Effects of graph operations on star pairwise compatibility graphs

TL;DR

This paper investigates how various graph operations influence the star number of star-$k$-pairwise compatibility graphs (PCGs), providing new bounds and exact values for specific graph classes like lobsters and acyclic graphs.

Contribution

It introduces the effects of graph operations on star-$k$-PCGs and determines the star number for lobsters and bounds for acyclic graphs.

Findings

Star number of lobster graphs is determined.

Upper bounds for star number of acyclic graphs are established.

Graph operations like adding twins or pendant vertices affect the star number.

Abstract

A graph $G=(V,E)$ is defined as a star-$k$-PCG when it is possible to assign a positive real number weight $w$ to each vertex $V$, and define $k$ distinct intervals $I_1, I_2, \ldots I_k$, in such a way that there is an edge $uv$ in $E$ if and only if the sum of the weights of vertices $u$ and $v$ falls within the union of these intervals. The star-$k$-PCG class is connected to two significant categories of graphs, namely PCGs and multithreshold graphs. The star number of a graph $G$, is the smallest $k$ for which $G$ is a star-$k$-PCG. In this paper, we study the effects of various graph operations, such as the addition of twins, pendant vertices, universal vertices, or isolated vertices, on the star number of the graph resulting from these operations. As a direct application of our results, we determine the star number of lobster graphs and provide an upper bound for the star number of acyclic graphs.