Planarity of generalized ladder graphs

TL;DR

This paper characterizes which generalized ladder graphs are planar or outerplanar, including a solution to a problem about planar functigraphs, expanding understanding of graph planarity in specific graph classes.

Contribution

It provides a complete characterization of planar and outerplanar generalized ladder graphs, including the case of functigraphs, addressing a previously open problem.

Findings

Complete characterization of planar generalized ladder graphs

Complete characterization of outerplanar generalized ladder graphs

Solution to the planarity problem for functigraphs C(P_n,f)

Abstract

The Cartesian product of P_2 and P_n is called an n-ladder graph for a positive integer n. We call two paths P_m and P_n together with some edges each of which joins a vertex on P_m and a vertex on P_n a generalized (m,n)-ladder graph. In this paper, we completely characterize the planar generalized ladder graphs and the outerplanar generalized ladder graphs. A functigraph C(P_n ,f) is a generalized (n,n)-ladder graph. Consequently, our result solves the problem posed by A. Chen et al. (2011) to characterize planar functigraphs C(P_n ,f).