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Effects of graph operations on star pairwise compatibility graphs

Angelo Monti, Blerina Sinaimeri

TL;DR

The effects of various graph operations, such as the addition of twins, pendant vertices, universal vertices, universal vertices, or isolated vertices, on the star number of the graph resulting from these operations are studied.

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.

Effects of graph operations on star pairwise compatibility graphs

TL;DR

The effects of various graph operations, such as the addition of twins, pendant vertices, universal vertices, universal vertices, or isolated vertices, on the star number of the graph resulting from these operations are studied.

Abstract

A graph is defined as a star--PCG when it is possible to assign a positive real number weight to each vertex , and define distinct intervals , in such a way that there is an edge in if and only if the sum of the weights of vertices and falls within the union of these intervals. The star--PCG class is connected to two significant categories of graphs, namely PCGs and multithreshold graphs. The star number of a graph , is the smallest for which is a star--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.

Paper Structure

This paper contains 4 sections, 15 theorems, 9 equations, 6 figures.

Table of Contents

  1. Preliminaries
  2. Graph operations on star-$k$-PCGs
  3. The star number of acyclic graphs
  4. Conclusions and open problems

Key Result

Lemma 1

Let $G$ be a graph with $\gamma(G)=k$, then there exists a left-free $k$-witness $G^w$ if and only if there exists a right-free $k$-witness $G^{w'}$.

Figures (6)

  • Figure 1: $(a)$ a graph $G$; $(b)$ a star $2$-witness tree for $G$ with the corresponding intervals $I_1=[3,5]$ and $I_2=[7,7]$; $(c)$ a $1$-witness tree for $G$ with the corresponding interval $I_1=[3,7]$.
  • Figure 2: $(a)$ an example of a star-$2$-PCG $G$; $(b)$ a star witness tree with the corresponding intervals $I_1=[5,9]$ and $I_2=[12,12]$; $(c)$ the witness graph $G^w$ and the corresponding intervals $I_1=[5,9]$ and $I_2=[12,12]$.
  • Figure 3: (\ref{['fig:universal_a']}) An example of a star-$1$-PCG for which no $1$-witness is free and where the addition of a universal vertex necessarily increases the number of intervals (see the graph $G_{27}$ in monti2024starkpcgs) and (\ref{['fig:universal_b']}) an example of a star-$1$-PCG for which no $1$-witness is free and where the addition of a universal vertex does not increase the number of intervals.
  • Figure 4: (a)-(b) An example of a star-$1$-PCG for which no $1$-witness is free and where the addition of a pedant vertex does not increase the number of intervals (b)-(c) an example of a star-$1$-PCG for which no $1$-witness is free and where the addition of a pendant vertex increases the number of intervals (see the graph $G_{70}$ in monti2024starkpcgs).
  • Figure 5: A right-free $2$-witness graph of a self-complementary graph $G$ with star number $2$monti2024starkpcgs.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Definition 1
  • Claim 1
  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • Definition 2
  • Lemma 2
  • ...and 20 more