Connected domination critical graphs with respect to relative complements

Xue-Gang Chen; Liang Sun

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 2, page 417-423
  • ISSN: 0011-4642

Abstract

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A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G . The minimum number of vertices in a connected dominating set of G is called the connected domination number of G , and is denoted by γ c ( G ) . Let G be a spanning subgraph of K s , s and let H be the complement of G relative to K s , s ; that is, K s , s = G ⊕ H is a factorization of K s , s . The graph G is k - γ c -critical relative to K s , s if γ c ( G ) = k and γ c ( G + e ) < k for each edge e ∈ E ( H ) . First, we discuss some classes of graphs whether they are γ c -critical relative to K s , s . Then we study k - γ c -critical graphs relative to K s , s for small values of k . In particular, we characterize the 3 - γ c -critical and 4 - γ c -critical graphs.

How to cite

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Chen, Xue-Gang, and Sun, Liang. "Connected domination critical graphs with respect to relative complements." Czechoslovak Mathematical Journal 56.2 (2006): 417-423. <http://eudml.org/doc/31038>.

@article{Chen2006,
abstract = {A dominating set in a graph $G$ is a connected dominating set of $G$ if it induces a connected subgraph of $G$. The minimum number of vertices in a connected dominating set of $G$ is called the connected domination number of $G$, and is denoted by $\gamma _\{c\}(G)$. Let $G$ be a spanning subgraph of $K_\{s,s\}$ and let $H$ be the complement of $G$ relative to $K_\{s,s\}$; that is, $K_\{s,s\}=G\oplus H$ is a factorization of $K_\{s,s\}$. The graph $G$ is $k$-$\gamma _\{c\}$-critical relative to $K_\{s,s\}$ if $\gamma _\{c\}(G)=k$ and $\gamma _\{c\}(G+e)<k$ for each edge $e\in E(H)$. First, we discuss some classes of graphs whether they are $\gamma _\{c\}$-critical relative to $K_\{s,s\}$. Then we study $k$-$\gamma _\{c\}$-critical graphs relative to $K_\{s,s\}$ for small values of $k$. In particular, we characterize the $3$-$\gamma _\{c\}$-critical and $4$-$\gamma _\{c\}$-critical graphs.},
author = {Chen, Xue-Gang, Sun, Liang},
journal = {Czechoslovak Mathematical Journal},
keywords = {connected domination number; connected domination critical graph relative to $K_\{s,s\}$ tree; connected domination number; tree},
language = {eng},
number = {2},
pages = {417-423},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Connected domination critical graphs with respect to relative complements},
url = {http://eudml.org/doc/31038},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Chen, Xue-Gang
AU - Sun, Liang
TI - Connected domination critical graphs with respect to relative complements
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 2
SP - 417
EP - 423
AB - A dominating set in a graph $G$ is a connected dominating set of $G$ if it induces a connected subgraph of $G$. The minimum number of vertices in a connected dominating set of $G$ is called the connected domination number of $G$, and is denoted by $\gamma _{c}(G)$. Let $G$ be a spanning subgraph of $K_{s,s}$ and let $H$ be the complement of $G$ relative to $K_{s,s}$; that is, $K_{s,s}=G\oplus H$ is a factorization of $K_{s,s}$. The graph $G$ is $k$-$\gamma _{c}$-critical relative to $K_{s,s}$ if $\gamma _{c}(G)=k$ and $\gamma _{c}(G+e)<k$ for each edge $e\in E(H)$. First, we discuss some classes of graphs whether they are $\gamma _{c}$-critical relative to $K_{s,s}$. Then we study $k$-$\gamma _{c}$-critical graphs relative to $K_{s,s}$ for small values of $k$. In particular, we characterize the $3$-$\gamma _{c}$-critical and $4$-$\gamma _{c}$-critical graphs.
LA - eng
KW - connected domination number; connected domination critical graph relative to $K_{s,s}$ tree; connected domination number; tree
UR - http://eudml.org/doc/31038
ER -

References

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  2. 10.1002/jgt.3190160305, J. Graph Theory 16 (1992), 221–231. (1992) MR1168581DOI10.1002/jgt.3190160305
  3. Domination critical graphs with respect to relative complements, Australas J. Combin. 18 (1998), 115–126. (1998) MR1658309
  4. Domination and total domination critical trees with respect to relative complements, Ars Combin. 59 (2001), 117–127. (2001) MR1832203
  5. Total domination critical graphs with respect to relative complements, Ars Combin. 64 (2002), 169–179. (2002) MR1914205
  6. Total domination edge critical graphs, Utilitas Math. 54 (1998), 229–240. (1998) MR1658130
  7. Renu Laskar, Connected domination in graphs, Graph Theory and Combinatorics (1984), 209–217. (1984) MR0777177
  8. The connected domination number of a graph, Math. Phys. Sci. 13 (1979), 607–613. (1979) MR0575817

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