Path-Additions of Graphs

TL;DR

This paper investigates how the operation of path-addition affects various classes of graphs, identifying which classes are closed under this operation and which are not, with implications for graph theory and visualization.

Contribution

It characterizes the closure properties of multiple graph classes under path-addition, revealing new insights into their structural robustness and limitations.

Findings

Non-planar, right angle crossing, and interval graphs are closed under path-addition.

Planar subclasses and bounded treewidth graphs are not closed under path-addition.

Some graph classes lose their properties when path-addition is applied.

Abstract

Path-addition is an operation that takes a graph and adds an internally vertex-disjoint path between two vertices together with a set of supplementary edges. Path-additions are just the opposite of taking minors. We show that some classes of graphs are closed under path-addition, including non-planar, right angle crossing, fan-crossing free, quasi-planar, (aligned) bar 1-visibility, and interval graphs, whereas others are not closed, including all subclasses of planar graphs, bounded treewidth, k-planar, fan-planar, outer-fan planar, outer-fan-crossing free, and bar (1,j)-visibility graphs.