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Weak Roman domination stable graphs upon edge-addition | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 3، دوره 8، شماره 3، آذر 2023، صفحه 467-481 اصل مقاله (406.58 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2022.27765.1348 | ||
| نویسندگان | ||
| Roushini Leely Pushpam1؛ NAGARAJAN SRILAKSHMI* 2 | ||
| 1University of Madras | ||
| 2DEPARTMENT OF MATHEMATICS DHANRAJ BAID JAIN COLLEGE | ||
| چکیده | ||
| A Roman dominating function (RDF) on a graph $G$ is a function $f: V(G) \to \{0, 1, 2\}$ such that every vertex with label 0 has a neighbor with label 2. A vertex $u$ with $f(u)=0$ is said to be undefended if it is not adjacent to a vertex with $f(v)>0$. The function $f:V(G) \to \{0, 1, 2\}$ is a weak Roman dominating function (WRDF) if each vertex $u$ with $f(u) = 0$ is adjacent to a vertex $v$ with $f(v) > 0$ such that the function $f^{\prime}: V(G) \to \{0, 1, 2\}$ defined by $f^{\prime}(u) = 1$, $f^{\prime}(v) = f(v) - 1$ and $f^{\prime}(w) = f(w)$ if $w \in V - \{u, v\}$, has no undefended vertex. A graph $G$ is said to be Roman domination stable upon edge addition, or just $\gamma_R$-EA-stable, if $\gamma_R(G+e)= \gamma_R(G)$ for any edge $e \notin E(G)$. We extend this concept to a weak Roman dominating function as follows: A graph $G$ is said to be weak Roman domination stable upon edge addition, or just $\gamma_r$-EA-stable, if $\gamma_r(G+e)= \gamma_r(G)$ for any edge $e \notin E(G)$. In this paper, we study $\gamma_r$-EA-stable graphs, obtain bounds for $\gamma_r$-EA-stable graphs and characterize $\gamma_r$-EA-stable trees which attain the bound. | ||
| کلیدواژهها | ||
| weak Roman domination؛ stability؛ edge addition | ||
| مراجع | ||
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