial type, which are generalization of triangular curves studied in [4].