Title: | Connected resolvability of graphs (English) |

Author: | Saenpholphat, Varaporn |

Author: | Zhang, Ping |

Language: | English |

Journal: | Czechoslovak Mathematical Journal |

ISSN: | 0011-4642 (print) |

ISSN: | 1572-9141 (online) |

Volume: | 53 |

Issue: | 4 |

Year: | 2003 |

Pages: | 827-840 |

Summary lang: | English |

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Category: | math |

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Summary: | For an ordered set $W=\lbrace w_1, w_2, \dots , w_k\rbrace $ of vertices and a vertex $v$ in a connected graph $G$, the representation of $v$ with respect to $W$ is the $k$-vector $r(v|W)$ = ($d(v, w_1)$, $d(v, w_2),\dots ,d(v, w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set for $G$ containing a minimum number of vertices is a basis for $G$. The dimension $\dim (G)$ is the number of vertices in a basis for $G$. A resolving set $W$ of $G$ is connected if the subgraph $<W>$ induced by $W$ is a nontrivial connected subgraph of $G$. The minimum cardinality of a connected resolving set in a graph $G$ is its connected resolving number $\mathop {\mathrm cr}(G)$. Thus $1 \le \dim (G) \le \mathop {\mathrm cr}(G) \le n-1$ for every connected graph $G$ of order $n \ge 3$. The connected resolving numbers of some well-known graphs are determined. It is shown that if $G$ is a connected graph of order $n \ge 3$, then $\mathop {\mathrm cr}(G) = n-1$ if and only if $G = K_n$ or $G = K_{1, n-1}$. It is also shown that for positive integers $a$, $b$ with $a \le b$, there exists a connected graph $G$ with $\dim (G) = a$ and $\mathop {\mathrm cr}(G) = b$ if and only if $(a, b) \notin \lbrace (1, k)\: k = 1\hspace{5.0pt}\text{or}\hspace{5.0pt}k \ge 3\rbrace $. Several other realization results are present. The connected resolving numbers of the Cartesian products $G \times K_2$ for connected graphs $G$ are studied. (English) |

Keyword: | resolving set |

Keyword: | basis |

Keyword: | dimension |

Keyword: | connected resolving set |

Keyword: | connected resolving number |

MSC: | 05C12 |

MSC: | 05C25 |

MSC: | 05C35 |

idZBL: | Zbl 1080.05507 |

idMR: | MR2018833 |

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Date available: | 2009-09-24T11:06:59Z |

Last updated: | 2020-07-03 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/127843 |

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Reference: | [1] P. Buczkowski, G. Chartrand, C. Poisson and P. Zhang: On $k$-dimensional graphs and their bases.Period. Math. Hungar. 46 (2003), 25–36. MR 1975342, 10.1023/A:1025745406160 |

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