Core Concepts
The author explores the burning number problem in graph theory, focusing on bounds and hardness, providing insights into various graph classes.
Abstract
The content delves into the burning number problem in graph theory, analyzing its complexity for different graph classes. It discusses structural results, proofs of theorems, tightness of upper bounds, and presents examples to illustrate key concepts.
The burning number problem is studied from algorithmic and structural perspectives. Theorems are proven regarding connected proper interval graphs, cubic graphs, and Pk-free graphs. The content highlights the NP-Completeness of the problem for specific graph classes and provides insights into variants like edge burning and total burning.
Key points include the definition of the burning number problem, conjectures related to it, complexity results for different graph classes, structural properties of graphs, proofs of theorems establishing NP-Completeness, tightness of upper bounds through examples, and implications for related problems like vertex cover.
Overall, the content offers a comprehensive analysis of the burning number problem in graph theory with a focus on bounds and hardness across various graph classes.
Stats
The decision problem of computing the burning number of an input graph is known to be NP-Complete for trees with maximum degree at most three and interval graphs.
Upper bound for the burning number of connected Pk-free graphs provided as ⌈(k+1)/2⌉.
The well-known conjecture states that all vertices of any graph can be burned in ⌈√n⌉ steps.
Complexity results discussed for edge burning and total burning variants in relation to the main problem.
Structural theorem presented regarding connected Pk-free graphs admitting a connected dominating set D.