The Pinnacle Sets of a Graph
Chassidy Bozeman, Christine Cheng, Pamela E. Harris, Stephen Lasinis,, Shanise Walker

TL;DR
This paper introduces the concept of pinnacle sets in graphs, characterizes their existence, explores their structure in specific graph classes, and studies the poset formed by these sets, highlighting computational complexity and algebraic properties.
Contribution
It defines pinnacle sets for graphs, characterizes their existence in connected graphs, and analyzes their structure and algebraic properties, including NP-completeness and lattice formations.
Findings
Determined pinnacle sets for complete graphs, bipartite graphs, cycles, and paths.
Proved that size-$k$ pinnacle set existence is NP-complete for connected graphs.
Established a join semilattice structure on pinnacle sets.
Abstract
We introduce and study the pinnacle sets of a simple graph with vertices. Given a bijective vertex labeling , the label of vertex is a pinnacle of if for all vertices in the neighborhood of . The pinnacle set of contains all the pinnacles of the labeled graph. A subset is a pinnacle set of if there exists a labeling such that is the pinnacle set of . Of interest to us is the question: Which subsets of are the pinnacle sets of ? Our main results are as follows. We show that when is connected, has a size- pinnacle set if and only if has an independent set of the same size. Consequently, determining if has a size- pinnacle set and determining if has a particular subset as a pinnacle set…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems
