2023/08/26 by Delio Jaramillo-Velez, Lisa Seccia · 1 citation
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Algebraic number #Binomial (polynomial) #Combinatorics #Commutative Algebra and Its Applications #Discrete mathematics #Domination analysis #Graph #Mathematical analysis #Mathematics #Polynomial and algebraic computation #Statistics #Upper and lower bounds #Vertex (graph theory)
paper · pdf · doi:10.1007/s13348-023-00412-w
crossref issued 2023/08/26 · crossref published 2023/08/26 · crossref published-online 2023/08/26 · openalex publication_date 2023/08/26 · crossref created 2023/08/26 · crossref deposited 2024/08/30 · crossref published-print 2024/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23 · crossref indexed 2026/08/06
Abstract The v-number of a graded ideal is an algebraic invariant introduced by Cooper et al., and originally motivated by problems in algebraic coding theory. In this paper we study the case of binomial edge ideals and we establish a significant connection between their v-numbers and the concept of connected domination in graphs. More specifically, we prove that the localization of the v-number at one of the minimal primes of the binomial edge ideal JG <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:math> of a graph G coincides with the connected domination number of the defining graph, providing a first algebraic description of the connected domination number. As an immediate corollary, we obtain a sharp combinatorial upper bound for the v-number of binomial edge ideals of graphs. Lastly, building on some known results on edge ideals, we analyse how the v-number of JG <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:math> behaves under Gröbner degeneration when G is a closed graph.