TL;DR
This paper characterizes connected closed graphs as those that are chordal, claw-free, and narrow, providing a precise structural condition related to their labeling and shortest paths.
Contribution
It establishes a complete characterization of connected closed graphs using graph properties like chordality, claw-freeness, and narrowness, linking labeling to structural features.
Findings
Closed graphs are exactly those that are chordal, claw-free, and narrow.
The narrow property involves vertices being close to all longest shortest paths.
The characterization connects graph labeling with structural graph properties.
Abstract
A graph is closed when its vertices have a labeling by [n] with a certain property first discovered in the study of binomial edge ideals. In this article, we prove that a connected graph has a closed labeling if and only if it is chordal, claw-free, and has a property we call narrow, which holds when every vertex is distance at most one from all longest shortest paths of the graph.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Rings, Modules, and Algebras · Graph Labeling and Dimension Problems