Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.16256 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913482265329664 |
|---|---|
| author | Koch, Garrison Shank, Nathan |
| author_facet | Koch, Garrison Shank, Nathan |
| contents | Given a graph $G=(V,E)$, the dominating number of a graph is the minimum size of a vertex set, $V' \subseteq V$, so that every vertex in the graph is either in $V'$ or is adjacent to a vertex in $V'$. A Roman Dominating function of $G$ is defined as $f:V \rightarrow \{0,1,2\}$ such that every vertex with a label of 0 in $G$ is adjacent to a vertex with a label of 2. The Roman Dominating number of a graph is the minimum total weight over all possible Roman Dominating functions. We consider the $k$-attack Roman Domination, particularly focusing on 2-attack Roman Domination. A Roman Dominating function of $G$ is a $k$-attack Roman Dominating function of $G$ if for all $j\leq k$, any subset $S$ of $j$ vertices all with label 0 must have at least $j$ vertices with label 2 in the open neighborhood of $S$. The $k$-attack Roman Dominating number of $G, \gkaRD{G}$, is the minimum total weight over all possible $k$-attack Roman Dominating functions. We find $\gtaRD{G}$ for particular graph class, discuss properties of $k$-attack Roman Domination, and make several connections with other domination ideas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_16256 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the $nk-attack Roman Dominating Number of a Graph Koch, Garrison Shank, Nathan Combinatorics Given a graph $G=(V,E)$, the dominating number of a graph is the minimum size of a vertex set, $V' \subseteq V$, so that every vertex in the graph is either in $V'$ or is adjacent to a vertex in $V'$. A Roman Dominating function of $G$ is defined as $f:V \rightarrow \{0,1,2\}$ such that every vertex with a label of 0 in $G$ is adjacent to a vertex with a label of 2. The Roman Dominating number of a graph is the minimum total weight over all possible Roman Dominating functions. We consider the $k$-attack Roman Domination, particularly focusing on 2-attack Roman Domination. A Roman Dominating function of $G$ is a $k$-attack Roman Dominating function of $G$ if for all $j\leq k$, any subset $S$ of $j$ vertices all with label 0 must have at least $j$ vertices with label 2 in the open neighborhood of $S$. The $k$-attack Roman Dominating number of $G, \gkaRD{G}$, is the minimum total weight over all possible $k$-attack Roman Dominating functions. We find $\gtaRD{G}$ for particular graph class, discuss properties of $k$-attack Roman Domination, and make several connections with other domination ideas. |
| title | On the $nk-attack Roman Dominating Number of a Graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2305.16256 |