Degree Monotone Paths and Graph Operations

TL;DR

This paper investigates how the length of the longest degree monotone path in a graph changes under various graph operations, providing bounds and sharpness results for different modifications and graph classes.

Contribution

It introduces bounds on the degree monotone path length after graph modifications and explores the behavior under graph products, extending previous work on this parameter.

Findings

Bounds established for $mp(G)$ after edge addition, deletion, contraction, and subdivision.

Analysis of $mp(G)$ under vertex addition and deletion.

Results on $mp(G imes H)$ for Cartesian product and join, with sharp bounds.

Abstract

A path $P$ in a graph $G$ is said to be a degree monotone path if the sequence of degrees of the vertices of $P$ in the order in which they appear on $P$ is monotonic. The length of the longest degree monotone path in $G$ is denoted by $mp(G)$. This parameter was first studied in an earlier paper by the authors where bounds in terms of other parameters of $G$ were obtained. In this paper we concentrate on the study of how $mp(G)$ changes under various operations on $G$. We first consider how $mp(G)$ changes when an edge is deleted, added, contracted or subdivided. We similarly consider the effects of adding or deleting a vertex. We sometimes restrict our attention to particular classes of graphs. Finally we study $mp(G \times H)$ in terms of $mp(G)$ and $mp(H)$ where $\times$ is either the Cartesian product or the join of two graphs. In all these cases we give bounds on the parameter $mp$ of the modified graph in terms of the original graph or graphs and we show that all the bounds are sharp.