| Title:
|
On signed distance-$k$-domination in graphs (English) |
| Author:
|
Xing, Huaming |
| Author:
|
Sun, Liang |
| Author:
|
Chen, Xuegang |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
56 |
| Issue:
|
1 |
| Year:
|
2006 |
| Pages:
|
229-238 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The signed distance-$k$-domination number of a graph is a certain variant of the signed domination number. If $v$ is a vertex of a graph $G$, the open $k$-neighborhood of $v$, denoted by $N_k(v)$, is the set $N_k(v)=\lbrace u\mid u\ne v$ and $d(u,v)\le k\rbrace $. $N_k[v]=N_k(v)\cup \lbrace v\rbrace $ is the closed $k$-neighborhood of $v$. A function $f\: V\rightarrow \lbrace -1,1\rbrace $ is a signed distance-$k$-dominating function of $G$, if for every vertex $v\in V$, $f(N_k[v])=\sum _{u\in N_k[v]}f(u)\ge 1$. The signed distance-$k$-domination number, denoted by $\gamma _{k,s}(G)$, is the minimum weight of a signed distance-$k$-dominating function on $G$. The values of $\gamma _{2,s}(G)$ are found for graphs with small diameter, paths, circuits. At the end it is proved that $\gamma _{2,s}(T)$ is not bounded from below in general for any tree $T$. (English) |
| Keyword:
|
signed distance-$k$-domination number |
| Keyword:
|
signed distance-$k$-dominating function |
| Keyword:
|
signed domination number |
| MSC:
|
05C69 |
| idZBL:
|
Zbl 1164.05427 |
| idMR:
|
MR2207014 |
| . |
| Date available:
|
2009-09-24T11:32:27Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128061 |
| . |
| Reference:
|
[1] J. H. Hattingh, M. A. Henning, and E. Ungerer: Partial signed domination in graphs.Ars Combin. 48 (1998), 33–42. MR 1623038 |
| Reference:
|
[2] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater: Fundamentals of Domination in Graphs.Marcel Dekker, New York, 1998. MR 1605684 |
| Reference:
|
[3] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater: Domination in Graphs: Advanced Topics.Marcel Dekker, New York, 1998. MR 1605685 |
| Reference:
|
[4] M. A. Henning: Domination in regular graphs.Ars Combin. 43 (1996), 263–271. Zbl 0881.05101, MR 1415996 |
| . |
