Splitting invariants describe how a plane curve “splits” by the pull-back under a Galois cover over the projective plane whose branch locus contains no component of the plane curve. They enable us to distinguish the embedded topology of several plane curves with the same fundamental group of the complements. 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. We prove the invariance of the G-combinatorial type under certain homeomorphisms based on the arguments of graph manifolds by Waldhausen and plumbing graphs by Neumann. Furthermore, we distinguish the embedded topology of quasi-triangular curves by the G-combinatorial type, which are generalization of triangular curves studied by Artal Bartolo et al. (2020, Collect. Math. 71, 427–441).

In the study of plane curves, one of the problems is to classify the embedded topology of plane curves in the complex projective plane that have a given fixed combinatorial type, where the combinatorial type of a plane curve is data equivalent to the embedded topology in its tubular neighborhood. A pair of plane curves with the same combinatorial type but distinct embedded topology is called a Zariski pair. In this paper, we consider Zariski pairs consisting of conic-line arrangements that arise from Poncelet’s closure theorem. We study unramified double covers of the union of two conics that are induced by a $2m$-sided Poncelet transverse. As an application, we show the existence of families of Zariski pairs of degree $2m+6$ for $m\geq 2$ that consist of reducible curves having two conics and $2m+2$ lines as irreducible components.

In this paper, we study how to distinguish the embedded topology of a smooth quartic and its bitangent lines. In order to do this, we introduce the concept of two-graphs and switching classes from graph theory. This new method improves previous results about a quartic and three bitangent lines considered by E. Artal Bartolo and J. Vallès, four bitangent lines considered by the authors and H. Tokunaga, and enables us to distinguish the embedded topology of a smooth quartic and five or more bitangent lines. As an application, we obtain a new Zariski 5-tuple and a Zariski 9-tuple for arrangements consisting of a smooth quartic and five of its bitangent lines and six of its bitangent lines, respectively.