{"title":"Outer independent total double Italian domination number","authors":"S. M. Sheikholeslami, L. Volkmann","doi":"10.56415/csjm.v32.02","DOIUrl":null,"url":null,"abstract":"If $G$ is a graph with vertex set $V(G)$, then let $N[u]$ be the closed neighborhood of the vertex $u\\in V(G)$. A total double Italian dominating function (TDIDF) on a graph $G$ is a function $f:V(G)\\rightarrow\\{0,1,2,3\\}$ satisfying (i) $f(N[u])\\ge 3$ for every vertex $u\\in V(G)$ with $f(u)\\in\\{0,1\\}$ and (ii) the subgraph induced by the vertices with a non-zero label has no isolated vertices. A TDIDF is an outer-independent total double Italian dominating function (OITDIDF) on $G$ if the set of vertices labeled $0$ induces an edgeless subgraph. The weight of an OITDIDF is the sum of its function values over all vertices, and the outer independent total double Italian domination number $\\gamma_{tdI}^{oi}(G)$ is the minimum weight of an OITDIDF on $G$. In this paper, we establish various bounds on $\\gamma_{tdI}^{oi}(G)$, and we determine this parameter for some special classes of graphs.","PeriodicalId":42293,"journal":{"name":"Computer Science Journal of Moldova","volume":null,"pages":null},"PeriodicalIF":0.2000,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computer Science Journal of Moldova","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.56415/csjm.v32.02","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
If $G$ is a graph with vertex set $V(G)$, then let $N[u]$ be the closed neighborhood of the vertex $u\in V(G)$. A total double Italian dominating function (TDIDF) on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2,3\}$ satisfying (i) $f(N[u])\ge 3$ for every vertex $u\in V(G)$ with $f(u)\in\{0,1\}$ and (ii) the subgraph induced by the vertices with a non-zero label has no isolated vertices. A TDIDF is an outer-independent total double Italian dominating function (OITDIDF) on $G$ if the set of vertices labeled $0$ induces an edgeless subgraph. The weight of an OITDIDF is the sum of its function values over all vertices, and the outer independent total double Italian domination number $\gamma_{tdI}^{oi}(G)$ is the minimum weight of an OITDIDF on $G$. In this paper, we establish various bounds on $\gamma_{tdI}^{oi}(G)$, and we determine this parameter for some special classes of graphs.