{
  "abstract": "On a network with a fixed set of verified sources, discounted averaging induces an equilibrium support $h_i^S=\\mathbb{E}_i[λ^{T_S}]$, the discounted probability that a random walk reaches $S$ before attenuation. We define the \\emph{discounted hitting domination number} $δ_{λ,τ}(G)$ as the minimum number of sources required to guarantee $h_i^S\\geτ$ at every vertex. Although this potential is known through penalized and group hitting probabilities, the associated minimum-cardinality uniform-coverage problem appears to be new. Aggregate support is monotone submodular, while the uniform-floor problem is an exact submodular-cover problem. Moreover, if $λ^{r+1}<τ\\le\\left(\\fracλΔ\\right)^r$, then $δ_{λ,τ}(G)$ equals the distance-$r$ domination number. This yields NP-completeness and APX-completeness at $(λ,τ)=(1/4,1/14)$ on graphs of maximum degree three. For spiders, we obtain an exact finite-state characterization and a polynomial-time algorithm for every fixed rational pair $(λ,τ)$, and show that the branching vertex need not belong to a minimum source set. Finally, an exact mixed-integer linear formulation certifies optimal placements on a real network and a synthetic graph and demonstrates substantial differences from degree, closeness, and classical domination.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31535",
  "author_total": 3,
  "authors": [
    {
      "name": "Julian D. Allagan"
    },
    {
      "name": "Kevin Pereyra"
    },
    {
      "name": "William A. Massey"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2gfpk2wc19a8m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31535",
  "fulltext_url": null,
  "id": "2gfpk2wc19a8m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31535",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214605750",
  "pdf_url": "https://arxiv.org/pdf/2609.31535",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2gfpk2wc19a8m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31535",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31535"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Discounted Hitting Domination on Graphs with Submodularity, Complexity and Exact Algorithms",
  "title_text": "Discounted Hitting Domination on Graphs with Submodularity, Complexity and Exact Algorithms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/2gfpk2wc19a8m",
  "venue": "arXiv",
  "versions": []
}