Structural classification of graphs with γ(G)=γ_t(G)=2
Given a finite graph with no isolated vertices and domination number , determine exactly which graphs satisfy ; equivalently, characterize all finite graphs without isolated vertices such that .
References
Primary source
Additional references
Progress summary
A newly posted paper claims to complete the classification, but its result has not yet been independently checked or peer reviewed.
The problem concerns graphs whose domination number and total domination number both equal . The claimed result separates triangle-free graphs from graphs of girth and excludes this equality from a prominent extremal diameter-two family.
September 2026 claimed classification
A newly posted paper claims a complete structural dictionary for the equality case, using degree-sum criteria, bipartiteness, vertex multiplication, and classifications of low-rank graphs. The paper also claims that the equality cannot occur in the relevant extremal diameter-two family; these claims remain unverified.
Current status (as of September 2026): A September 2026 arXiv paper claims the structural classification and the extremal-family exclusion, but independent verification is absent.
Sources
- arxiv.org
- eudml.org
- av.tib.eu
- quantamagazine.org
- quantamagazine.org
- quantamagazine.org
- quantamagazine.org
- quantamagazine.org
- quantamagazine.org
- ar5iv.labs.arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- quantamagazine.org
- cdn.openai.com
Solutions 0
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