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bibkey: teimoorifaal2026brestricted authors: H. Teimoori Faal year: 2026 title: “A Bivariate B-Restricted Clique Polynomial: From Local Neighborhoods to Global Expansion” doi: null url: https://arxiv.org/abs/2602.24151v1 claim: “Open Problem 1 asks whether an r-connected K_{r+3}-free non-chordal graph can have a B-restricted clique polynomial that fails to be real-stable.” strata_touched:

  • D5/S3/Combinatorics/Graph/CliquePolynomial/FaalNonStable license: citation-only triage: anchor

Verified locator

DOI: null

Source: https://arxiv.org/abs/2602.24151v1

H. Teimoori Faal, arXiv:2602.24151v1. Page numbers refer to the numbered PDF pages.

  • Definition 2.1, page 4: “For G=(V,E) and B ⊆ V, define” (C_B(G;x,y) := \sum_{K \subseteq V,\ K\text{ clique}} x^{|K|} y^{|K\cap B|}). The empty clique contributes 1; Example 2.3 uses this convention.
  • Definition 2.4, page 4: “A polynomial (f(x,y) \in \mathbb{R}[x,y]) is real stable if it is not identically zero and” (f(x,y) \neq 0) “for all ((x,y) \in \mathbb{C}^2) such that (\operatorname{Im}(x) > 0) and (\operatorname{Im}(y) > 0).”
  • Definition 4.2, page 7: “A graph is r-connected if removal of fewer than r vertices leaves it connected.”
  • Definition 4.3, page 7: “A graph is chordal if every induced cycle has length 3.”
  • Theorem 4.8, page 8: “Let r≥1. If G is r-connected, K_{r+3}-free, and chordal, then” (C_B(G;x,y)) “is real-stable.”
  • Section 7, Open Problems, item 1, page 18: “Necessity of Conditions: Are the conditions of r-connectivity and chordality also necessary? Is there an r-connected K_{r+3}-free non-chordal graph for which C_B(G;x,y) fails to be real-stable?”

Crossref title search returned no matching work or DOI. The arXiv version is the verified locator; no publisher DOI is asserted.

Scope

The second question of Open Problem 1 has a positive answer for every integer r≥1. Put a=max(r−2,0), take the join K_a ∨ C₄, and mark one cycle vertex. Its polynomial is ((1+x)^a(1+2x)(1+x+xy)), which vanishes at ((x,y)=(i,-1+i)) in the product of the upper half-planes. The family is r-connected, K_{r+3}-free and non-chordal. For r=1 it is C₄, which is 2-connected; no exact-connectivity assertion is made.

Theorem 4.8 is a source assertion, not a premise of this settlement. Its complete-graph counterexample and the question of which marked sets preserve stability are separate questions.