The thickness of amalgamations of graphs

TL;DR

This paper investigates the thickness of graphs formed by various amalgamation operations, providing bounds and formulas that relate the thickness of the resulting graph to the original graphs, with applications in VLSI design.

Contribution

It introduces new bounds and formulas for the thickness of graphs obtained through vertex, bar, edge, and 2-vertex amalgamations, expanding understanding of graph planarity properties.

Findings

Derived formulas for thickness after vertex-amalgamation.

Established bounds for thickness after edge-amalgamation.

Connected graph amalgamation operations to planarity measures.

Abstract

The thickness $\theta(G)$ of a graph $G$ is the minimum number of planar spanning subgraphs into which the graph $G$ can be decomposed. As a topological invariant of a graph, it is a measurement of the closeness to planarity of a graph, and it also has important applications to VLSI design. In this paper, the thickness of graphs that are obtained by vertex-amalgamation and bar-amalgamation of any two graphs whose thicknesses are known are obtained, respectively. And the lower and upper bounds for the thickness of graphs that are obtained by edge-amalgamation and 2-vertex-amalgamation of any two graphs whose thicknesses are known are also derived, respectively.