The Pinnacle Sets of a Graph

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 $G$ with $n$ vertices. Given a bijective vertex labeling $\lambda\,:\,V(G)\rightarrow [n]$, the label $\lambda(v)$ of vertex $v$ is a pinnacle of $(G, \lambda)$ if $\lambda(v)>\lambda(w)$ for all vertices $w$ in the neighborhood of $v$. The pinnacle set of $(G, \lambda)$ contains all the pinnacles of the labeled graph. A subset $S\subseteq[n]$ is a pinnacle set of $G$ if there exists a labeling $\lambda$ such that $S$ is the pinnacle set of $(G,\lambda)$. Of interest to us is the question: Which subsets of $[n]$ are the pinnacle sets of $G$? Our main results are as follows. We show that when $G$ is connected, $G$ has a size-$k$ pinnacle set if and only if $G$ has an independent set of the same size. Consequently, determining if $G$ has a size-$k$ pinnacle set and determining if $G$ has a particular subset $S$ as a pinnacle set are NP-complete problems. Nonetheless, we completely identify all the pinnacle sets of complete graphs, complete bipartite graphs, cycles and paths. We also present two techniques for deriving new pinnacle sets from old ones that imply a typical graph has many pinnacle sets. Finally, we define a poset on all the size-$k$ pinnacle sets of $G$ and show that it is a join semilattice. If, additionally, the poset has a minimum element, then it is a distributive lattice. We conclude with some open problems for further study.