{
  "abstract": "We study the problem of counting \\emph{Italian dominating functions} (IDFs) of each possible weight on paths, trees, and cycles. An Italian dominating function on a graph $G=(V,E)$ is a function $f\\colon V\\to\\{0,1,2\\}$ such that every vertex $v$ with $f(v)=0$ satisfies $\\sum_{u\\in N(v)}f(u)\\ge 2$; the \\emph{Italian domination polynomial} $D_I(G,x)=\\sum_k d_I(G,k)\\,x^k$ records the number $d_I(G,k)$ of such functions of weight~$k$. This polynomial is the Italian analogue of the well-studied domination and Roman domination polynomials, and computing its coefficients on a fixed graph class is, in essence, a \\emph{constrained integer composition} problem: passing to the deficit variables $y(v)=2-f(v)$, each coefficient $d_I(G,2n-j)$ counts the number of compositions of $j$ into $n$ parts, each at most~$2$, that additionally satisfy a local adjacency capacity constraint. We make this connection explicit and use it to derive closed-form expressions for $d_I(C_n,2n-j)$, $j=0,\\ldots,5$, as trinomial coefficients minus explicitly enumerated forbidden configurations. On the algorithmic side, we give dynamic programming algorithms that compute \\emph{all} coefficients of $D_I$ on path graphs ($O(n^2)$ time, $O(n)$ space), on general trees ($O(n^2)$ time, $O(n)$ space, via a post-order merge with a deficit parameter), and on cycle graphs ($O(n^2)$ time, $O(n)$ space, via boundary conditioning), together with formal correctness proofs and complexity analyses. The known values $γ_I(P_n)=\\lfloor n/2\\rfloor+1$ and $γ_I(C_n)=\\lceil n/2\\rceil$ are recalled with references rather than reproved. All numerical results are verified against brute-force enumeration, and complete tables are provided.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00108",
  "author_total": 2,
  "authors": [
    {
      "name": "Pingping Shao"
    },
    {
      "name": "Chengye Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a1mav7xk3rte8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00108",
  "fulltext_url": null,
  "id": "a1mav7xk3rte8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00108",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.00108",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a1mav7xk3rte8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00108",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00108"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counting Weight-$k$ Italian Dominating Sets on Trees and Cycles",
  "title_text": "Counting Weight-k Italian Dominating Sets on Trees and Cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/a1mav7xk3rte8",
  "venue": "arXiv",
  "versions": []
}