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On the Roman Domination Polynomials | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 15، دوره 9، شماره 3، آذر 2024، صفحه 595-605 اصل مقاله (376.35 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28145.1456 | ||
| نویسندگان | ||
| Seyed Hosein Mirhoseini؛ Nader Jafari Rad* | ||
| Department of Mathematics, Shahed University, Tehran, Iran | ||
| چکیده | ||
| A Roman dominating function (RDF) on a graph $G$ is a function $ f:V(G)\to \{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. The weight of an RDF $f$ is the sum of the weights of the vertices under $f$. The Roman domination number, $\gamma_R(G)$ of $G$ is the minimum weight of an RDF in $G$. The Roman domination polynomial of a graph $G$ of order $n$ is the polynomial $RD(G,x)=\sum_{i=\gamma_R(G)}^{2n} d_R(G,i) x^{i}$, where $d_R(G,i)$ is the number of RDFs of $G$ with weight $i$. In this paper we prove properties of Roman domination polynomials and determine $RD(G,x)$ in several classes of graphs $G$ by new approaches. We also present bounds on the number of all Roman domination polynomials in a graph. | ||
| کلیدواژهها | ||
| Roman domination polynomial Roman dominating function؛ Roman domination number | ||
| مراجع | ||
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