On the power propagation time of a graph

TL;DR

This paper investigates bounds and properties of power propagation time in graphs, including its generalizations, effects of graph modifications, and characterizations of specific graph classes with extremal propagation times.

Contribution

It provides Nordhaus-Gaddum bounds, analyzes the impact of edge modifications, and characterizes graphs with extremal $k$-power propagation times, advancing understanding of propagation dynamics.

Findings

Established bounds on sum of power propagation times for a graph and its complement.

Characterized graphs with specific $k$-power propagation times.

Identified all trees with power propagation time $n-3$.

Abstract

In this paper, we give Nordhaus-Gaddum upper and lower bounds on the sum of the power propagation time of a graph and its complement, and we consider the effects of edge subdivisions and edge contractions on the power propagation time of a graph. We also study a generalization of power propagation time, known as $k-$power propagation time, by characterizing all simple graphs on $n$ vertices whose $k-$power propagation time is $n-1$ or $n-2$ (for $k\geq 1$) and $n-3$ (for $k\geq 2$). We determine all trees on $n$ vertices whose power propagation time ($k=1$) is $n-3$, and give partial characterizations of graphs whose $k-$power propagation time is equal to 1 (for $k\geq 1$).