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The extended irregular domination problem | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 27 آذر 1403 اصل مقاله (1.05 M) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.30046.2289 | ||
| نویسندگان | ||
| Lorenzo Mella1؛ Anita Pasotti* 2 | ||
| 1Dip. di Scienze Fisiche, Informatiche, Matematiche, Universit`a degli Studi di Modena e Reggio Emilia, Via Campi 213/A, I-41125 Modena, Italy | ||
| 2DICATAM - Sez. Matematica, Universit`a degli Studi di Brescia, Via Branze 43, I-25123 Brescia, Italy | ||
| چکیده | ||
| In this paper we introduce a new domination problem strongly related to the following one recently proposed by Broe, Chartrand and Zhang. One says that a vertex $v$ of a graph $\Gamma$ labeled with an integer $\ell$ dominates the vertices of $\Gamma$ having distance $\ell$ from $v$. An irregular dominating set of a given graph $\Gamma$ is a set $S$ of vertices of $\Gamma$, having distinct positive labels, whose elements dominate every vertex of $\Gamma$. Since it has been proven that no connected vertex transitive graph admits an irregular dominating set, here we introduce the concept of an \emph{extended} irregular dominating set, where we admit that precisely one vertex, labeled with 0, dominates itself. Then we present existence or non existence results of an extended irregular dominating set $S$ for several classes of graphs, focusing in particular on the case in which $S$ is as small as possible. We also propose two conjectures. | ||
| کلیدواژه‌ها | ||
| Dominating set؛ vertex transitive graph؛ starter | ||
| مراجع | ||
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